Abstract. -We give an extensive introduction to the current literature on the CREMONA groups over the field of complex numbers, mostly of rank 2, with an emphasis on group theoretical and dynamical questions.

After a short introduction which explains in an informal style some selected results and techniques Chapter 2 gives a description of the hyperbolic space on which the CREMONA group in two variables acts, and which has turned out to provide some of the key techniques to understand the plane CREMONA group. In Chapter 3 the ZARISKI topology is described. Chapter 4 gives an overview of various presentations of the plane CREMONA group. Chapter 5 treats some group theoretical properties of the plane CREMONA group. Chapter 6 surveys some results about finite (mostly abelian) subgroups of the plane CREMONA group. Chapter 7 surveys results about various subgroups using techniques that rely on the base-field being uncountable. Chapter 8 gives a big variety of important results that can be deduce from the action of the plane CREMONA group on the hyperbolic space, such as the TITS alternative or the non-simplicity of the group. Chapter 9 gives an introduction to some notions from dynamics and their relationship to the plane CREMONA group.

PREFACE

The main purpose of the present treatise is to draw a portrait of the n-dimensional Cremona group Bir(P n C ). The study of this group started in the XIXth century; the subject has known a lot of developments since the beginning of the XXIth century. Old and new results are discussed; unfortunately we will not be exhaustive. The Cremona group is approached through the study of its subgroups: algebraic, finite, normal, nilpotent, simple, torsion subgroups are evoked but also centralizers of elements, representation of lattices, subgroups of automorphisms of positive entropy etc Let us introduce birational self maps of the plane and the plane Cremona group from a geometrical point of view.

A plane collineation is a one-to-one map from P 2 C to itself such that the images of collinear points are themselves collinear. Such maps leave the projective properties of curves unaltered. In advancing beyond such properties let us introduce other maps of the plane to itself that establish relations between curves of differents orders and possessing different sets of singularities. The most general rational map of the plane is defined by equations of the form φ : (z 0 : z 1 : z 2 ) φ 0 (z 0 , z 1 , z 2 ) : φ 1 (z 0 , z 1 , z 2 ) : φ 2 (z 0 , z 1 , z 2 )

where φ 0 , φ 1 and φ 2 are homogeneous polynomials of degree n without common factor of positive degree. Such a map makes correspond to a point p with coordinates (p 0 : p 1 : p 2 ) a point φ(p) = q with coordinates (q 0 : q 1 : q 2 ) where δq 0 = φ 0 (p 0 , p 1 , p 2 ), δq 1 = φ 1 (p 0 , p 1 , p 2 ), δq 2 = φ 2 (p 0 , p 1 , p 2 ) (0.0.1) with δ in C * . Consider the net of curves Λ φ defined by the equation

αφ 0 + βφ 1 + γφ 2 = 0
where α, β and γ are arbitrary parameters. As p describes a line in P 2 C , then q = φ(p) describes a curve C of Λ φ . The curves of the net Λ φ are thus correlated by φ with the lines of the plane.

PREFACE

Conversely given any net Λ of curves such as Λ φ a linear representation of the curves of Λ on the lines of the plane is equivalent to a rational map of the plane.

The curves of Λ φ may have base-points p i common to them all. Each such point is a common zero of φ 0 , φ 1 and φ 2 , so the equations (0.0.1) to determine its corresponding point are illusory. Conversely each point, termed a base-point of φ, which renders equation (0.0.1) illusory is a base-point of Λ φ . In other words

Theorem. -The base-points of any rational map are the base-points of the associated net of curves.

Any two general curves C and C of Λ φ define a pencil of curves C + αC of the net. Denote by n the number of free intersections of C and C not occuring at the base-points p i of Λ φ ; denote by r 1 , r 2 , . . ., r n these points. The integer n is called the grade of Λ φ .

To curves of the arbitrary pencil C + αC there correspond by the map φ lines of a pencil L + αL . Furthermore if the base-point of the latter pencil is q, then clearly every point r i corresponds to q. Conversely if any two points of the plane have the same preimage q, then they belong to the same free intersection set of some pencil in Λ φ .

Theorem. -Let φ be a rational self map of the plane. Let Λ φ be its associated net and let n be the grade of Λ φ . An arbitrary point q is the transform of n points r 1 , r 2 , . . ., r n which together form the free intersection set of a pencil of curves of Λ φ .

In other words the general rational map of the plane is a (n, 1) correspondence between the points p and q. And this means that, when the ratios of q 0 , q 1 , q 2 are given the equations (0.0.1) have in general n distinct solutions for the ratios of p 0 , p 1 and p 2 . If n = 1, i.e. if these equations have only one solution, (p 0 : p 1 : p 2 ) are rational functions of (q 0 : q 1 : q 2 ). In this case the equations of the reverse map will be of the form αp 0 = ψ 0 (q 0 , q 1 , q 2 ) αp 1 = ψ 1 (q 0 , q 1 , q 2 ) αp 2 = ψ 2 (q 0 , q 1 , q 2 ) where ψ 0 , ψ 1 and ψ 2 are homogeneous polynomials of degree n . A Cremona map is a rational map whose reverse is also rational, we also speak about birational self map of the plane. The plane Cremona group is the group of birational self maps of the plane. A homaloidal net of curves in the plane is one whose grade is 1. Equations (0.0.1) define a birational map φ if and only if the associated net Λ φ is homaloidal. Conversely from any given homaloidal net we can derive many birational self maps of the plane; if φ 0 , φ 1 and φ 2 are three independent linear combinations of φ 0 , φ 1 and φ 2 , the net α φ 0 + β φ 1 + γ φ 2 = 0 can also be expressed in the form α φ 0 + β φ 1 + γ φ 2 = 0 and the map defined by

(z 0 : z 1 : z 2 ) φ 0 (z 0 , z 1 , z 2 ) : φ 1 (z 0 , z 1 , z 2 ) : φ 2 (z 0 , z 1 , z 2 ) PREFACE v
is based on the same net. Moreover

Theorem. -To any birational self map of the plane there is associated a homaloidal net of curves.

Conversely any homaloidal net of curves generates an infinity of birational self maps of the plane, any of which is the product of any other by a plane collineation.

A collineation is the simplest kind of birational self map of the plane whose homaloidal net is composed of the lines of the plane.

The degree of a birational self map of the plane is the degree of the curves of its generating homaloidal net.

Let φ be a birational self map of the plane of degree n. Denote by n the degree of its inverse φ -1 . If the number of intersections of two curves C and C is denoted by C • C and if L and L are lines, then

n = L • Λ φ = φ(L) • φ(Λ φ ) = Λ φ -1 • L = n .

Hence

Theorem.

-A birational self map of the plane and its inverse have the same degree.

Let us finish this introduction by pointing out that this statement is not true in higher dimension:

P 3 C P 3
C (z 0 : z 1 : z 2 : z 3 ) (z 2 0 : z 0 z 1 : z 1 z 2 : z 0 z 3z 2 1 ) is a birational self map of P 3 C of degree 2 whose inverse P 3 C P 3 C (z 0 : z 1 : z 2 : z 3 ) z 2 0 z 1 : z 0 z 2 1 : z 2 0 z 2 : z 1 (z 0 z 3 + z 2 1 ) has degree 3. As we will see there are many other differences between the 2-dimensional Cremona group and the n-dimensional Cremona group, n ≥ 3.

Note that the study of Bir(P 2 C ) is central: if S is a complex rational surface, then its group of birational self maps is isomorphic to Bir(P 2 C ).

We now deal with the content of the manuscript. Chapter 1 contains introductory examples and the very basic techniques used to study birational maps of the projective plane. This chapter explains in particular the importance of divisors and linear systems in the study of the plane Cremona groups.

Chapter 2 builds up on Chapter 1 by explaining how to blow-up all points in P 2 C and subsequent blown-up surfaces. It gives rise to an infinite hyperbolic space on which the Cremona group acts. This space plays a fundamental role in the study of Cremona groups, as it allows to apply tools from geometric group theory to study subgroups of the Cremona group, as well as degree growth and dynamical behaviours of birational maps. vi PREFACE Chapter 3 presents two natural topologies on the Cremona group and their properties, and the notion of algebraic subgroups of the Cremona groups. The construction of one of the topologies -the Zariski topology -is defined via the concept of morphisms. It links to the concept of an algebraic group acting on a variety, which is discussed in this chapter as well.

Chapter 4 adresses a very basic and classical interest while dealing with a group: finding a "nice" and generating set and "nice" structures of the group, such as an amalgamated structure. This is quite an important topic in research on Cremona groups because for the plane Cremona group there are "nice" generating sets, and many statements are proven by using them. In higher dimensions no nice generating sets are known: this is one of the many reasons why working with Cremona groups in higher dimensions is very hard.

Chapter 5 discusses other group geometric properties of plane Cremona groups. While Chapter 2 presents a representation of the Cremona group in terms of isometries of an infinite hyperbolic space this chapter deals with linear representations (there are none) and representations of subgroups of SL(n, Z), n ≥ 3, inside the plane Cremona group.

Chapter 6 deals with results on finite subgroups of the plane Cremona groups. They have been of much interest for a very long time, and a short overview of the progress made in the last 80 years is given. The chapter focuses on the classification results of finite abelian and finite cyclic subgroups by Blanc and Dolgachev and Iskovskikh.

Chapter 7 is an extension of Chapter 6; it deals with infinite abelian subgroups of the plane Cremona group. It then moves on the related topic of endomorphisms of Cremona groups, subject already mentioned in Chapter 5.

Chapter 8 picks up the topic of Chapter 2 which is the action of the plane Cremona group on an infinite hyperbolic space by isometries. The action and its properties have been very fruitful and has played a vital role in many recent results on the plane Cremona group.

Chapter 9 has a more dynamical flavour. We first give three answers to the question "when is a birational self map of P 2 C birationally conjugate to an automorphism ?" We then recall some constructions of automorphisms of rational surfaces with positive entropy. And then we realize SL(2, Z) as a subgroup of automorphisms of a rational surface with the property that every element of infinite order has positive entropy.

CHAPTER 1 INTRODUCTION

This chapter is devoted to recalls and first definitions. In the first section morphisms between varieties, blow-ups, Cremona groups and bubble space are introduced, the Zariski theorem, base-points, indeterminacy points are recalled, ans examples of subgroups of the Cremona group are given, among them the group of automorphisms of P n C , the Jonquières group, the group of monomial maps. The second section is devoted to divisors (prime divisors, Weil divisors, principal divisors, Picard group) and intersection theory.

The third section deals with a geometric definition of birational maps of the complex projective plane.

First definitions and examples

Denote by P n C the complex projective space of dimension n. A rational map φ :

V 1 ⊂ P n C V 2 ⊂ P k C
between two smooth projective complex varieties V 1 and V 2 is a regular map on a non-empty Zariski open subset of V 1 such that the image of the points where φ is well defined is contained in V 2 . If φ is well defined on V 1 we say that φ is a morphism or a regular map, otherwise we denote by Ind(φ) the set where φ is not defined, and call it the indeterminacy set of φ. A birational map between V 1 and V 2 is a rational map that admits an inverse which is rational. In other words it is an isomorphism between two non-empty Zariski open subsets of V 1 and V 2 .

Example 1. -Let us give an example of a birational morphism. Let p be a point on a smooth algebraic surface S. We say that π : Y → S is a blow-up of p if Y is a smooth surface, π |Y {π -1 (p)} : Y {π -1 (p)} → S {p} is an isomorphism, and π -1 (p) P 1 C . We call π -1 (p) the exceptional divisor.

If π : Y → S and π : Y → S are two blow-ups of the same point p, then there exists an isomorphism ϕ : Y → Y such that π = π • ϕ. We can thus speak about the blow-up of p ∈ S.

Let us describe the blow-up of (0 : 0 : 1) in P 2 C endowed with the homogeneous coordinates (z 0 : z 1 : z 2 ). Consider the affine chart z 2 = 1, i.e. let us work in C 2 with coordinates (z 0 , z 1 ). Set V = (z 0 , z 1 ), (u : v) ∈ C 2 × P 1 C | z 0 v = z 1 u . Let π : V → C 2 be the morphism given by the first projection. Then π -1 (0, 0) = (0, 0), (u : v) | (u : v) ∈ P 1 C , so π -1 (0, 0) P 1 C ; if p = (z 0 , z 1 ) is a point of C 2 {(0, 0)}, then π -1 (p) = ((z 0 , z 1 ), (z 0 : z 1 )) ∈ V {π -1 (0, 0)}, and π |V {π -1 (0,0)} is an isomorphism, the inverse being (z 0 , z 1 ) → (z 0 , z 1 ), (z 0 : z 1 ) .

In other words V = Bl (0,0) P 2 C is the surface obtained by blowing up the complex projective plane at (0 : 0 : 1), π is the blow up of (0 : 0 : 1), and π -1 (0, 0) is the exceptional divisor.

Let V be a complex algebraic variety, and let Bir(V ) be the group of birational maps of V . The group Bir(P n C ) is called the Cremona group. If we fix homogeneous coordinates (z 0 : z 1 : . . . : z n ) of P n C every element φ ∈ Bir(P n C ) can be described by homogeneous polynomials of the same degree φ 0 , φ 1 , . . ., φ n ∈ C[z 0 , z 1 , . . . , z n ] without common factor of positive degree: φ : (z 0 : z 1 : . . . : z n ) φ 0 (z 0 , z 1 , z 2 , . . . , z n ) : φ 1 (z 0 , z 1 , z 2 , . . . , z n ) : . . . : φ n (z 0 , z 1 , z 2 , . . . , z n ) .

The degree of φ is the degree of the φ i 's. In the affine chart z 0 = 1, the map φ is given by (ϕ 1 , ϕ 2 , . . . , ϕ n ) where for any 1 ≤ i ≤ n

ϕ i = φ i (1, z 1 , z 2 , . . . , z n ) φ 0 (1, z 1 , z 2 , . . . , z n ) ∈ C(z 1 , z 2 , . . . , z n ).
The subgroup of Bir(P n C ) consisting of elements φ such that all the ϕ i are polynomials as well as the entries of φ -1 is exactly the group Aut(A n C ) of polynomial automorphisms of the affine space A n C .

Let S be a smooth projective surface. The bubble space B(S) is, roughly speaking, the set of all points that belong to S, or are infinitely near to S. Let us be more precise: consider all surfaces Y above S, i.e. all birational morphisms π : Y → S ; we identify p 1 ∈ Y 1 and p 2 ∈ Y 2 if π -1 1 • π 2 is a local isomorphism in a neighborhood of p 2 that maps p 2 onto p 1 . The bubble space B(S) is the union of all points of all surfaces above S modulo the equivalence relation generated by these identifications. A point p ∈ B(S) ∩ S is a proper point. All points in B(S)

that are not proper are called infinitely near.

Let S and S be two smooth projective surfaces. Let φ : S S be a birational map. By Zariski's theorem (see for instance [START_REF] Beauville | Complex algebraic surfaces[END_REF]) we can write φ = π 2 • π -1 1 where π 1 : Y → S and π 2 : Y → S are finite sequences of blow-ups. We may assume that there is no (-1)-curve in Y contracted by both π 1 and π 2 . We then say that π 2 • π -1 1 is a minimal resolution of φ. The base-points Base(φ) of φ are the points blown up by π 1 . The proper base-points of φ are precisely the indeterminacy points of φ.

A birational morphism π : S → S induces a bijection π • : B(S) → B(S ) Base(π -1 ). A birational map of smooth projective surfaces φ : S S induces a bijection

φ • : B(S) Base(φ) → B(S ) Base(φ -1 ) by φ • = (π 2 ) • • (π 1 ) -1 • where π 2 • π -1 1 is a minimal resolution of φ.
Let us now give some subgroups of the Cremona group:

-First consider the automorphism group of P n C . It is the subgroup formed by regular maps, i.e. maps well defined on P n C and whose inverse is also well defined on P n C : Aut(P n C ) = φ ∈ Bir(P n C ) | Base(φ) = Base(φ -1 ) = / 0 .

To any M = a i, j 0≤i, j≤n ∈ PGL(n + 1, C) corresponds an element of Bir(P n C ) of degree 1:

(z 0 : z 1 : . . . : z n ) → n ∑ j=0 a 0, j z j : n ∑ j=0 a 1, j z j : . . . : n ∑ j=0 a n, j z j and vice-versa. Such elements are biregular. Furthermore Bezout theorem implies that all biregular maps are linear. We thus have the following isomorphism Aut(P n C ) PGL(n + 1, C). -The n-dimensional subgroup of Aut(P n C ) consisting of diagonal automorphisms is denoted by D n . Note that D n is the torus of highest rank of Bir(P n C ) (1) .

-Start with the surface P 1 C × P 1 C considered as a smooth quadric in P 3 C ; its automorphism group contains PGL(2, C) × PGL(2, C). By the stereographic projection the quadric is birationally equivalent to the plane, so that Bir(P 2 C ) contains also a copy of PGL(2, C) × PGL(2, C).

If G is a semi-simple algebraic group, H is a parabolic subgroup of G, and V = G H is a homogeneous variety of dimension n, then V is rational. Once a birational map π : V P n C is given, π • G • π -1 determines an algebraic subgroup of Bir(P n C ). -A fibration of a surface S is a rational map π : S C, where C is a curve, such that the general fibers are one-dimensional. Two fibrations π 1 : S C and π 2 : S C are identified if there exists an open dense subset U ⊂ S that is contained in the domains of π 1 and π 2 such that π 1|U and π 2|U define the same set of fibers. We say that a group G preserves a fibration π if G permutes the fibers. A rational fibration of a rational surface S is a rational map π : S P 1 C such that the general fiber is rational. The following statement due to Noether and Enriques says that, up to birational maps, there exists only one rational fibration of P 2 C :

Theorem 1.1 ( [START_REF] Beauville | Complex algebraic surfaces[END_REF]). -Let S be a surface. Let π : S C be a rational fibration. Then there exists a birational map φ : C × P 1 C S such that π • φ is the projection onto the first factor.

The Jonquières subgroup J of Bir(P 2 C ) is the subgroup of elements that preserve the pencil of lines through the point (0 : 0 : 1) ∈ P 2 C . Any subgroup of Bir(P 2 C ) that preserves a rational fibration is conjugate to a subgroup of J (Theorem 1.1).

With respect to affine coordinates (z 0 : z 1 : 1) an element of J is of the form

(z 0 , z 1 ) αz 0 + β γz 0 + δ , A(z 0 )z 1 + B(z 0 ) C(z 0 )z 1 + D(z 0 )
where α β γ δ belongs to PGL(2, C) and A B C D to PGL(2, C(z 0 )). This induces an isomorphism J PGL(2, C) PGL(2, C(z 0 )).

-Let M = (a i, j ) 1≤i, j≤n ∈ M(n, Z) be a n × n matrix of integers. The matrix M determines a rational self map of P n C given in the affine chart z 0 = 1 by .

φ M : (z 1 , . . . , z n ) → z a 1,
The map φ M is birational if and only if M belongs to GL(n, Z). This yields an injective homomorphism GL(n, Z) → Bir(P n C ) whose image is called the group of monomial maps and is denoted Mon(n, C).

-The well known result of Noether and Castelnuovo states that Theorem 1.2 ( [START_REF] Castelnuovo | Le trasformationi generatrici del gruppo cremoniano nel piano[END_REF][START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]). -The group Bir(P 2 C ) is generated by the involution σ 2 : (z 0 : z 1 : z 2 ) (z 1 z 2 : z 0 z 2 : z 0 z 1 ) and the group Aut(P 2 C ) = PGL(3, C). For n ≥ 3 the Cremona group is not generated by PGL(n + 1, C) and Mon(n, C) (see [START_REF] Hudson | Cremona Transformations in Plane and Space[END_REF][START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]). In other words the subgroup PGL(n + 1, C), Mon(n, C) is a strict subgroup of Bir(P n C ). The finite index subgroup of PGL(n + 1, C), Mon(n, C) generated by PGL(n + 1, C) and the involution σ n : (z 0 : z 1 : . . . :

z n )    n ∏ i=0 i =0 z i : n ∏ i=0 i =1 z i : . . . : n ∏ i=0 i =n z i   
has been studied in [START_REF] Blanc | The group of Cremona transformations generated by linear maps and the standard involution[END_REF]D 15b]. The group G(n, C) = σ n , PGL(n + 1, C) "looks like" G(2, C) = Bir(P 2 C ) in the following sense ([D 15b]): there is no non-trivial finite dimensional linear representation of G(n, C) over any field; the group G(n, C) is perfect, i.e. G(n, C), G(n, C) = G(n, C); the group G(n, C) equipped with the Zariski topology is simple; let ϕ be an automorphism of Bir(P n C ); there exist an automorphism κ of the field C and a birational self map

ψ of P n C such that ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ G(n, C).
We will deal with the Noether and Castelnuovo theorem in §4.3.1 and §4.3.2; the Hudson and Pan theorem in §4.3.3; the fact that there is no non-trivial finite dimensional linear representation of G(2, C) over any field in §5.1; the fact that Bir(P 2 C ) = G(2, C) is perfect in §5.2; the fact that Bir(P 2 C ) = G(2, C) equipped with the Zariski topology is simple in §3.4; the description of Aut(Bir(P 2 C )) = Aut(G(2, C)) in §7.1.

Divisors and intersection theory

Let V be an algebraic variety.

A prime divisor on V is an irreducible closed subset of V of codimension 1. For instance if V is a surface, then the prime divisors of V are the irreducible curves that lie on it; if V is the complex projective space, then the prime divisors are given by the zeros locus of irreducible homogeneous polynomials.

A Weil divisor on V is a formal finite sum of prime divisors with integer coefficients:

m ∑ i=1 a i D i m ∈ N, a i ∈ Z, D i prime divisor of V .
Let us denote by Div(V ) the set of all Weil divisors of V . Let f ∈ C(V ) * be a rational function, and let D be a prime divisor. The multiplicity ν f (D) of f at D is defined by Since ν f (D) is zero for all but finitely many D the divisor div( f ) belongs to Div(V ). Divisors obtained like that are called principal divisors. The set of principal divisors form a subgroup of Div(V ); indeed div( f g) = div( f ) + div(g) for any f , g ∈ C(V ) * .

ν f (D) = k > 0 if f vanishes on D at the order k; ν f (D) = -k if f
Let us introduce an equivalence relation on Div(V ). Two divisors D, D are linearly equivalent if D -D is a principal divisor. The set of equivalence classes corresponds to the quotient of Div(V ) by the subgroup of principal divisors. The Picard group of V is the group of isomorphism classes of line bundles on V ; it is denoted Pic(V ). When V is smooth the quotient of Div(V ) by the subgroup of principal divisors is isomorphic to Pic(V ).

Example 2. -Let us determine Pic(P n C ). Consider the morphism of groups θ : Div(P n C ) → Z which associates to any divisor D of degree d the integer d. Note that ker θ is the subgroup of principal divisors of P n C : let D = ∑ a i D i be an element of ker θ where each D i is a prime divisor given by an homogeneous polynomial f i ∈ C[z 0 , z 1 , . . . , z n ] of some degree d i . Since ∑ a i d i = 0, f = ∏ f a i i belongs to C(P n C ) * . By construction D = div( f ) hence D is a principal divisor. Conversely any principal divisor is equal to div( f ) where f = g/h for some homogeneous polynomials g, h of the same degree. Thus any principal divisor belongs to ker θ.

Since Pic(P n C ) is the quotient of Div(P n C ) by the subgroup of principal divisors, we get by restricting θ to the quotient an isomorphism between Pic(P n C ) and Z. As an hyperplane is sent on 1 we obtain that Pic(P n C ) = ZH where H is the divisor of an hyperplane. Let us now assume that dimV = 2; set V = S. We can define the notion of intersection: If C is an irreducible curve on S, the strict transform C of C is C = π -1 (C {p}).

If C ⊂ S is a curve and if p is a point of S, let us define the multiplicity m p (C) of C at p.

Recall that if V is a quasi-projective variety, and if q is a point of V , then O q,V denotes the set of equivalence classes of pairs (U, ϕ) where ϕ belongs to C[U], and U ⊂ V is an open subset such that q ∈ U. Let m be the maximal ideal of O p,S . If f is a local equation of C, then m p (C)

is the integer k such that f belongs to m k m k+1 .

Example 3. -Assume that S is a rational surface. There exists a neighborhood U of p in S with U ⊂ C 2 . We can assume that p = (0, 0) in this affine neighborhood and that C is a curve described by the equation n ∑ i=1 P i (z 0 , z 1 ) = 0 where P i is an homogeneous polynomial of degree i.

The multiplicity m p (C) is the lowest i such that P i is not equal to 0. The following properties hold: In particular C • D ≥ 0.

Let C be a curve on S, and let p be a point of S. Take local coordinates z 0 , z 1 at p such that p = (0, 0). Set k = m p (C). The curve C is thus given by P k (z 0 , z 1 ) + P k+1 (z 0 , z 1 ) + . . . + P r (z 0 , z 1 ) = 0 where the P i 's denote homogeneous polynomials of degree i. The blow up of p can be viewed as (u, v) → (uv, v), and the pull-back of C is given by v k p k (u, 1) + vp k+1 (u, 1) + . . . + v r-k p r (u, 1) = 0.

In other words the pull-back of C decomposes into k times the exceptional divisor E = π -1 (0, 0) = (v = 0) and the strict transform. We can thus state: Lemma 1.5 ( [START_REF] Hartshorne | Algebraic geometry[END_REF]). -Let S be a smooth surface. Let π : Bl p S → S be the blow-up of a point p ∈ S. If C is a curve on S, if C is its strict transform and if E = π -1 (p) is the exceptional divisor, then π * (C) = C + m p (C)E.

We also have the following statement:

Proposition 1.6 ( [START_REF] Hartshorne | Algebraic geometry[END_REF]). -Let S be a smooth surface, let p be a point of S, and let π : Bl p S → S be the blow-up of p. Denote by E ⊂ Bl p S the exceptional divisor π -1 (p) P 1 C . Then Pic(Bl p S) = π * Pic(S) + ZE.

The intersection form on Bl p S is induced by the intersection form on S via the following formulas: If V is an algebraic variety, then the nef cone Nef(V ) is the cone of divisors D such that D •C ≥ 0 for any curve C in V .

A geometric definition of birational maps

Let φ be the element of Bir(P 2 C ) given by φ : (z 0 :

z 1 : z 2 ) φ 0 (z 0 , z 1 , z 2 ) : φ 1 (z 0 , z 1 , z 2 ) : φ 2 (z 0 , z 1 , z 2 )
where the φ i 's are homogeneous polynomials of the same degree ν, and without common factor of positive degree. The linear system Λ φ of φ is the strict pull-back of the system O P 2

C

(1) of lines of P 2 C by ϕ. Remarks 1.7. -If A is an automorphism of P 2 C , then Λ φ = Λ A•φ . The degree of the curves of Λ φ is ν.

Example 4. -The linear system associated to σ 2 is the linear system of conics passing through (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1).

Remark 1.8. -Let us define the linear system of a divisor and then mention the connection between the linear system of a divisor and the linear system of a birational map. Let D be a divisor on a surface S. Denote by |D| the set of all effective divisors on S linearly equivalent to D. Every non-vanishing section of O S (D) defines an element of |D|, namely its divisor of zeros; conversely every element of |D| is the divisor of zeros of a non-vanishing section of O S (D), defined up to scalar multiplication. Hence |D| can be naturally identified with the projective space associated to the vector space H 0 (O S (D)). A linear subspace P of |D| is called a linear system on S; of course equivalently P can be defined by a vector subspace of H 0 (O S (D)). The subspace P is complete if P = |D|. The dimension of P is its dimension as a projective space. A one-dimensional linear system is a pencil . A curve C is a fixed component of P if every divisor of P contains C. The fixed part of P is the biggest divisor that is contained in every element of P. A point p of S is a base-point of P if every divisor of P contains p. If the linear system has no fixed part, then it has only a finite number of fixed points; this number is bounded by D 2 for D ∈ P.

Let S be a surface. Then there is a bijection between rational maps φ : S P n C such that φ(S) is contained in no hyperplane and linear systems on S without fixed part and of dimension n .

This correspondence is constructed as follows: to the map φ we associate the linear system φ * |H| where |H| is the system of hyperplanes in P n C . Conversely let P be a linear system on S with no fixed part; denote by P the projective dual space to P. Define a rational map φ : S P by sending p ∈ S to the hyperplane in P consisting of the divisors passing through p: the map φ is defined at p if and only if p is not a base-point of P.

If p 1 is a point of indeterminacy of φ, then denote by π 1 : Bl p 1 P 2 C → P 2 C the blow-up of p 1

and by E 1 the associated exceptional divisor. The map ϕ 1 = φ • π 1 is a birational map from

) = Z ⊕ ZE 1 ⊕ ZE 2 ⊕ . . . ⊕ ZE r , 2 = • , E 2 i = E i • E i = -1, E i • E j = 0 ∀ 1 ≤ i = j ≤ r, E i • = 0 ∀ 1 ≤ i ≤ r.
The curves of Λ φ pass through the p i 's with multiplicity m p i (φ). Applying r times Lemma 1.5 the elements of Λ ϕ r are equivalent to

νL - r ∑ i=1 m p i (φ)E i
where L is the pull-back of a generic line in P 2 C . As a result the curves of Λ ϕ r have self intersection ν 2 -r ∑ i=1 m p i (φ) 2 . Note that all the members of a linear system are linearly equivalent and that the dimension of Λ ϕ r is 2; the self intersection has thus to be non-negative by Proposition 1.4. As a consequence the number r exists; in other words φ has a finite number of base-points. By construction ϕ r : Bl p 1 ,p 2 ,...,p r P 2 C → P 2 C is a birational morphism which is the blow-up of the base-points of φ -1 . Consider a general line L of P 2 C that does not pass through p 1 , p 2 , . . ., p r . Its pull-back ϕ -1 r (L) corresponds to a smooth curve on Bl p 1 ,p 2 ,...,p r P 2 C which has self-intersection 1 and genus 0. Hence (ϕ -1 r (L)) 2 = 1, ϕ -1 r (L) • K Bl p 1 ,p 2 ,...,p r P 2 C = -3.

As the elements of Λ ϕ r are equivalent to νL -r ∑ i=1 m p i (φ)E i and since K Bl p 1 ,p 2 ,...,p r P 2

C = -3L + r ∑ i=1 E i
the following equalities hold:

         r ∑ i=1 m p i (φ) = 3(ν -1), r ∑ i=1 m p i (φ) 2 = ν 2 -1.
Examples 1.

-If ν = 2, then r = 3 and m p 1 (φ) = m p 2 (φ) = m p 3 (φ) = 1. If ν = 3, then r = 5 and m p 1 (φ) = 2, m p 2 (φ) = m p 3 (φ) = m p 4 (φ) = m p 5 (φ) = 1.

CHAPTER 2 AN ISOMETRIC ACTION OF THE CREMONA GROUP ON AN INFINITE DIMENSIONAL HYPERBOLIC SPACE

If S is a projective surface, the group Bir(S) of birational self maps of S acts faithfully by isometries on a hyperbolic space H ∞ (S) of infinite dimension. After recalling some notions of hyperbolic geometry in the first section of this chapter we describe this construction in the second section. Let us now give an outline of it before heading into details. Let S be a projective surface. If π : Y → S is a birational morphism, then one obtains an embedding π * : NS(S) → NS(Y ) of Néron-Severi groups. If π 1 : Y 1 → S and π 2 : Y 2 → S are two birational morphisms, then π 2 is above π 1 if π -1 1 • π 2 is a morphism, one can always find a third birational morphism π 3 : Y 3 → S that is above π 1 and π 2 .

Hence the inductive limit of all groups NS(Y i ) for all surfaces Y i above S is well-defined; this limit Z(S) is the Picard-Manin space of S. The intersection forms on Y i yield to a scalar product , on Z(S).

Consider all surfaces Y above S, i.e. all birational morphisms π : Y → S. We identify p 1 ∈ Y 1 and p 2 ∈ Y 2 if π -1 1 • π 2 is a local isomorphism in a neighborhood of p 2 that maps p 2 onto p 1 .

The bubble space B(S) of S is the union of all points of all surfaces above S modulo the equivalence relation generated by these identifications. If p belongs to B(S), then we denote by e p the divisor class of the exceptional divisor of the blow up of p. The equalities e p •e p = -1 and e p • e p = 0 hold by Proposition 1.6

The Néron-Severi group NS(S) is naturally embedded as a subgroup of the Picard-Manin space; this finite dimensional lattice is orthogonal to e p for any p ∈ B(S). More precisely

Z(S) = NS(S)

p∈B(S)

Ze p .

As a result any element v of Z(S) can be written as a finite sum

v = w + ∑ p∈B(S)
m p e p .

There is a completion process for which the completion Z(S) of Z(S)

⊗ Z R is Z(S) = w + ∑ p∈B(S)
m p e p | w ∈ NS(R, S), ∑ p∈B(S) m 2 p < ∞ .

The intersection form extends as a scalar product with signature (1, ∞) on this space. The hyperbolic space H ∞ (S) of S is defined by H ∞ (S) = w ∈ Z(S), | w, w = 1, w, a > 0 for all ample classes a ∈ NS(S) .

It is an infinite dimensional analogue of the classical hyperbolic space H n . One can define a complete distance dist on

H ∞ (S) by cosh(dist(v, w)) = v, w .
Geodesics are intersection of H ∞ (S) with planes. The projection of H ∞ (S) to the projective space P(Z(S)) is one to one, and the boundary of its image is the projection of the cone of isotropic vectors of Z(S):

∂H ∞ (S) = R + v | v ∈ Z(S), v, v = 0, v, a > 0 for all ample classes a ∈ NS(S) .
The important fact is that Bir(S) acts faithfully on Z(S) by continuous linear endomorphisms preserving the intersection form, the effective cone, the nef cone, Z(S) and also H ∞ (S).

If φ is an element of Bir(S), we denote by φ * its action on Z(S): it is a linear isometry with respect to the intersection form; we also denote by φ * the isometry of H ∞ (S) induced by this endomorphism of Z(S). Let f be an isometry of H ∞ (S); the translation length of f is

L( f ) = inf dist(v, f (v)) | v ∈ H ∞ (S) .
If this infimum is a minimum, then either it is equal to 0, f has a fixed point in H ∞ (S), and f is elliptic; or it is positive, and f is loxodromic.

When the infimum is not realized, L( f ) is equal to 0, and f is parabolic. This classification into three types holds for all isometries of H ∞ (S). For isometries φ * induced by birational maps φ of S there is a dictionary between this classification and the geometric properties of φ. We give this dictionary in the third section.

Some hyperbolic geometry

Consider a real Hilbert space H of dimension n. Let e 0 be a unit vector of H , and let e ⊥ 0 be the orthogonal complement of the space Re 0 . Denote by (e i ) i∈I an orthonormal basis of e ⊥ 0 . A scalar product with signature (1, n -1) can be defined on H by setting

u, v = a 0 b 0 -∑ i∈I a i b i for any two elements u = a 0 e 0 + ∑ i∈I a i e i and v = b 0 e 0 + ∑ i∈I b i e i of H . The set v ∈ H | v, v = 1
defines a hyperboloid with two connected components. Let H n-1 be the connected component of this hyperboloid that contains e 0 . A metric can be defined on H n-1 by

d(u, v) := arccosh( u, v ).
Remark 2.1. -A useful model for H 2 is the Poincaré model: H 2 is identified to the upper half-plane z ∈ C | Im(z) > 0 with its Riemanniann metric given by ds 2 = x 2 +y 2 y 2 . Its group of orientation preserving isometries coincides with PSL(2, R), acting by linear fractional transformations.

Let (H , ., . ) be a real Hilbert space of infinite dimension. Let e 0 be a unit vector of H , and let e ⊥ 0 be its orthogonal complement. Any element v of H can be written in a unique way as it has signature (1, ∞). Let H ∞ be the hyperboloid given by

H ∞ = x ∈ H | B(x,x) = 1, B(e 0 , x) > 0 .
We consider on H ∞ the distance d defined by cosh d(x, y) = B(x,y). The space (H ∞ , d) is a complete metric space of infinite dimension.

2.1.1. δ-hyperbolicity and CAT(-1) spaces. -Let (X, d) be a geodesic metric space. Let x, y, z be three points of X. We denote by [p, q] the segment with endpoints p and q. A geodesic triangle with vertices x, y, z is the union of three geodesic segments

[x, y], [y, z] and [z, x]. Let δ ≥ 0. If for any point m ∈ [x, y] there is a point in [y, z] ∪ [z, x]
at distance less than δ of m, and similarly for points on the other edges, then the triangle is said do be δ-slim. A δ-hyperbolic space is a geodesic metric space whose all of geodesic triangles are δ-slim. A δ-hyperbolic space is called Gromov hyperbolic space.

Examples 2. -Metric trees are 0-hyperbolic: all triangles are tripods. The hyperbolic plane is (-2)-hyperbolic. In fact the incircle of a geodesic triangle is the circle of largest diameter contained in the triangle, and any geodesic triangle lies in the interior of an ideal triangle, all of which are isometric with incircles of diameter 2 log 3 (see [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF]). The space R 2 endowed with the euclidian metric is not δ-hyperbolic (for instance because of the existence of homotheties).

Let us now introduce CAT(-1) spaces (1) . Let (X, d X ) be a geodesic metric space. Consider a geodesic triangle T in X determined by the three points x, y, z and the data of three geodesics between two of these three points. A comparison triangle of T in the metric space

(X , d X ) is a triangle T such that    d X (x, y) = d X (x , y ) d X (x, z) = d X (x , z ) d X (y, z) = d X (y , z ) Let p be a point of [x, y] ⊂ T . A point p ∈ [x , y ] ⊂ T is a comparison point of p if d X (x , p ) = d X (x, p). The triangle T satisfies the CAT(-1) inequality if for any (x, y) ∈ T 2 d X (x, y) ≤ ||x -y || H 2
where T is a comparison triangle of T in H 2 and x ∈ T (resp. y ∈ T ) is a comparison point of x (resp. y).

The space X is CAT(-1) if all its triangles satisfy the CAT(-1) inequality.

Remark 2.2. -The CAT(-1) spaces are Gromov hyperbolic, but the converse is false.

Set H >0 = v ∈ H | v, v > 0 .
The image of v by the map

η : H >0 → H ∞ v → v v, v
is called the normalization of v. Geometrically η associates to a point v ∈ H >0 the intersection of H ∞ with the line through v. Note that if the intersection of H with a vectorial subspace of dimension n + 1 of H is not empty, then it is a copy of H n . In particular there exists a unique geodesic segment between two points of H ∞ obtained as the intersection of H ∞ with the plane that contains these two points. Hence any triangle of H ∞ is isometric to a triangle of H 2 . As a result H ∞ is CAT(-1) and δ-hyperbolic for the same constant δ as H 2 .

(1) The terminology corresponds to the initials of E. Cartan, A. Alexandrov and V. Toponogov.

2.1.2. Boundary of H ∞ . -Let (X, d) be a geodesic metric space. Let T be a geodesic triangle of X given by x, y, z ∈ X and geodesic segments between two of these three points. The triangle T satisfies the CAT(0) inequality if for any

(x, y) ∈ T 2 d X (x, y) ≤ ||x -y || R 2 where x ∈ T (resp. y ∈ T ) is a comparison point of x (resp. y) and T is a comparison triangle of T in R 2 .
The space X is CAT(0) if all its triangles satisfy the CAT(0) inequality.

Remark 2.3. -A CAT(-1) space is a CAT(0) space. In particular H ∞ is a CAT(0) space.

Since H ∞ is a CAT(0), complete metric space there exists a notion of boundary at infinity that generalizes the notion of boundary of finite dimensional Riemann varieties which are complete, simply connected and with negative curvature. The boundary of H ∞ is defined by h is unique up to scalar multiplication. There is another unique isotropic eigenline Rv - h corresponding to the eigenvalue 1 λ . On the orthogonal complement of Rv - h ⊕ Rv + h the isometry h acts as a rotation with respect to , . The boundary points determined by v - h and v + h are the two fixed points of h in H n ∪ ∂H n ; the first one is an attracting fixed point α(h), the second one is a repelling fixed point ω(h).

∂H ∞ = v ∈ H | v, v = 0, v, e 0 > 0 . A point of ∂H ∞ is called point at infinity.
To an isometry h of H n one can associate the translation length of h:

L(h) = inf d(h(p), p) |p ∈ H n .
The isometry h is elliptic if and only if L(h) = 0, and the infimum is achieved, i.e. h has a fixed point in H n . The isometry h is parabolic if and only if L(h) = 0, and the infinimum is not achieved. The isometry h is loxodromic if and only if L(h) > 0. In that case exp(L(h)) is the largest eigenvalue of h and d(p, h n (p)) grows like nL(h) as n goes to infinity for any point p ∈ H n .

2.2. The isometric action of Bir(S) on an infinite dimensional hyperbolic space 2.2.1. The Picard-Manin space. -Let S be a smooth, irreducible, projective, complex surface. As we see in Chapter 1 the Picard group Pic(S) is the quotient of the abelian group of divisors by the subgroup of principal divisors ( [START_REF] Hartshorne | Algebraic geometry[END_REF]). The intersection between curves extends to a quadratic form, the so-called intersection form:

Pic(S) × Pic(S) → Z, (C, D) → C • D
The quotient of Pic(S) by the subgroup of divisors E such that E • D = 0 for all divisor classes D is the Néron-Severi group NS(S). In case of rational surfaces we have NS(S) = Pic(S). The Néron-Severi group is a free abelian group, and its rank, the Picard number is finite. The pull-back of a birational morphism π : Y → S yields an injection from Pic(S) into Pic(Y ); we thus get an injection from NS(S) into NS(Y ). The morphism π : Y → S can be written as a finite sequence of blow ups. Let e 1 , e 2 , . . ., e k ⊂ Y be the class of the irreducible components of the exceptional divisor of π, that is the classes contracted by π. We have the following decomposition

NS(Y ) = NS(S) ⊕ Ze 1 ⊕ Ze 2 ⊕ . . . ⊕ Ze k (2.2.1)
which is orthogonal with respect to the intersection form.

Consider π 1 : Y → S and π 2 : Y → S two birational morphisms of smooth projective surfaces. We say that π 1 is above π 2 if π -1 2 • π 1 is a morphism. For any two birational morphisms π 1 : Y → S and π 2 : Y → S there exists a birational morphism π 3 : Y → S that lies above π 1 and π 2 .

Let us consider the set of all birational morphisms of smooth projective surfaces π : Y → S. The corresponding embeddings of the Néron-Severi groups NS(S) → NS(Y ) form a directed family; the direct limit Let p be a point of the bubble space of S. Denote by e p the divisor class of the exceptional divisor of the blow-up of p in the corresponding Néron-Severi group. One deduces from (2.2.1) the following decomposition

Z(S) := lim

Z(S) = NS(S) ⊕ p∈B(S)
Ze p .

Furthermore according to Proposition 1.6 the following properties hold e p • e p = -1 e p • e q = 0 for all p = q 2.2.2. The hyperbolic space H ∞ (S). -Let S be a smooth projective surface, and let Z(S) be its Picard-Manin space. We define Z(S) to be the completion of the real vector space Z(S) ⊗ R

Z(S) = v + ∑ p∈B(S) m p e p | v ∈ NS(S) ⊗ R, m p ∈ R, ∑ p∈B(S) m 2 p < ∞ .
The intersection form extends continuously to a quadratic form on Z(S) with signature (1, ∞). Let Isom(Z(S)) be the group of isometries of Z(S) with respect to the intersection form. The set of vectors v ∈ Z(S) such that v, v = 1 is a hyperboloid. The subset

H ∞ (S) = v ∈ Z(S) | v, v = 1, v, e 0 > 0
is the sheet of that hyperboloid containing ample classes of NS(S, R). Let Isom(H ∞ (S)) be the subgroup of Isom(Z(S)) that preserves H ∞ (S). The space H ∞ (S) equipped with the distance defined by cosh

(d(v, v )) = v, v
is isometric to a hyperbolic space H ∞ . Let ∂H ∞ (S) be the boundary of H ∞ (S). To simplify we will often write H ∞ (resp. ∂H ∞ ) instead of H ∞ (S) (resp. ∂H ∞ (S)).

An isometric action of Bir(S).

-Let us now describe the action of Bir(S) on H ∞ (see [START_REF] Yu | of North-Holland Mathematical Library[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). Let φ : Y → S be a birational morphism of smooth projective surfaces.

Denote by p 1 , p 2 , . . ., p n ∈ B(S) the points blown up by φ. Denote by e p i the irreducible component of the exceptional divisor contracted to p i . One has

NS(Y ) = NS(S) ⊕ Ze p 1 ⊕ Ze p 2 ⊕ . . . ⊕ Ze p n .
The morphism φ induces the isomorphism φ * : Z(Y ) → Z(S) defined by

   φ * (e p ) = e φ • (p) ∀ p ∈ B(Y ) Base(φ) φ * (e p i ) = e p i ∀ 1 ≤ i ≤ n φ * (D) = D ∀ D ∈ NS(S) ⊂ NS(Y )
Let φ : Y S be a birational map of smooth projective surfaces. Let π 2 • π -1 1 be a minimal resolution of φ. The map φ induces an isomorphism φ * : Z(Y ) → Z(S) defined by

φ * = (π 2 ) * • (π 1 ) -1 * .
Let S be a smooth projective surface. Any element φ of Bir(S) induces an isomorphism φ * : Z(S) → Z(S), and φ * yields an automorphism of Z(S) ⊗ R which extends to an automorphism of the completion Z(S) and preserves the intersection form.

Let φ be a birational self map of P 2 C . Assume that φ has degree d. Then the base-point e 0 , i.e. the class of a line in P 2 C , is mapped by φ * to the finite sum

de 0 -∑ i m i e p i
where each m i is a positive integer and e p i are the classes of the exceptional divisors corresponding to the base-points of φ -1 . For instance if φ = σ 2 is the standard Cremona involution, then (σ 2 ) * e 0 = 2e 0 -e p 1 -e p 2 -e p 3

where p 1 = (1 : 0 : 0), p 2 = (0 : 1 : 0) and p 3 = (0 : 0 : 1).

         d 2 = 1 + n ∑ j=1 m 2 j 3d -3 = n ∑ j=1 m j
Example 5. -Let us understand the isometry (σ 2 ) * . Denote by p 1 , p 2 and p 3 the basepoints of σ 2 , and set S = Bl p 1 ,p 2 ,p 3 P 2 C . The involution σ 2 lifts to an automorphism σ 2 on S. The Néron-Severi group NS(S) of S is the lattice of rank 4 generated by the class e 0 , coming from the class of a line in P 2 C , and the classes e i = e p i given by the three exceptional divisors. The action of σ 2 on NS(S) is given by

       ( σ 2 ) * e 0 = 2e 0 -e 1 -e 2 -e 3 ( σ 2 ) * e 1 = e 0 -e 2 -e 3 ( σ 2 ) * e 2 = e 0 -e 1 -e 3 ( σ 2 ) * e 3 = e 0 -e 1 -e 2
Then ( σ 2 ) * coincides on NS(S) with the reflection with respect to e 0 -e 1 -e 2 -e 3 :

( σ 2 ) * (p) = p + p, e 0 -e 1 -e 2 -e 3 ∀ p ∈ NS(S)
Let us blow up all points of S; we thus obtain a basis of Z(P 2 C ):

Z(P 2 C ) = NS(S) p∈B(S)
Ze p .

The isometry (σ 2 ) * of Z(P 2 C ) acts on NS(S) as the reflection ( σ 2 ) * and permutes each vector e p with e σ 2 (p) .

Types and degree growth

Consider an ample class h ∈ NS(S, R) with self-intersection 1. The degree of φ ∈ Bir(S) with respect to the polarization h is defined by

deg h φ = φ * (h), h = cosh(d(h, φ * h)).

Note that if S = P 2

C and h = e 0 is the class of a line, then deg h φ is the degree of φ as defined in Chapter 1.

A birational map φ of a projective surface S is virtually isotopic to the identity if there is a positive iterate φ n of φ and a birational map ψ : Z S such that ψ -1 • φ n • ψ is an element of Aut(Z) 0 ; a Jonquières twist if φ preserves a one parameter family of rational curves on S, but φ is not virtually isotopic to the identity; a Halphen twist if φ preserves a one parameter family of genus one curves on S, but φ is not virtually isotopic to the identity.

Furthermore the Jonquières twists (resp. Halphen twists) preserve a unique fibration ([DF01]).

Remark 2.5. -If φ is a Jonquières (resp. Halphen) twist, then, after conjugacy by a birational map ψ : Z S, φ permutes the fibers of a rational (resp. genus one) fibration π : Z → B. If z is the divisor class of the generic fiber of the fibration, then z is an isotropic vector in Z(S)

fixed by φ * . In particular φ * can not be loxodromic.

Let C and C be two smooth cubic curves in the complex projective plane. By Bezout theorem C and C intersect in nine points denoted p 1 , p 2 , . . ., p 9 . There is a pencil of cubic curves passing through these nine points. Let us blow up p 1 , p 2 , . . ., p 9 . We get a rational surface S with a fibration π : S → P 1 C whose fibers are genus 1 curves. More generally let us consider a pencil of curves of degree 3m for m ∈ Z + , blow up its base-points and denote by S the surface we get. Such a pencil of genus 1 curves is called a Halphen pencil , and such a surface is called a Halphen surface of index m .

Definition. -A surface S is a Halphen one if | -mK S | satisfies the three following properties it is one-dimensional, it has no fixed component, it is base-point free.

According to [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF] up to birational conjugacy every pencil of genus 1 curves of P 2 C is a Halphen pencil, Halphen surfaces are the only examples of rational elliptic surfaces.

Lemma 2.6 ([Ure]

). -Let S be a Halphen surface. Let φ be an element of Bir(S) that preserves the Halphen pencil. Then φ belongs to Aut(S).

Up to conjugacy by birational maps every pencil of genus 1 curves of P 2 C is a Halphen pencil and Halphen surfaces are the only examples of rational elliptic surfaces ([CD12a]) so Lemma 2.6 implies:

Corollary 2.7. -A subgroup G of Bir(P 2 C ) that preserves a pencil of genus 1 curves is conjugate to a subgroup of the automorphism group of some Halphen surface.

Proof of Lemma 2.6. -The Halphen pencil is defined by a multiple of the class of the anticanonical divisor -K S . As a result any birational map of a Halphen surface that preserves the Halphen fibration preserves the class of the canonical divisor K S . Assume by contradiction that φ is not an automorphism. Take a minimal resolution of φ

Z η 0 0 π Ð Ð S φ G G S
Denote by E i and F i the total pull backs of the exceptional curves. On the one hand

K Z = η * (K S ) + ∑ E i ,
and on the other hand

K Z = π * (K S ) + ∑ F i .
The map φ preserves K S , so η * (K S ) = π * (K S ), and hence ∑ E i = ∑ F i . By assumption φ is not an automorphism, i.e. ∑ E i contains at least one (-1)-curve E k . Hence both

E k • ∑ E i = -1 and E k • ∑ F i = -1
hold. This implies that E k is contained in the support of ∑ F i : contradiction with the minimality of the resolution.

Remark 2.8. -The automorphism groups of Halphen surfaces are studied in [START_REF] Gizatullin | Rational G-surfaces[END_REF] and in [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF].

On the contrary Jonquières twists are not conjugate to automorphisms of projective surfaces ([DF01, BD15]).

Let S be a projective complex surface with a polarization H. Let φ : S S be a birational map. The dynamical degree of φ is defined by

λ(φ) = lim n→+∞ deg H (φ n ) 1/n .
Definitions. -An element φ of Bir(P 2 C ) is called elliptic, (resp. parabolic, resp. loxodromic) if the corresponding isometry φ * is elliptic (resp. parabolic, resp. loxodromic). . The map φ is loxodromic if and only if λ(φ) > 1. As a consequence when φ ∈ Bir(P 2 C ), λ(φ) > 1, the isometry φ * preserves a unique geodesic line Ax(φ) ⊂ H ∞ called the axis of φ. This line is the intersection of H ∞ with a plane P φ ⊂ Z(P 2 C ) which intersects the isotropic cone of Z(P

(p) = λ(φ) -1 α+λ(φ)ω √ 2 one obtains exp(L(φ * )) + 1 exp(L(φ * )) = 2cosh(d(p, φ * (p))) = 2 p, φ * (p) = λ(φ) + 1 λ(φ) .
The translation length is thus equal to log λ(φ). Consequently λ(φ) does not depend on the polarization and is invariant under conjugacy.

There is a correspondence between the dynamical behavior of a birational map φ of S, in particular its degree, and the type of the induced isometry on H ∞ : Theorem 2.9 ([Giz80, DF01, Can99]). -Let S be a smooth projective complex surface with a fixed polarization H. Let φ : S S be a birational map. Then one of the following holds:

φ is elliptic, (deg H φ n ) n is
(z 0 , z 1 ) → (αz 0 + P(z 1 ), βz 1 + γ) | α, β ∈ C * , γ ∈ C, P ∈ C[z 1 ] is elliptic.
Any element of J of the form

(z 0 , z 1 ) z 0 , a(z 0 )z 1 + b(z 0 ) c(z 0 )z 1 + d(z 0 ) with (tr M) 2 det M ∈ C(z 0 ) C where M = a(z 0 ) b(z 0 ) c(z 0 ) d(z 0 ) is a Jonquières twist ([CD12b]).
Consider the family of birational self maps of P 2 C ) given in the affine chart z 2 = 1 by

φ ε : (z 0 , z 1 ) z 1 + 1 -ε, z 0 z 1 -ε z 1 + 1 . If ε = -1, then φ ε is elliptic; ε ∈ {0, 1}, then φ ε is a Jonquières twist; ε ∈ 1 2 , 1 3 , then φ ε is a Halphen twist; ε ∈ k≥4 1 k
, then φ ε is loxodromic.

This family has been introduced in [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF].

If φ : (z 0 , z 1 ) (z 1 , z 0 +z 2 1 ), then deg(φ n ) = (deg φ) n = 2 n . If ψ : (z 0 , z 1 ) (z 2 0 z 1 , z 0 z 1 ), then deg ψ n ∼ 3+ √ 5 2 n ; in particular deg(ψ n ) = (deg ψ) n .
Let us finish with a more geometric example. Consider the elliptic curve

E = C Z[i]
. The linear action of the group GL(2, Z[i]) on the complex plane preserves the lattice

Z[i] × Z[i]
. This yields to an action of GL(2, Z[i]) by regular automorphisms on the abelian surface S = E × E. Since this action commutes with (z 0 , z 1 ) → (iz 0 , iz 1 ) one gets a morphism from PGL(2,

Z[i]) to Aut S (z 0 , z 1 ) → (iz 0 , iz 1 ) . As S (z 0 , z 1 ) → (iz 0 , iz 1 ) is rational one obtains an embedding of PGL(2, Z[i]) into Bir(P 2 C ).
Any element virtually isotopic to the identity is regularizable, that is birationally conjugate to an automorphism. What can we say about two birational maps virtually isotopic to the identity ? We will see that if they commute they are simultaneously regularizable. Before proving it let us introduce a new notion.

Definitions. -An element φ ∈ Bir(P 2 C ) is algebraically stable if deg φ n = (deg φ) n for all n ≥ 0.
More generally if S is a compact complex surface, then φ ∈ Bir(S) is algebraically stable if (φ * ) n = (φ n ) * for all n ≥ 0.

A geometric characterization of algebraically stable maps is the following: φ ∈ Bir(S) is algebraically stable if and only if there is no curve C ⊂ S such that φ k (C) ∈ Ind(φ) for some integer k. Let us give an idea of the fact that this geometric characterization is equivalent to the Definition when

S = P 2 C . If φ k C Ind(φ) ⊂ Ind(φ), then all the components of φ • φ k have a common factor that defines the equation of C . Then deg(φ • φ k ) < (deg φ)(deg φ k ). The converse holds.
Diller and Favre proved the following result:

Proposition 2.10 ([DF01]
). -Let S be a compact complex surface, and let φ be a birational self map of S. There exists a composition of finitely many point blow-ups that lifts φ to an algebraically stable map.

Before giving the proof, let us give its idea. Assume that φ is not algebraically stable. In other words there exist a curve C ⊂ S and an integer k such that C is blown down onto p 1 and p k = φ k-1 (p 1 ) belongs to Ind(φ). The idea of Diller and Favre to get an algebraically stable map is the following: after blowing up the points p i = φ i (p 1 ), i = 1, . . ., k, the orbit of C consists of curves. Doing this for any element of Exc(φ) whose an iterate belongs to Ind(φ) one gets the statement (note that the cardinal of Exc(φ) is finite, so the process ends).

Proof. -Let us write φ as follows φ = φ n • φ n-1 • . . . • φ 1 where

φ i : S i-1 → S i ; S 0 = S n = S; and
(i) either φ i blows up a point p i = Ind(φ i ) ∈ S i , and we denote by

V i+1 = Exc(φ -1 i ) ⊂ S i+1 the exceptional divisor of φ -1 i ; (ii) or φ i blows down the exceptional divisor E i ⊂ S i ; in this case we set q i+1 := φ i (E i ) ∈ S i+1 .
For any j ∈ N set S j := S j mod n and φ j := φ j mod n . Assume that φ is not algebraically stable. Then there exist integers 1

≤ i ≤ N such that φ i blows down E i and φ N-1 • φ N-2 • . . . • φ i (E i ) = p N ∈ Ind(φ N ).
Choosing a pair (i, N) of minimal length we can assume that for all i < j ≤ N

m j := φ j • φ j-1 • . . . • φ i (E i ) = φ j • φ j-1 • . . . • φ i+1 (q i+1 )
does not belong to Ind(φ i ) ∪ Exc(φ i ).

First blow up S N at m N = p N . Then φ N lifts to a biholomorphism φ N of Bl p N S N ; φ N-1 either blows up the two distinct points m N-1 and p N-1 or blows up m N-1 and blows down E N-1 / ∈ m N-1 ;

∑ Card φ j Exc(φ j ) = ∑ Card φ j Exc( φ j ) . Remark that modifying S N means modifying S N+n , S N-n , . . . nevertheless blowing up a point m j does not interfere with the behavior of the map φ j around m N+n , m N-n , . . . (indeed if j 1 = j 2 mod n but j 1 = j 2 , then the points m j 1 , m j 2 of S 1 = S 2 are distinct), and these points can be blow up independently.

Similarly blow up m N-1 , m N-2 , . . ., m i+2 . At each step ∑ Card(φ j (Exc(φ j ))) remains constant. Let us finish by blowing up m i+1 = φ i (E i ); the situation is then different: φ i becomes a biholomorphism φ i . The number of components of Exc(φ i ) thus reduces from 1 to 0. As a consequence ∑ Card φ j Exc( φ j ) = ∑ Card φ j Exc(φ j ) -1.

(2.3.1) Repeating finitely many times the above argument either we produce an algebraically stable map φ = φ N • φ N-1 • . . . • φ 1 , or thanks to (2.3.1) we eleminate all exceptional components of the factors of φ. In both cases we get an algebraically stable map.

Lemma 2.11 ([D 06a]). -Let φ, ψ be two birational self maps of a compact complex surface S. Assume that φ and ψ are both virtually isotopic to the identity. Assume that φ and ψ commute.

There exist a surface Y and a birational map ζ : Y S such that

ζ -1 • φ • ζ ∈ Aut(Y ) 0 for some integer , ζ -1 • ψ • ζ is algebraically stable.
Proof. -Since φ is virtually isotopic to the identity we can assume that up to birational conjugacy and finite index φ is an automorphism of S. Let N(ψ) be the minimal number of blow-ups needed to make ψ algebraically stable (such a N(ψ) exists according to Proposition 2.10). If N(ψ) = 0, then ζ = id suits. Assume that Lemma 2.11 holds when N(ψ) ≤ j. Consider a pair (φ, ψ) of birational self maps of S such that φ and ψ are both virtually isotopic to the identity, φ and ψ commute,

N(ψ) = j + 1.
Since ψ is not algebraically stable there exists a curve C blown down by ψ and such that ψ q (C) is a point of indeterminacy p of ψ for some integer q. The maps ψ and φ commute, so an iterate φ k of φ fixes the irreducible components of Ind(ψ). Let us blow up p via π. On the one hand π -1 • φ k • π is an automorphism because p is fixed by φ k and on the other hand

N(π -1 • ψ • π) = j.
One can thus conclude by induction that there exist a surface Y and a birational map

ζ : Y S such that ζ -1 • φ • ζ ∈ Aut(Y ) 0 for some integer and ζ -1 • ψ • ζ is algebraically stable.
Proposition 2.12 ([D 06a]). -Let φ, ψ be two birational self maps of a surface S. Assume that φ and ψ are both virtually isotopic to the identity. Assume that φ and ψ commute.

Then there exist a surface Z and a birational map π : Z S such that

π -1 • φ • π and π -1 • ψ • π belong to Aut(Z); π -1 • φ k • π and π -1 • ψ k • π belong to Aut(Z) 0 for some integer k.
Proof. -By assumption there exist a surface S, a birational map η : S S and an integer n such that η -1 • φ • η belongs to Aut(S) and η -1 • φ n • η belongs to Aut(S) 0 . Let us now work on S; to simplify denote by φ the automorphism η -1 • φ n • η and by ψ the birational map

η -1 • ψ • η.
According to Lemma 2.11 there exist a surface Y , a birational map υ : Y S and an integer such that

ζ -1 • φ • ζ belongs to Aut(Y ) 0 and ζ -1 • ψ • ζ is algebraically stable. Set φ = ζ -1 • φ i • ζ and ψ = ζ -1 • ψ • ζ.
To get an automorphism from ψ let us blow down curves in Exc(ψ -1 ). But curves blown down by ψ -1 are of self-intersection < 0 and φ fixes such curves since φ is isotopic to the identity. We conclude by using the fact that Card Exc(ψ -1 ) is finite.

On the hyperbolicity of graphs associated to the Cremona group

To reinforce the analogy between the mapping class group and the plane Cremona group Lonjou looked for a graph analogous to the curve graph and such that the Cremona group acts on it trivially in [START_REF] Lonjou | Sur l'hyperbolicité de graphes associés au groupe de Cremona[END_REF].

A candidate is the graph introduce by Wright (Chapter 4 §4.2.2 and [START_REF] Wright | Two-dimensional Cremona groups acting on simplicial complexes[END_REF]).

As we have recalled in Chapter 4 §4.2.2 the complex C is a simplicial complex of dimension 2 and 1-connected on which Bir(P 2 C ) acts. Since Lonjou is interested in the Gromov hyperbolicity property, she is only interested in the 1-skeleton of C. She proved that the diameter of this non-locally finite graph is infinite ([Lon19b, Corollary 2.7]). She then focuses on the following question "Is this graph Gromov hyperbolic ?" (2) The answer is no: Theorem 2.13 ( [START_REF] Lonjou | Sur l'hyperbolicité de graphes associés au groupe de Cremona[END_REF]). -The Wright graph is not Gromov hyperbolic.

The first point of the proof is to note that the Wright graph is quasi-isometric to a graph related to the system of generators of Bir(P 2 C ) given by PGL(3, C) and the Jonquières maps. It is an analogue of the Cayley graph in the case of a finitely generated group. The vertices of this graph are the elements of Bir(P 2 C ) modulo pre-composition by an automorphism of P 2 C . An edge connects two vertices if there exists a Jonquières map that permutes the two vertices. The distance between two vertices φ, ψ in Bir(P 2 C ) is the minimal number of Jonquières maps needed to decompose ψ -1 • φ (in [START_REF] Blanc | Length in the Cremona group[END_REF] Blanc and Furter called this integer the translation length of ψ -1 • φ. They gave an algorithm to compute this length. They also got that the diameter of the Wright graph is infinite).

The second point is to prove that this graph contains a subgraph quasi-isometric to Z 2 (see [Lon19b, Theorem 2.12]). She took two Halphen twists that commute. They generate a

(2) Minosyan and Osin note that if the answer to this question is yes, the results of [START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF] allow to give a new proof of the non-simplicity of Bir(P 2 C ) (see [START_REF] Minasyan | Acylindrical hyperbolicity of groups acting on trees[END_REF][START_REF] Minasyan | Correction to: Acylindrical hyperbolicity of groups acting on trees[END_REF]).

subgroup isomorphic to Z 2 . Using some results of [START_REF] Blanc | Length in the Cremona group[END_REF] she established that the action of this subgroup on one of the vertices of the graph induces the desired graph (3) . Then Lonjou constructed two graphs associated to a Voronï tessellation of the Cremona group introduced in [START_REF] Lonjou | Pavage de Voronoï associé au groupe de Cremona[END_REF]; she proved that one of these graphs is quasi-isometric to the Wright graph; the second one is Gromov hyperbolic.

(3) The Cayley graph of the modular group of a compact surface of genus g ≥ 2 is not Gromov hyperbolic; indeed, this group has subgroups isomorphic to Z 2 (for instance generated by two Dehn twists along two disjoint closed curves).

CHAPTER 3 ALGEBRAIC SUBGROUPS OF THE CREMONA GROUP

The first section of this chapter deals with the algebraic structure of the n-dimensional Cremona group, the fact that it is not an algebraic group of infinite dimension if n ≥ 2, the obstruction to this, which is of a topological nature. By contrast, the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group is recalled. More precisely in [START_REF] Browder | Mathematical developments arising from Hilbert problems[END_REF] Shafarevich asked "Can one introduce a universal structure of an infinite dimensional group in the group of all automorphisms (resp. all birational automorphisms) of arbitrary algebraic variety ?" We will see that the answer to this question is no ( [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). For any algebraic variety V defined over C there is a natural notion of families of elements of Bir(P n C ) parameterized by V . These are maps V (C) → Bir(P n C ) compatible with the structures of algebraic varieties. Note that Bir(P 1 C ) PGL(2, C) and families V Bir(P 1 C ) correspond to morphisms of algebraic varieties. If n ≥ 2 the set Bir d (P n C ) of all birational maps of P n C of degree d has the structure of an algebraic variety defined over C such that the families V → Bir d (P n C ) correspond to morphisms of algebraic varieties ( [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). So Bir(P n C ) decomposes into a disjoint infinite union of algebraic varieties, having unbounded dimension. Blanc and Furter established the following statement:

Theorem 3.1 ([BF13]). -Let n ≥ 2.
There is no structure of algebraic variety of infinite dimension on Bir(P n C ) such that families V → Bir(P n C ) would correspond to morphisms of algebraic varieties.

The lack of structure come from the degeneration of maps of degree d into maps of smaller degree. A family of birational self maps of P 2 C of degree d which depends on a parameter t may degenerate for certain values of t onto a non-reduced expression of the type P id = P(z 0 : z 1 : z 2 )

where P denotes an homogeneous polynomial of degree d -1. Consider for instance the family φ a,b,c : (z 0 :

z 1 : z 2 ) z 0 (az 2 2 + cz 0 z 2 + bz 2 0 ) : z 1 (az 2 2 + (b + c)z 0 z 2 + (a + b)z 2 0 ) : z 2 (az 2 2 + cz 0 z 2 + bz 2 0 )
parameterized by the nodal plane cubic a 3 + b 3 = abc. The family (φ a,b,c ) is globally defined by formulas of degree 3, but each element φ a,b,c has degree ≤ 2 and there is no global parameterization by homogeneous formulas of degree 2. In fact the obstruction to a positive answer to Shafarevich question comes only from the topology:

Theorem 3.2 ([BF13]
). -There is no C-algebraic variety of infinite dimension that is homeomorphic to Bir(P n C ). In 2010 in the question session of the workshop "Subgroups of the Cremona group" in Edinburgh, Serre asked the following question "Is it possible to introduce such topology on Bir(P 2 C ) that is compatible with PGL(3, C) and PGL(2, C) × PGL(2, C) ?" We will see that Blanc and Furter gave a positive answer to this question:

Theorem 3.3 ([BF13]
). -Let n ≥ 1 be an integer. There is a natural topology on Bir(P n C ), called the Euclidean topology, such that:

Bir(P n
C ), endowed with the Euclidean topology, is a Hausdorff topological group, the restriction of the Euclidean topology to algebraic subgroups in particular to PGL(n+ 1, C) and PGL(2, C) n is the classical Euclidean topology.

In the literature an algebraic subgroup G of Bir(V ) corresponds to taking an algebraic group G and a morphism G → Bir(V ) that is a group morphism and whose schematic kernel is trivial. We will see that in the case of V = P n C one can give a more intrinsic definition (Corollary 3.11) which corresponds to taking closed subgroups of Bir(P n C ) of bounded degree and that these two definitions agree (Lemma 3.12).

An element φ ∈ Bir(P n C ) is algebraic if it is contained in an algebraic subgroup G of Bir(P n C ). It is equivalent to say that the sequence (deg φ n ) n∈N is bounded. According to [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF] the group G is thus an affine algebraic group. As a consequence φ decomposes as φ = φ s • φ u where φ s is a semi-simple element of G and φ u an unipotent element of G. This decomposition does not depend on G (see [START_REF] Popov | Tori in the Cremona groups[END_REF]). In particular there is a natural notion of semi-simple and unipotent elements of Bir(P n C ). As we will see G could even by chosen to be the abelian algebraic subgroup φ i | i ∈ Z of Bir(P n C ). In all linear algebraic groups the set of unipotent elements is closed; Popov asked if it is the case in the context of the Cremona group. A natural and related question is the following one: is the set Bir(P n C ) alg of algebraic elements of Bir(P n C ) closed ? The second section deals with the answers to these questions (Theorem 3.31).

In the third section the classification of maximal algebraic subgroups of the plane Cremona group is given.

In the fourth section we give a sketch of the proof of the fact that Bir(P n C ) is topologically simple when endowed with the Zariski topology, i.e. it does not contain any non-trivial closed normal strict subgroup. The main ingredients of the proof are some clever deformation arguments.

The fifth section is devoted to a modern proof of the regularization theorem of Weil which says that for every rational action ρ of an algebraic group G on a variety X there exist a variety Y with a regular action µ of G and a dominant rational map φ : X Y with the following properties: for any (g, p) ∈ G × X such that ρ is defined in (g, p); φ is defined in p and ρ(g, p); µ is defined in (g, φ(p))

we have φ(ρ(g, p)) = µ(g, φ(p)).

Topologies and algebraic subgroups of

→ ψ • ϕ, ψ → ϕ • ψ, ψ → ψ -1
are homeomorphisms of Bir(P n C ) with respect to the Zariski topology. Indeed let V be an irreducible algebraic variety. If f , g : V × P n C → V × P n C are two Vbirational maps inducing morphisms V → Bir(P n C ), then f • g and f -1 are again V -birational maps that induce morphisms V → Bir(P n C ).

Let ). Take a positive integer d. Let W d be the set of equivalence classes of non-zero (n + 1)uples (φ 0 , φ 1 , . . . , φ n ) of homogeneous polynomials φ i ∈ C[z 0 , z 1 , . . . , z n ] of degree d where (φ 0 , φ 1 , . . . , φ n ) is equivalent to (λφ 0 , λφ 1 , . . . , λφ n ) for any λ ∈ C * . We denote by (φ 0 : φ 1 : . . . : φ n ) the equivalence class of (φ 0 , φ 1 , . . . , φ n ). Let H d ⊆ W d be the set of elements φ = (φ 0 : φ 1 : . . . : φ n ) ∈ W d such that the rational map ψ φ : P n C P n C given by (z 0 : z 1 : . . . :

z n ) φ 0 (z 0 , z 1 , . . . , z n ) : φ 1 (z 0 , z 1 , . . . , z n ) : . . . : φ n (z 0 , z 1 , . . . ,
(ϕ, φ) ∈ W d n-1 ×W d | ϕ • φ = P id for some P ∈ C[z 0 , z 1 , . . . , z n ] d n .
If P is nonzero, then the rational maps ψ φ and ψ ϕ are birational and inverses of each other.

If P is zero, then ψ φ contracts the entire set P n C onto a strict subvariety included in the set ϕ 1 = ϕ 2 = . . . = ϕ n = 0 .

In particular for any pair (ϕ, φ) of Y the rational map ψ φ is birational if and only if its Jacobian is nonzero.

As a consequence any element φ ∈ H d corresponds to at least one pair (ϕ, φ) in Y (indeed according to [START_REF] Bass | The Jacobian conjecture: reduction of degree and formal expansion of the inverse[END_REF] the inverse of a birational self map of P n C of degree d has degree ≤ d n-1 ).

The description of Y shows that it is closed in Consider the H d -rational map φ defined by

W d n-1 × W d .
f : H d × P n C H d × P n C (ϕ, z) (ϕ, ϕ(z)). Set J = det ∂ϕ i ∂x j 0≤i, j≤n . Let V ⊂ H d × P n
C be the open set where J is not zero.

Claim 3.6 ([BF13]

). -The restriction f |V of f to V is an open immersion.

Hence π d is a morphism and it follows from the construction of H d that the image of π d is Bir ≤d (P n C ). Let φ be an element in Bir(P n C ) ≤d . It corresponds to a birational self map ψ φ of P n C given by ψ φ : (z 0 : z 1 : . . . : z n ) (φ 0 (z 0 , z 1 , . . . , z n ) : φ 1 (z 0 , z 1 , . . . , z n ) : . . . : φ n (z 0 , z 1 , . . . , z n ))

for some homogeneous polynomials of degree k ≤ d having no common divisor. Then

(π d ) -1 (ψ φ ) = (ϕ 0 : ϕ 1 : . . . : ϕ n ) ∈ W d | ϕ i φ j = ϕ j φ i ∀ 1 ≤ i < j ≤ n ⊂ H d .
This ). -Let V be an irreducible algebraic variety, and let υ : V → Bir(P n C ) be a morphism. There exists an open affine covering (V i ) i∈I of V such that for each i there exist an integer d i and a morphism υ i :

V i → H d i such that υ |V i = π d i • υ i .
Proof. -Consider a morphism τ : V → Bir(P n C ) given by a V -birational map φ :

V × P n C V × P n C
which restricts to an open immersion on an open set U. Take a point p 0 in V . Let V 0 ⊂ V be an open affine set containing p 0 . Take an element w 0 = (p 0 , y) of U. Let us fix homogeneous coordinates (z 0 : z 1 : . . . : z n ) on P n C such that y = (1 : 0 : 0 : . . . : 0), φ(w 0 ) does not belong to the plane z 0 = 0. Let us denote by A n C ⊂ P n C the affine set where z 0 = 1;

x 1 = z 1 z 0 x 2 = z 2 z 0 . . . x n = z n z 0
are natural affine coordinates of A n C . The map φ restricts to a rational map of V 0 ×P n C defined at w 0 . Its composition with the projection on the i-th coordinate is a rational function on V 0 × A n C defined at w 0 . Hence φ |V 0 ×A n C can be written in a neighborhood of w 0 as

(v, x 1 , x 2 , . . . , x n ) → R 1 Q 1 , R 2 Q 2 , . . . , R n Q n for some R i , Q i in C[V ][x 1 , x 2 , . . . ,
x n ] such that none of the Q i vanish at w 0 . As a result φ is given in a neighborhood of w 0 by v, (z 0 : z 1 : . . . : z n ) → (P 0 : P 1 : . . . : P n )

where the

P i ∈ C[V 0 ][z 0 , z 1 , . . . , z n
] are homogeneous polynomials of the same degree d 0 such that not all vanish at w 0 . Denote by U 0 the set of points of (V × P n C ) ∩ U where at least one of the P i does not vanish; U 0 is an open subset of V × P n C . Its projection pr 1 (U 0 ) on V is an open subset of V 0 containing p 0 . There thus exists an affine open subset A 0 ⊆ pr 1 (U 0 ) containing p 0 . The n-uple (P 0 , P 1 , . . . , P n ) yields to a morphism υ 0 :

A 0 → H d . By construction υ | A 0 = π d • υ 0 . If
V i → H d i with υ |V i = π d i • υ i . As π -1 d i (S) is closed and υ -1 (S) ∩ V i = υ -1 i (π -1 d i (F)) one gets that υ -1 (S) ∩ V i is closed in V i for any i. As a result υ -1 (S) is closed.
We will now prove the second assertion. According to the first assertion it suffices to prove that

π -1 Bir ≤d (P n C ) = π -1 (π d (H d ))
is closed in H for any . This follows from Lemma 3.5.

Finally let us prove the third assertion. The surjectivity follows from the construction of H d and π d (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). Since π d is a morphism, π d is continuous. Let S ⊆ H d be a closed subset. According to Lemma 3.5 the set π -1 (π d (S)) is closed in H for any . The first assertion allows to conclude.

The first and third assertions of Corollary 3.8 imply: Proposition 3.9 ( [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). -The Zariski topology of Bir(P n C ) is the inductive limit topology given by the Zariski topologies of Bir ≤d (P

G × G → G, (ϕ, ψ) G → G, ϕ → ϕ -1
give rise to morphisms of algebraic varieties K × K → K and K → K.

This gives G a unique structure of algebraic group.

Corollary 3.11 ([BF13]

). -Let G be a subgroup of Bir(P n C ). Assume that G is closed for the Zariski topology, of bounded degree.

Then there exist an algebraic group K together with a morphism K → Bir(P n C ) inducing a homeomorphism π : K → G such that: π is a group homomorphism and for any irreducible algebraic variety V the morphisms V → Bir(P n C ) having their image in G correspond, via π, to the morphisms of algebraic varieties V → K.

Proof This allows to reduced to the connected case; Proposition 3.10 allows to conclude.

Lemma 3.12 ([BF13]

). -Let A be an algebraic group and ρ : A → Bir(P n C ) be a morphism that is also a group homomorphism.

Then the image G of A is a closed subgroup of Bir(P n C ) of bounded degree. If π : K → G is the homeomorphism constructed in Corollary 3.11, there exists a unique morphism of algebraic groups ρ : A → K such that ρ = π • ρ.

Proof. -Lemma 3.7 asserts that G = ρ(A) has bounded degree. The closure G of G is a subgroup of Bir(P n C ); indeed inversion being a homeomorphism G

-1 = G -1 = G. Similarly translation by g ∈ G is a homeomorphism thus gK = gK = K, that is GG ⊂ G. In turn, if g ∈ G, then Gg ⊂ G, so Gg = Gg ⊂ G.
As a result G is a subgroup of Bir(P n C ). According to Corollary 3.11 there exist a canonical homeomorphism K → G where K is an algebraic group and a lift ρ : A → H of the morphism ρ : A → Bir(P n C ) whose image is contained in G. As ρ is a group homomorphism ρ is a morphism of algebraic groups hence its image is closed, so im ρ = K. Therefore, G = G.

Proposition 3.13 ([BF13]

). -Any algebraic subgroup of Bir(P n C ) is affine.

Sketch of the proof. -Let G be an algebraic subgroup of Bir(P n C ). One can show that G is linear, and this reduces to the connected case. By the regularization theorem of Weil (see §3.5) the group G acts by automorphisms on some (smooth) rational variety V . Assume that α V : V → A(V ) is the Albanese morphism. According to the Nishi-Matsumura theorem the induced action of G on A(V ) factors through a morphism A(G) → A(V ) with finite kernel (see for instance [START_REF] Brion | Some basic results on actions of nonaffine algebraic groups[END_REF]). But V is rational hence A(V ) is trivial and so does A(G). The structure theorem of Chevalley asserts that G is affine (see for instance [START_REF] Rosenlicht | Some basic theorems on algebraic groups[END_REF]).

Let us finish by some examples:

The Cremona group in one variable Bir(P 1 C ) coincides with the group of linear projective transformations PGL(2, C); it is an algebraic group of dimension 3. In dimension 2 the Cremona group contains the two following algebraic subgroups:

• ). -Let us recall that W d is a projective space and H d is locally closed in W d for the Zariski topology (Lemma 3.5). Let us put the Euclidean topology on W d : the distance between (p 0 : p 1 : . . . : p n ) and (q 0 : q 1 : . . . :

q n ) is (see [Wey39]) ∑ i< j |p i q j -p j q i | 2 ∑ i |p i | 2 ∑ i |q i | 2
We then put the induced topology on H d . The behavior of the Zariski topology on Bir(P n C ) leads to:

Definition. -
. Recall that if f : X → Y is a quotient map between topological spaces, A is a subspace of X, A is open and A = f -1 ( f (A)), then the induced map A → f (A) is a quotient map ([Bou98, Chapter I, §3.6]). Set H d,d = (π d ) -1 (Bir d (P n C )).
As (π d ) -1 (Bir

d (ϕ i ) is constant equal to some m ≤ d. Assume m = d, then (π d ) -1 (π d (ϕ i )) = {ϕ i } for each i. As a result each ϕ i belongs to F, so ϕ belongs to F ⊂ F as wanted. Suppose m < d. Set k = d -m ≥ 1.
For any i there exists a non-zero homogeneous polynomial a i ∈ C[z 0 , z 1 , . . . , z n ] of degree k such that

ϕ i = a i f i,0 : a i f i,1 : . . . : a i f i,n
and ( f i,0 : f i,1 : . . . : f i,n ) ∈ W m corresponds to a birational map of degree m < d. Each a i is defined up to a constant and P(C[z 0 , z 1 , . . . , z n ]) is compact, so, taking a subsequence if needed, we can suppose that (a i ) i∈N converges to a non-zero homogeneous polynomial a ∈ C[z 0 , z 1 , . . . , z n ] of degree k.

Taking a subsequence if needed we can assume that {( f i,0 : f i,1 : . . . : f i,n )} i∈N converges to an element ( f 0 : f 1 : . . . : f n ) of the projective space W m . Since (ϕ i ) i∈N converges to ϕ we get that ϕ = (a f 0 : a f 1 : . . .

: a f n ) in H d .
As ϕ i belongs to F = (π d ) -1 (π d (F)) for any i there exists

ϕ i in F such that π d (ϕ i ) = π d (ϕ i ). Consequently ϕ i = b i f i,0 : b i f i,1 : . . . : b i f i,n
for some non-zero homogeneous polynomial b i ∈ C[z 0 , z 1 , . . . , z n ] of degree k. As before we can assume that (b i ) i∈N converges to a non-zero homogeneous polynomial b ∈ C[z 0 , z 1 , . . . , z n ] of degree k. The sequence (ϕ i ) i∈N converges to (b f 0 : b f 1 : . . . : b f n ) and F is closed, thus (b f 0 : b f 1 : . . . : b f n ) belongs to F. This implies that ϕ = (a f 0 : a f 1 : . . . : a f n ) belongs to F.

We can thus state:

Lemma 3.16 ([BF13]). -Let d ≥ 1
ι d : H d → H d+1 , ( f 0 : f 1 : . . . : f n ) → (z 0 f 0 : z 0 f 1 : . . . : z 0 f n ).
It is a morphism of algebraic varieties that is a closed immersion. As a result it is continuous and closed with respect to the Euclidean topology. The diagram

H d ι d G G π d H d+1 π d+1 Bir ≤d (P n C ) ι d G G Bir ≤d+1 (P n C ) commutes.
The continuity of ι d implies the continuity of ι d : let U be an open subset of Bir ≤d+1 (P n C ); the equality

(π d ) -1 ((ι d ) -1 (U)) = (π d+1 • ι d ) -1 (U) shows that (π d ) -1 ((ι d ) -1 (U)) is open in H d , that is (ι d ) -1 (U)
I d : Bir ≤d (P n C ) → Bir ≤d n-1 (P n C ), φ → φ -1 is continuous. Proof. -As in Lemma 3.5 we consider the set Y ⊂ W d n-1 ×W d defined by Y = (ϕ, φ) ∈ W d n-1 ×W d | ϕ • φ = P id for some P ∈ C[z 0 , z 1 , . . . , z n ] d . Let U ⊂ W d (resp. U ⊂ W d n-1 )
L = Y ∩ (W d n-1 × U) = Y ∩ (U × U) is locally closed in the algebraic variety W d n-1 ×W d .
The projection on the first factor is a morphism η 1 : L → H d n-1 which is not surjective in general. The projection on the second factor induces a surjective morphism η 2 : L → H d . By construction the diagram

H d π d L ⊂ W d n-1 ×W d η 2 o o η 1 G G H d n-1 π d-1 Bir ≤d (P n C ) I d G G Bir ≤d n-1 (P n C ) commutes.
Let us prove that η 2 is a closed map for the Euclidean topology. The set W d n-1 is compact, so the second projection

W d n-1 ×W d → W d is a closed map. Its restriction η 2 : Y → W d to the closed subset Y of W d n-1 × W d is a closed map. Since L = (η 2 ) -1 (H d ), we get that η 2 is a closed map (1) .
As the diagram is commutative for any F ⊂ Bir ≤d n-1 (P n C ) we have

η 2 (π d n-1 • η 1 ) -1 (F) = (I d • π d ) -1 (F);
furthermore this set corresponds to elements (φ 0 : φ 1 : . . . : φ n ) ∈ W d such that the rational map ψ φ is the inverse of an element of F. Assume that F is closed in Bir ≤d n-1 (P n C ). The maps η 1 and π d n-1 are continuous for the Euclidean topology hence (π

d n-1 • η 1 ) -1 (F) is closed in L. Lemma 3.16 asserts that π -1 d (I -1 d (F)) = η 2 (π d n-1 • η 1 ) -1 (F) is closed in H d and I -1 d (F) is closed in Bir ≤d (P n C ).
Let us introduce the map I defined by

I : Bir(P n C ) → Bir(P n C ), φ → φ -1 .
The degree of the inverse of a birational self map of P n C of degree d has degree at most d n-1 .

Consequently for any d ≥ 1 the map I restricts to an injective map

I d : Bir ≤d (P n C ) → Bir ≤d n-1 (P n C ).
According to Lemma 3.18 the map I d is continuous. The definition of the topology of Bir(P n C )

implies that I is continuous. Since I = I -1 one has:

Corollary 3.19 ([BF13]
). -The map

I : Bir(P n C ) → Bir(P n C ), φ → φ -1
is a homeomorphism.

Let us now look at the composition of two birational maps.

Lemma 3.20 ([BF13]

). -For any d, k ≥ 1 the map

χ d,k : Bir ≤d (P n C ) × Bir ≤k (P n C ) → Bir ≤dk (P n C ), (φ, ψ) → φ • ψ is continuous.
(1) Let us recall that if ϕ : A → B is a continuous closed map between topological spaces and C is any subset of B, then ϕ induces a continuous closed map ϕ -1 (C) → C.

Proof. -Let us consider the map χ d,k : H d × H k → H dk given by (φ 0 : φ 1 : . . . : φ n ), (ψ 0 : ψ 1 : . . . :

ψ n ) → φ n (ψ 0 , ψ 1 , . . . , ψ n )) : . . . : φ n (ψ 0 , ψ 1 , . . . , ψ n )).
The diagram

H d × H k π d ×π k χ d,k G G H dk π dk Bir ≤d (P n C ) × Bir ≤k (P n C ) I d G G Bir ≤dk (P n C ) commutes.
The map χ d,k is a morphism of algebraic varieties, so is continuous for the Euclidean topology. Therefore, if F is a closed subset of Bir ≤dk (P n C ), then

(π dk • χ d,k ) -1 (F) is closed in H d × H k . But the diagram is commutative, so (π d • χ d,k )(F) = (π d × π k ) -1 (χ d,k ) -1 (F) .
The product of two proper maps is proper ([Bou98, Chapter 1, §10.1]); as a consequence π d × π k is proper and hence closed. This implies that

π d × π k is a quotient map. Hence (χ d,k ) -1 (F) is closed
(ϕ i ) i∈N is a sequence of elements of K with deg ϕ i+1 > deg ϕ i for each i. Let us consider K = ϕ i | i ∈ N .
On the one hand it is a closed subset of the compact set K; hence it is compact. On the other hand the intersection of any subset of K with Bir ≤d (P n C ) is closed, so K is an infinite set endowed with the discrete topology; in particular it cannot be compact: contradiction.

f m,k : (z 1 , z 2 , . . . , z n ) z 1 + 1 k z m 2 , z 2 , . . . , z n .
Fixing m we note that the sequence ( f m,k ) k≥1 converges to the identity; in particular f m,k belongs to U when k is large enough.

Lemma 3.28. -For n ≥ 2 the topological space Bir(P n C ) is not metrisable.

Proof. -Consider the inclusion

C[z 2 ] → Aut(C n ) ⊂ Bir(P n C ) P → (z 1 , z 2 , . . . , z n ) (z 1 + P(z 2 ), z 2 , z 3 , . . . , z n )
Observe that C[z 2 ] is closed in Bir(P n C ) and that for any d the induced topology on C[z 2 ] ≤d is the topology as a vector space (or as an algebraic group). The induced topology on C[z 2 ] is thus the inductive limit topology given by

C[z 2 ] ≤1 ⊂ C[z 2 ] ≤2 ⊂ . . .
For any sequence = ( n ) n∈N of positive integers the set ) is bounded (see [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF] for more details or Chapter 4, §4.3.3). The fact that Bir(P n C ) is not compactly generated follows from Lemma 3.26.

U = d ∑ i=0 a i X i | |a i | < 1 i is open in C[z 2 ].
Remark 3.30. -Theorem 3.1 holds for any field, Theorem 3.2 holds for any algebraically closed field, and Theorem 3.3 holds for (locally compact) local field.

Algebraic elements of the Cremona group

The goal of this section is the study of algebraic elements; in particular we will show that the set of all these elements is a countable union of closed subsets but it is not closed.

In this section the considered topology is the Zariski topology.

). It contains φ k | k ∈ Z ; thus it contains Ω. As a result φ k (Ω) = Ω for any k ∈ Z. Set M = ψ ∈ Bir(P n C ) | ψ(Ω) ⊂ Ω = ω∈Ω Ωω -1 .
As M is closed and contains

φ k | k ∈ Z , the set M contains Ω. Therefore, M is closed under composition. Similarly the set ψ -1 | ψ ∈ Ω is closed in Bir(P n C ) and contains φ k | k ∈ Z . The set Ω is then a subgroup of Bir(P n C ). Let us now prove that Ω is abelian. The centralizer Cent(ϕ) = ψ ∈ Bir(P n C ) | ψ • ϕ = ϕ • ψ of an element ϕ of Bir(P n C
) is the preimage of the identity by the continuous map

Bir(P n C ) → Bir(P n C ) ψ → ψ • ϕ • ψ -1 • ϕ -1 . Since a point of Bir(P n C ) is closed (Lemma 3.5), Cent(φ) is closed. The closed subgroup Cent(φ) of Bir(P n C ) contains φ j | j ∈ Z hence it contains Ω. Conse- quently each element of Ω commutes with φ. The set ψ ∈ Bir(P n C ) | ψ • ω = ω • ψ ∀ ω ∈ Ω = ω∈Ω Cent(ω)
is closed and contains φ j | j ∈ Z , so contains Ω. Therefore, Ω is abelian.

Proposition 3.33 ([Bla16]). -Let φ be an element of Bir(P n C ). If the sequence (deg φ k ) k∈N is unbounded, then φ is not contained in any algebraic sub- group of Bir(P n C ). If the sequence (deg φ k ) k∈N is bounded, then φ j | j ∈ Z is an abelian algebraic sub- group of Bir(P n C ).
A direct consequence is the following result: [START_REF] Blanc | Algebraic elements of the Cremona groups[END_REF]). -Let φ be a birational self map of P n C . The following assertions are equivalent: the map φ is algebraic; the sequence (deg φ k ) k∈N is bounded, i.e. φ is elliptic.

Corollary 3.34 ([
Proof of Proposition 3.33. -The first assertion follows from Lemma 3.12.

Let us now focus on the second assertion. Assume that the sequence (deg φ k ) k∈N is bounded. According to [START_REF] Bass | The Jacobian conjecture: reduction of degree and formal expansion of the inverse[END_REF] one has for any k

deg φ -k ≤ (deg φ k ) n-1 .
As a consequence the set φ j | j ∈ Z is contained in Bir(P n C ) ≤d for some d, and so does the closure Ω of φ j | j ∈ Z}. Lemma 3.32 allows to conclude.

Proposition 3.35. -For any k, d ∈ N set Bir(P n C ) k,d = φ ∈ Bir(P n C ) | deg φ k ≤ d and Bir(P n C ) ∞,d = φ ∈ Bir(P n C ) | deg φ k ≤ d ∀ k ∈ N Then the set Bir(P n C ) k,d is closed in Bir(P n C ); the set Bir(P n C ) ∞,d = i∈N Bir(P n C ) i,
(P n C ) → Bir(P n C ), ϕ → ϕ k is continuous (Remark 3.4); the set Bir(P n C ) k,d is thus closed in Bir(P n C ).
The first assertion clearly implies the second one. The third assertion follows from Corollary 3.34.

Let us now deal with the first assertion of Theorem 3.31. Assume n ≥ 2. Consider the morphism ρ :

A 1 C → Bir(P n C ) given by a → (z 0 : z 1 : . . . : z n ) (z 0 z 1 : z 1 (z 1 + z 0 ) : z 2 (z 1 + az 0 ) : z 3 z 1 : z 4 z 1 : . . . : z n z 1 .
It is clearly injective. Let ρ : P 1 C → W 2 be the closed embedding given by

(α : β) → (αz 0 z 1 : αz 1 (z 1 + z 0 ) : z 2 (z 1 + az 0 ) : αz 3 z 1 : αz 4 z 1 : . . . : αz n z 1 ).
Note that ρ((0 : 1)) does not belong to H 2 . However for any t ∈ A 1 C one has pr 2 ( ρ((1 : t)) = ρ(t). The restriction to A 1 C ; thus it yields a closed embedding A 1 C → H 2 . According to Corollary 3.8 the restriction of π 2 to ρ(P 1 C {(0 : 1)}) is an homeomorphism. Proposition 3.36. -For t ∈ C the following conditions are equivalent:

-ρ(t) is algebraic, -ρ(t) is unipotent, -ρ(t) is conjugate to ρ(0) : (z 1 , z 2 , . . . , z n ) → (z 1 + 1, z 2 , . . . , z n ),
t belongs to the subgroup of (C, +) generated by 1. The pull-back by ρ of the set of algebraic elements is not closed.

Proof. -A direct computation yields to

ρ(a) k : (z 1 , z 2 , . . . , z n ) → z 1 + k, z 2 (z 1 + a)(z 1 + a + 1) . . . (z 1 + a + k -1) z 1 (z 1 + 1) . . . (z 1 + m -1) , z 3 , z 4 , . . . , z n
In particular the second coordinate of ρ(a

) k (z 1 , z 2 , . . . , z n ) is z 2 k-1 ∏ i=0 (z 1 + a + i) k-1 ∏ i=0 (z 1 + i)
If a does not belong to the subgroup of (C, +) generated by 1, then the degree growth of ρ(a) k is linear which implies that ρ(a) is not algebraic. If a belongs to the subgroup of (C, +) generated by 1, then

deg ρ(a) k ≤ |k| + 1 ∀ k ∈ N.
As a consequence ρ(a) is algebraic. Furthermore ρ(a) is conjugate to

ρ(0) : (z 1 , z 2 , . . . , z n ) → (z 1 + 1, z 2 , . . . , z n ) via (z 1 , z 2 , . . . , z n ) z 1 , z 2 z 1 (z 1 + 1) . . . (z 1 + a -1) , z 3 , z 4 , . . . , z n if a > 0 or via (z 1 , z 2 , . . . , z n ) z 1 , z 2 z 1 (z 1 -1) . . . (z 1 + a), z 3 , z 4 , . . . , z n if a < 0 (2)
. In particular ρ(a) is unipotent.

(2) Let us recall that a belongs to the subgroup of (C, +) generated by 1.

The second assertion follows from the first one and the fact that the subgroup of (C, +) generated by 1 is not closed.

Finally let us prove the second assertion of Theorem 3.31. Assume n ≥ 2. Consider the morphism ρ :

A 1 C × (A 1 C {0}) → Bir(P n C ) given by (a, ξ) → (z 0 : z 1 : . . . : z n ) (z 0 (z 1 + z 0 ) : ξz 1 (z 1 + z 0 ) : z 2 (z 1 + az 0 ) : z 3 (z 1 + z 0 ) : . . . : z n (z 1 + z 0 ) .
It is injective. Let ρ : P 2 C → W 2 be the closed embedding given by

(α : β : γ) αz 0 (z 1 + z 0 ) : γz 1 (z 1 + z 0 ) : z 2 (αz 1 + βz 0 ) : αz 3 (z 1 + z 0 ) : . . . : αz n (z 1 + z 0 ) .
Note that

(z 0 : z 1 : . . . : z n ) αz 0 (z 1 + z 0 ) : γz 1 (z 1 + z 0 ) : z 2 (αz 1 + βz 0 ) : αz 3 (z 1 + z 0 ) : . . . : αz n (z 1 + z 0 )
is a birational map if and only if αγ = 0. This yields a closed embedding

A 1 C × (A 1 C {0}) → H 2 , (a, ξ) → ρ((1 : a : ξ)).
Furthermore pr 2 ( ρ(1 : a : ξ)) = ρ(a, ξ). Proposition 3.33 says that the restriction of π 2 to the image is a homeomorphism.

Proposition 3.37. - For (a, ξ) ∈ A 1 C × (A 1 C {0}) the following conditions are equi- valent: -ρ(a, ξ) is algebraic, -ρ(a, ξ) is semi-simple, -ρ(a, ξ) is conjugate to ρ(1, ξ) : (z 1 , z 2 , . . . , z n ) → (ξz 1 , z 2 , z 3 , . . . , z n ),
there exists k ∈ Z such that a = ξ k . The pull-back by ρ of the set of algebraic elements is not closed.

Proof. - Note that ρ(a, ξ) k : (z 1 , z 2 , . . . , z n ) ξ k z 1 , z 2 (z 1 + a)(ξz 1 + a) . . . (ξ k z 1 + a) (z 1 + 1)(ξz 1 + 1) . . . (ξ k-1 z 1 + 1) , z 3 , z 4 , . . . , z n .
In particular the second coordinate of ρ(a, ξ) k is

z 2 k-1 ∏ i=0 (ξ i z 1 + a) k-1 ∏ i=0 (ξ i z 1 + 1)
.

If a does not belong to ξ ⊂ (C, •), then the degree growth of ρ(a, ξ) k is linear hence ρ(a, ξ) is not algebraic. If a belongs to ξ ⊂ (C, •), then a = ξ k for some k ∈ Z and for any j ∈ N

deg ρ(a, ξ) j ≤ |k| + 1, so ρ(a, ξ) is algebraic. Remark that ρ(a, ξ) is conjugate to ρ(1, ξ) via (z 1 , z 2 , . . . , z n ) z 1 , z 2 z 1 (z 1 + 1) . . . (z 1 + a -1) , z 3 , z 4 , . . . , z n if k > 0 and via (z 1 , z 2 , . . . , z n ) z 1 , z 2 z 1 (z 1 -1) . . . (z 1 + a), z 3 , z 4 , . . . , z n if k < 0.
The second assertion follows from the first one and the fact that

(a, ξ) ∈ A 1 C × (A 1 C {0}) | a = ξ k for some k ∈ Z is not closed. Remark 3.38.
-Note that all the results of this section hold for Bir(P n k ) whose k is an algebraically closed field of characteristic 0.

Classification of maximal algebraic subgroups of Bir(P 2

C ) In [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF] the author gives a complete classification of maximal algebraic subgroups of the plane Cremona group and provides algebraic varieties that parametrize the conjugacy classes. The algebraic subgroups of Bir(P n C ) have been studied for a long time. Enriques established in [START_REF] Enriques | Sui gruppi continui di transformazioni cremoniani nel piano[END_REF] the complete classification of maximal connected algebraic subgroups of Bir(P 2 C ): every such subgroup is the conjugate of the identity component of the automorphism group of a minimal rational surface. A modern proof was given in [START_REF] Umemura | On the maximal connected algebraic subgroups of the Cremona group[END_REF]. The case of Bir(P 3 C ) was treated by Enriques and Fano and more recently by Umemura ([Ume80, Ume82b, Ume82a]). Demazure has studied the smooth connected subgroups of Bir(P n C ) that contain a split torus of dimension n (see [START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF]). Only a few results are known for non-connected subgroups even in dimension 2. Nevertheless there are a lot of statements in the case of finite subgroups which are algebraic ones ([Wim96, BB00, dF04, BB04, Bea07, Isk05, DI09, Bla07b, Bla07a]) and we deal with in Chapter 6. But these results do not show which finite groups are maximal algebraic subgroups. As mentioned in [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF] there are some remaining open questions like the description of the algebraic varieties that parameterize conjugacy classes of finite subgroups G of Bir(P 2 C ).

Blanc gives an answer to this question for abelian finite subgroups G of Bir(P 2 C ) whose elements do not fix a curve of positive genus ([Bla09a]); finite cyclic subgroups of Bir(P 2 C ) (see [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]); maximal algebraic subgroups of Bir(P 2 C ) (see [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF]). Before specifying Blanc results let us recall some notions. If S is a projective smooth rational surface and G a subgroup of Aut(S) we say that

(G, S) is a pair. A birational map ϕ : X Y is G-equivariant if the inclusion ϕ • G • ϕ -1 ⊂ Aut(Y ) holds. The pair (G, S) is minimal if every birational G-equivariant morphism ϕ : S
S where S is a projective, smooth surface, is an isomorphism. A morphism π : S → P 1

C is a conic bundle if all generic fibers of π are isomorphic to P 1 C and if there exists a finite number of singular fibers which are the transverse union of two curves isomorphic to P 1 C .

3.3.1. del Pezzo surfaces and their automorphism groups. -A del Pezzo surface is a smooth projective surface S such that the anti-canonical divisor -K S is ample. Let us recall the classification of del Pezzo surfaces. The number d = K 2 S is called the degree of S. By Noether's formula 1 ≤ d ≤ 9. For d ≥ 3, the anticanonical linear system | -K S | maps S onto a non-singular surface of degree

d in P d C . If d = 9, then S P 2 C . If d = 8, then S P 1 C × P 1 C or S F 1 . For d ≤ 7 a del Pezzo surface S is isomorphic to the blow up of n = 9 -d points in P 2 C
in general position, that is no three of them are colinear, no six are on the same conic, if n = 8, then the points are not on a plane cubic which has one of them as its singular point.

There exist ([Dol12, Chapter 8]) a unique isomorphism class of del Pezzo surfaces of degree 5 (resp. 6, resp. 7, resp. 9), two isomorphism classes of del Pezzo surfaces of degree 8, and infinitely many isomorphism classes of del Pezzo surfaces of degree 1, (resp. 2, resp. 3, resp. 4).

We will see that automorphism groups of del Pezzo surfaces are algebraic subgroups of Bir(P 2 C ) and that they are finite if and only if the degree of the corresponding surface is ≤ 5. If S is a del Pezzo surface of degree 5, then Aut(S) = S 5 . Automorphism groups of del Pezzo surfaces of degree ≤ 4 are described in [DI09, §10]. In particular the authors got the following:

Theorem 3.39 ([DI09]
). -If the automorphism group of a del Pezzo surface is finite, then it has order at most 648.

Lemma 3.40 ([Ure]

). -If the automorphism group of a del Pezzo surface is finite, then it can be embedded into GL(8, C).

Proof. -Let S be a del Pezzo surface such that Aut(S) is finite. Then deg S ≤ 5 and S is isomorphic to Bl p 1 ,p 2 ,...,p r P 2 C where 4 ≤ r = 9deg S ≤ 8 and p 1 , p 2 , . . ., p r are general points of P 2 C . Denote by e 0 the pullback of the class of a line and by e p i the class of the exceptional line E p i corresponding to the point p i . The dimension of the Néron-Severi space NS(S) ⊗ R is r + 1 and e 0 , e p 1 , e p 2 , . . ., e p r is a basis of NS(S) ⊗ R. Note that the equality e p i • e p i = -1 implies that E p i is the only representative of e p i on S.

If ϕ ∈ Aut(S) acts as the identity on NS(S) ⊗ R, then ϕ preserves the exceptional lines E p i for 1 ≤ i ≤ r. Hence ϕ induces an automorphism of P 2 C that fixes p 1 , p 2 , . . ., p r . As r ≥ 4 and as the p i are in general position the induced automorphism of P 2

C is the identity. The action of Aut(S) on NS(S) ⊗ R is thus faithful and we get a faithful representation

Aut(S) → GL(r + 1, C).
Any element ϕ of Aut(S) fixes K S ; as a result the one-dimensional subspace R • K S of NS(S) ⊗ R is fixed. By projecting the orthogonal complement of K S in NS(S) ⊗ R we obtain a faithful representation of Aut(S) into GL(r, C).

A del Pezzo surface of degree 6 is isomorphic to the blow up of the complex projective plane in three general points, i.e. isomorphic to the surface

S 6 = (z 0 : z 1 : z 2 ), (a : b : c) ∈ P 2 C × P 2 C | az 0 = bz 1 = cz 2 .
The automorphism group of S 6 is isomorphic to (C * ) 2 S 3 × Z 2Z where S 3 acts by permuting the coordinates of the two factors simultaneously, Z 2Z exchanges the two factors and d ∈ (C * ) 2 acts as follows

d • (z 0 : z 1 : z 2 ), (a : b : c) = d(z 0 : z 1 : z 2 ) : d -1 (a : b : c) .

In other words Aut

(S 6 ) is conjugate to S 3 × Z 2Z D 2 ⊂ GL(2, Z) D 2 . Lemma 3.41 ([Ure]
). -The group Aut(S 6 ) can be embedded in GL(6, C).

Proof. -Consider the rational map

φ : P 2 C P 6 C , (z 0 : z 1 : z 2 ) (z 2 0 z 1 : z 2 0 z 2 : z 0 z 2 1 : z 2 1 z 2 : z 0 z 2 2 : z 1 z 2 2 : z 0 z 1 z 2 ).
The rational action of (S 3 × Z 2Z ) D 2 on φ(P 2 C ) extends to a regular action on P 6 C that preserves the affine space given by z 6 = 0. This yields an embedding of (S 3 × Z 2Z ) D 2 into GL(6, C).

3.3.2. Hirzebruch surfaces and their automorphism groups. -Let us introduce the Hirzebruch surfaces. Consider the surface F 1 obtained by blowing up (1 : 0 : 0) ∈ P 2 C ; it is a compactification of C 2 which has a natural fibration corresponding to the lines z 1 = constant. The divisor at infinity is the union of two rational curves which intersect in one point: one of them is the strict transform of the line at infinity in P 2 C , it is a fiber denoted by f 1 ; the other one, denoted by s 1 , is the exceptional divisor which is a section for the fibration. Furthermore f 2 1 = 0 and s 2 1 = -1. More generally for any n, F n is a compactification of C 2 with a rational fibration and such that the divisor at infinity is the union of two transversal rational curves: a fiber f n and a section s n of self-intersection -n. These surfaces are called Hirzebruch surfaces. One can go from F n to F n+1 as follows. Consider the surface F n . Set p = s n ∩ f n . Let p 1 be the blow up of p ∈ F n and let p 2 be the contraction of the strict transform

f n of f n . One goes from F n to F n+1 via p 2 • p -1
1 . We can also go from

F n+1 to F n via p 2 • p 1 -1
where p 1 is the blow-up of a point q such that q ∈ f n+1 , q ∈ s n+1 ; p 2 is the contraction of the strict transform f n+1 of f n+1 .

We will say that both p 2 • p -1 1 and p 2 • p 1 -1 are elementary transformations.

The n-th Hirzebruch surface

F n = P O P 1 C ⊕ O P 1 C (n) is isomorphic to the hypersurface ([x 0 , x 1 ], [y 0 , y 1 , y 2 ]) ∈ P 1 C × P 2 C | x n 0 y 1 -x n 1 y 2 = 0 of P 1 C × P 2 C . Their automorphism groups are Aut(P 2 C × P 1 C ) = (PGL(2, C) × PGL(2, C)) (z 0 , z 1 ) → (z 1 , z 0 ) , Aut(P 2 C ) = PGL(3, C) and Aut(F n ) = (z 0 , z 1 ) → az 0 + P(z 1 ) (γz 1 + δ) n , αz 1 + β γz 1 + δ α β γ δ ∈ PGL(2, C), a ∈ C * , P ∈ C[z 1 ], deg P ≤ n .
In other words as soon as n ≥ 2 the group Aut(

F n ) is isomorphic to C[z 0 , z 1 ] n GL(2, C) µ n where µ n ⊂ GL(2, C) is the subgroup of n-torsion elements in the center of GL(2, C). Lemma 3.42 ([Ure]). -If n ≥ 2 is even, then GL(2, C) µ n is isomorphic as an algebraic group to PGL(2, C) × C * . If n is odd, then GL(2, C) µ n is isomorphic as an algebraic group to PGL(2, C).
In particular all finite subgroups of Aut(F n ) can be embedded into PGL(2, C) × PGL(2, C) as soon as n ≥ 2.

3.3.3. Automorphism groups of exceptional conic bundles. -An exceptional conic bundle S is a conic bundle with singular fiber above 2n points in P 1

C and with two sections s 1 and s 2 of self-intersection -n, where n ≥ 2 (see [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF]).

Lemma 3.43 ([Bla09b]). -Let π : S → P 1 C be an exceptional conic bundle. Then Aut(S, π) is isomorphic to a subgroup of PGL(2, C) × PGL(2, C). 3.3.4. Z 2Z 2 -conic bundles. -A conic bundle π : S → P 1 C is a Z 2Z 2 -conic bundle if the group Aut S P 1 C is isomorphic to Z 2Z 2 ,
each of the three involutions of Aut S P 1

C

fixes pointwise an irreducible curve C such that π : C → P 1 C is a double covering that is ramified over a positive even number of points.

The automorphism group Aut(S, π) of a Z 2Z 2 -conic bundle is finite; its structure is given by the following exact sequence ( [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF])

1 -→ V -→ Aut(S, π) -→ H V -→ 1 where V Z 2Z
2 and H V is a finite subgroup of Aut(P 1 C ). Note that we also have the following property:

Lemma 3.44 ([Ure]). -Let G ⊂ Bir(P 2
C ) be an infinite torsion group. Assume that for any finitely generated subgroup Γ ⊂ G there exists a Z 2Z 2 -conic bundle S → P 1 C such that Γ is conjugate to a subgroup of Aut(S, π). Then any finitely generated subgroup of G is isomorphic to a subgroup of PGL(2, C) × PGL(2, C).

Blanc results. -First Blanc proved:

Theorem 3.45 [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF]). -Every algebraic subgroup of Bir(P 2 C ) is contained in a maximal algebraic subgroup of Bir(P 2 C ). The maximal algebraic subgroups of the plane Cremona group are the conjugate of the groups G = Aut(S, π) where S is a rational surface and π : S → Y is a morphism such that 1. Y is a point, G = Aut(S) and S is one of the following:

P 2 C , P 1 C × P 1 C ; a del Pezzo surface of degree 1, 4, 5 or 6; a del Pezzo surface of degree 3 (resp. 2) such that the pair (Aut(S), S) is minimal and such that the fixed points of the action of Aut(S) on S are lying on exceptional curves; 2. Y P 1

C and π is one of the following conic bundles: the fibration by lines of the Hirzebruch surface F n for n ≥ 2; an exceptional conic bundle with at least 4 singular fibers;

a Z 2Z 2 -conic bundle such that S is not a del Pezzo surface.

Moreover, in all these cases, the pair (G, S) is minimal and the fibration π : S → Y is a G-Mori fibration which is birationally superrigid . This means that two such groups G = Aut(S, π) and G = Aut(S , π ) are conjugate if and only if there exists an isomorphism S → S which sends fibers of π onto fibers of π .

Then Blanc described more precisely the structure of these minimal algebraic subgroups of Bir(P 2 C ). Furthermore he provides algebraic varieties that parameterize the conjugacy classes of these groups:

Theorem 3.46 ([Bla09b]
). -The maximal algebraic subgroups of Bir(P 2 C ) belong up to conjugacy to one of the eleven following families:

(1) Aut(P 2 C ) PGL(3, C); (2) Aut(P 1 C × P 1 C ) PGL(2, C) 2 Z 2Z ; (3) Aut(S) (C * ) 2 S 3 × Z 2Z
where S is the del Pezzo surface of degree 6;

(4) Aut(F n ) C n+1 GL(2, C) µ n where µ n is the n-th torsion of the center of GL(2, C) with n ≥ 2; (5) Aut(S, π) where (S, π) is an exceptional conic bundle with singular fibers over a set ∆ ⊂ P 1 C of 2n distinct points, n ≥ 2; the projection of Aut(S, π) onto PGL(2, C) gives an exact sequence

1 -→ C * Z 2Z -→ Aut(S, π) -→ H ∆ -→ 1
where H ∆ is the finite subgroup of PGL(2, C) formed by elements that preserve ∆; (6) Aut(S) S 5 where S is the del Pezzo surface of degree 5;

1 -→ Z 3Z -→ Aut(S) -→ H Γ -→ 1 where H Γ is the group of automorphisms of P 2
C that preserve Γ, H Γ contains a subgroup isomorphic to Z 3Z 2 ; the Clebsch cubic surface whose automorphism group is isomorphic to S 5 ; a cubic surface given by z 3 0 + z 0 (z 2 1 + z 2 2 + z 2 3 ) + λz 1 z 2 z 3 = 0 for some λ ∈ C, 9λ 3 = 8λ, 8λ 3 = -1 and whose automorphism group is isomorphic to S 4 ; (9) Aut(S) Z 2Z H S where S is a del Pezzo surface of degree 2 which is a double cover of a smooth quartic Q S ⊂ P 2 C such that H S = Aut(Q S ) acts without fixed point on the quartic without its bitangent points; (10) Aut(S) where S is a del Pezzo surface of degree 1, double cover of a quadratic cone Q, ramified along a curve Γ S of degree 6, complete intersection of Q with a cubic surface of P 3 C . We have the following exact sequence 1 -→ Z 2Z -→ Aut(S) -→ H S -→ 1 where H S denotes the automorphism group of Q preserving the curve Γ S ;

(11) Aut(S, π) where (S, π) is a Z 2Z 2 -conic bundle such that S is not a del Pezzo surface. The projection of Aut(S, π) onto PGL(2, C) gives the following exact sequence

1 -→ V -→ Aut(S, π) -→ H V -→ 1 where V Z 2Z 2
contains three involutions fixing an hyperelliptic curve ramified over points of p 1 , p 2 , p 3 ⊂ P 1 C and H V ⊂ Aut(P 1 C ) is the finite subgroup preserving the set p 1 , p 2 , p 3 .

The eleven families are disjoint and the conjugacy classes in any family are parameterized respectively by

(1) , (2), (3), (6) the point;

(4) there is only one conjugacy class for any integer n ≥ 2;

(5) for any integer n ≥ 2 the set of 2n points of P 1 C modulo the action of Aut(P 1 C ) = PGL(2, C); (7) the isomorphism classes of del Pezzo surfaces of degree 4; (8) the isomorphism classes of cubic surfaces given respectively by the isomorphism classes of elliptic curves; for the Clebsch surface there is only one isomorphism class;

by the classes of λ ∈ C | 9λ 3 = 8λ, 8λ 3 = -1 modulo the equivalence λ ∼ -λ. (9) the isomorphism classes of smoooth quartics of P 2 C having automorphism groups acting without fixed points on the quartic without its bitangent points; (10) the isomorphism classes of del Pezzo surfaces of degree 1; (11) the triplets of ramification p 1 , p 2 , p 3 ⊂ P 1 C that determine Z 2Z 2 conic bundles on surfaces that are not del Pezzo ones, modulo the action of P 1 C . The approach of Blanc used the modern viewpoint of Mori's theory and Sarkisov's program, aiming a generalization in higher dimension: he described each maximal algebraic subgroup of the classification as a G-Mori fibration; he then proved that any algebraic subgroup is contained in one of the groups of the classification; he also showed that any group of the classification is a minimal G-fibration that is furthermore superrigid.

× P 1 C ) PGL(2, C) × PGL(2, C) Z 2Z ⊂ GL(6, C). If deg S = 8, then S is isomorphic either to F 0 = P 1 C × P 1 C or to F 1 . On the one hand Aut(P 1 C × P 1 C ) PGL(2, C) × PGL(2, C) Z 2Z ⊂ GL(6, C).
and on the other hand Aut(F 1 ) is not a maximal algebraic subgroup of Bir(P 2 C ) (Theorem 3.46). If deg S = 9, then S P 2 C and Aut(S) = PGL(3, C) ⊂ GL(8, C).

Closed normal subgroups of the Cremona group

As we have seen we can endow the Cremona group with a natural Zariski topology induced by morphisms V → Bir(P n C ) where V is an algebraic variety. In [START_REF] Browder | Mathematical developments arising from Hilbert problems[END_REF] Mumford discussed properties of Bir(P 2 C ) and in particular asked if it is a simple group with respect to the Zariski topology, i.e. if every closed normal subgroup of Bir(P 2 C ) is trivial. Blanc and Zimmermann provided an affirmative answer to Mumford question: Let us mention that Bir(P n C ), n ≥ 2, is not simple as an abstract group (for n = 2 see [CL13] or §8.6, for n ≥ 3 see [BLZar]). Furthermore there is an analogue of Theorem 3.48 when Bir(P n C ) is endowed with the Euclidean topology: Theorem 3.50 ( [START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). -Let n ≥ 2 be an integer. The topological group Bir(P n C ) is topologically simple when endowed with the Euclidean topology.

The proof of Theorem 3.50 is similar to the proof of Theorem 3.48, so we will just focus on this last one.

Sketch of the proof of

(n, C) → C * f n | f ∈ C *
whose kernel is the group PSL(n, C). Denote by id the identity matrix of size (n -1) × (n -1) and consider the morphism

ρ : A 1 C {0} → PGL(n, C), t → t 0 0 id . Note that ρ -1 (PSL(n, C)) contains t n |t ∈ A 1 C which is an infinite subset of A 1 C and is thus dense in A 1 C . Therefore the closure of PSL(n, C) contains ρ(A 1 C {0}). Any element of PGL(n, C) is equal to some ρ(t) modulo PSL(n, C) hence PSL(n, C) is dense in PGL(n, C).
Let N be a non-trivial normal subgroup of PGL(n, C). Let f be a non-trivial element of N. Let us prove that N contains PSL(n, C). The center of PGL(n, C) being trivial one can replace f by α

• f • α -1 • f -1 where α ∈ PGL(n, C) does not commute with f , and assume that f belongs to N ∩ PSL(n, C). But PSL(n, C) is a simple group ([Die71, Chapitre II, §2]) so PSL(n, C) ⊂ N.
The first two points imply that PGL(n, C) does not contain any non-trivial normal strict subgroup which is closed with respect to the Zariski topology.

We will now focus on Bir(P n C ) as soon as n ≥ 2. Proposition 3.52 ( [START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). -Let φ be an element of Bir(P n C ). Let p be a point of P n C such that φ induces a local isomorphism at p, and fixes p. Then there exist morphisms ν :

A 1 C {0} → Aut(P n C ) and υ : A 1 C → Bir(P n C ) such that: υ(t) = ν(t) -1 • φ • ν(t) for any t ∈ C, moreover ν(1) = id, so υ(1) = φ; υ(0) belongs to Aut(P n C
) and is the identity if and only if the action of φ on the tangent space is trivial.

Proof. -Up to conjugacy by an element of Aut(P n C ) we can assume that p = (1 : 0 : 0 : . . . : 0). In the affine chart z 0 = 1 one can write φ locally as

p 1,1 (z 1 , . . . , z n ) + . . . + p 1, (z 1 , . . . , z n ) 1 + q 1,1 (z 1 , . . . , z n ) + . . . + q 1, (z 1 , . . . , z n ) , . . . , p n,1 (z 1 , . . . , z n ) + . . . + p n, (z 1 , . . . , z n ) 1 + q n,1 (z 1 , . . . , z n ) + . . . + q n, (z 1 , . . . , z n )
where p i, j , q i, j are homogeneous of degree j. For each t ∈ C {0} the element

ν t : (z 1 , z 2 , . . . , z n ) → (tz 1 ,tz 2 , . . . ,tz n ) extends to a linear automorphism of P n C that fixes p. Hence the map t → ν -1 t • φ • ν t gives rise to a morphism Θ : A 1 C {0} → Bir(P n C
) and the image of Θ contains only conjugates of φ by linear automorphisms. Note that

Θ : t → p 1,1 (z 1 , . . . , z n ) + t p 1,2 (z 1 , . . . , z n ) + . . . + t -1 p 1, (z 1 , . . . , z n ) 1 + tq 1,1 (z 1 , . . . , z n ) + t 2 q 1,2 (z 1 , . . . , z n ) + . . . + t q 1, (z 1 , . . . , z n ) , . . . , p n,1 (z 1 , . . . , z n ) + t p n,2 (z 1 , . . . , z n ) + . . . + t -1 p n, (z 1 , . . . , z n ) 1 + tq n,1 (z 1 , . . . , z n ) + t 2 q n,2 (z 1 , . . . , z n ) + . . . + t q n, (z 1 , . . . , z n )
and Θ(0) corresponds to the linear part of Θ at p which is locally given by

p 1,1 (z 1 , . . . , z n ), . . . , p n,1 (z 1 , . . . , z n ) .
As φ is a local isomorphism at p, this linear part is an automorphism of P n C . Furthermore it is trivial if and only if the action of φ on the tangent space is trivial.

Let φ ∈ Bir(P n C ) {id}; it induces an isomorphism from U to V where U, V ⊂ P n C are two
non-empty open subsets. There exist a point p in U and two automorphisms α 1 , α 2 of P n C such that

ψ = α 1 • φ • α 2 fixes p, ψ = α 1 • φ • α 2 is a local isomorphism at p, D p ψ is not trivial.
According to Proposition 3.52 there exist morphisms ν :

A 1 C {0} → Aut(P n C ) and υ 1 : A 1 C → Bir(P n C ) such that υ 1 (t) = ν(t) -1 • ψ -1 • ν(t) for each t = 0, υ 1 (0) is an automorphism of P n C . Consider the morphism υ 2 : A 1 C → Bir(P n C ) defined by υ 2 (t) = α -1 1 • ψ • υ 1 (t) • υ 1 (0) -1 • α -1 2 . Since α 1 , α 2 , υ 1 (0) and ν(t) are automorphisms of P n C for all t = 0 υ 2 (t) = α -1 1 • ψ • ν(t) -1 • ψ -1 • ν(t) • υ 1 (0) -1 • α -1
2 belongs for any t = 0 to the normal subgroup of Bir(P n C ) generated by Aut(P n C ). As a consequence φ = υ 2 (0) belongs to the closure of the normal subgroup of Bir(P n C ) generated by Aut(P n C ). The normal subgroup of Bir(P n C ) generated by Aut(P n C ) is dense in Bir(P n C ) (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]).

In particular Bir(P n C ) does not contain any non-trivial closed normal strict subgroup. Indeed let {id} = N ⊂ Bir(P n C ) be a closed normal subgroup with respect to the Zariski topology. Then Aut(

P n C ) ⊂ N (see [BZ18, Prop. 3.3, Lemma 3.4]).
Since N is closed it contains the closure of the normal subgroup generated by Aut(P n C ) which is equal to Bir(P n C ). Furthermore, one has:

Theorem 3.53 ([BZ18]). -If n ≥ 1, the group Bir(P n C
) is connected with respect to the Zariski topology.

If n ≥ 2, the group Bir(P n C ) is path-connected, and thus connected with respect to the Euclidean topology.

Let us give an idea of the proof of this statement. We start with an example. Example 6. -Let n ≥ 2 and let α be an element of C * . Consider the birational self map of P n C given by

Φ : (z 0 : z 1 : . . . : z n ) z 0 (z 1 + αz 2 ) + z 1 z 2 z 1 + z 2 : z 1 : z 2 : . . . : z n .
The points p = (0 : 1 : 0 : 0 : . . . : 0) and q = (0 : 0 : 1 : 0 : 0 : . . . : 0) are fixed by Φ. Applying Proposition 3.52 to the points p and q we get two morphisms

Θ 1 , Θ 2 : A 1 C → Bir(P n C ) such that Θ 1 (0) : (z 0 : z 1 : . . . : z n ) → (z 0 + z 2 : z 1 : z 2 : . . . : z n ) ∈ Aut(P n C ), Θ 2 (0) : (z 0 : z 1 : . . . : z n ) → (αz 0 + z 1 : z 1 : z 2 : . . . : z n ) ∈ Aut(P n C ), Θ 1 (1) = Θ 2 (1) = Φ.
Proposition 3.54. -Let n ≥ 2 be an integer. For any φ, ψ ∈ Bir(P n C ) there is a morphism υ : P 1 C → Bir(P n C ) such that υ(0) = φ and υ(1) = ψ. Proof. -Up to composition with φ -1 one can assume that φ = id. Let us consider the subset S of Bir(P n C ) given by

S = ϕ ∈ Bir(P n C ) | ∃ ν : A 1 C → Bir(P n C
) morphism such that ν(0) = id and ν(1) = ϕ . Let φ (resp. ψ) be an element of S; denote by ν φ (resp. ν ψ ) the associated morphism. We define a morphism ν φ•ψ :

A 1 C → Bir(P n C ) by ν φ•ψ (t) = ν φ (t) • ν ψ (t) which satisfies ν φ•ψ (0) = id and ν φ•ψ (1) = φ • ψ. For any ϕ ∈ Bir(P n C ) it is also possible to define a morphism t → ϕ • ν φ (t) • ϕ -1 . Therefore, S is a normal subgroup of Bir(P n C ). Claim 3.55 ([BF13]). -The group S contains PSL(n + 1, C).
Take α, Φ, Θ 1 and Θ 2 as in Example 6; for i ∈ {1, 2} the morphisms

t → Θ i (t) • (Θ i (0)) -1 show that g • Θ 1 (0) -1 and g • Θ 2 (0) -1 belong to S, hence Θ 1 (0) • Θ 2 (0) -1 belong to S. But Θ 1 (0) belongs to PSL(n + 1, C) ⊂ S, so Θ 2 (0) belongs to S. Thus Aut(P n C ) = PGL(n + 1, C) is contained in S. Take φ ∈ Bir(P n C ) of degree d ≥ 2.
Let p be a point of P n C such that φ induces a local isomorphism at p. Consider an element A of PSL(n + 1, C) such that A • φ fixes p. There exists a morphism θ :

A 1 C → Bir(P n C ) such that θ(0) belongs to Aut(P n C ) and θ(1) = A • φ. Let us define θ : A 1 C → Bir(P n C ) by θ (t) = ρ(t) • θ(0) -1 . Then θ (1) = A • φ • θ(0) -1
. But A and θ(0) belong to Aut(P n C ) ⊂ S, so φ belongs to S.

Regularization of rational group actions

The aim of [START_REF] Kraft | Regularization of rational group actions[END_REF] is to give a modern proof of the regularization theorem of Weil which says:

Theorem 3.56 ([Wei55]
). -For every rational action of an algebraic group G on a variety X there exist a variety Y with a regular action of G and a G-equivariant birational map X Y .

In this section a variety is an algebraic complex variety, and an algebraic group is an algebraic C-group.

A rational map φ : 3.5.1. Rational group actions. -Let X and Z be two varieties. Let us recall that a map

X Y is called biregular in p ∈ X if there is an open neighborhood U ⊂ (X Base(φ)) of p such that φ |U : U → Y is an open immersion. As a result the subset X = p ∈ X | φ is biregular in p is open in X,
φ : Z → Bir(X) is a morphism if there exists an open dense set U ⊂ Z × X such that the induced map U → X, (q, p) → φ(q)(p) is a morphism of varieties; for every q ∈ Z the open set U q = p ∈ X | (q, p) ∈ U is dense in X;
for every q ∈ Z the birational map φ(q) : X X is defined on U q .

Equivalently there is a rational map φ : Z × X → X such that for every

q ∈ Z the open subset (Z × X Base(φ)) ∩ ({q} × X) is dense in {q} × X; the induced birational map φ q : X X, p → φ(q, p) is birational.
Recall that this definition allows to define the Zariski topology on Bir(X) (see §3.1).

We can now define rational group actions on varieties. Let G be an algebraic group, and let X be a variety. A rational action of G on X is a morphism ρ : G → Bir(X) which is a morphism of groups. In other words there is a rational map still denoted ρ

ρ : G × X X such that the open set ((G × X) Base(ρ)) ∩ ({g} × X) is dense in {g} × X for every g ∈ G;
the induced map ρ g : X X, p → ρ(g, p) is birational for every g ∈ G; the map g → ρ g is a group morphism.

Theorem 3.58 [START_REF] Kraft | Regularization of rational group actions[END_REF]). -Let ρ : G → Bir(X) be a rational action where X is affine. Assume that there exists a dense subgroup Γ ⊂ G such that ρ(Γ) ⊂ Aut(X). Then the G-action on X is regular.

Definition. -Let X and Y be two varieties. Let ρ be a rational G-action on X. Let µ be a rational G-action on Y .

A dominant rational map φ : X Y is G-equivariant if the following holds: for every (g, p) ∈ G × X such that ρ is defined in (g, p), φ is defined in p and in ρ(g, p), µ is defined in (g, φ(p)), we have φ(ρ(g, p)) = µ(g, φ(p)).

Remark 3.59. -The set of (g, p) ∈ G × X satisfying the previous assumptions is open and dense in G × X and has the property that it meets all {g} × X in a dense open set.

Let X be a variety with a rational action ρ : G × X X of an algebraic group G. Consider

ρ : G × X G × X, (g, p) → (g, ρ(g, p)). It is clear that (G × X) Base( ρ) = (G × X) Base(ρ). Furthermore ρ is birational with inverse ρ -1 (g, p) = (g, ρ(g -1 , p)), that is ρ -1 = τ • ρ • τ
where τ is the isomorphism

τ : G × X → G × X, (g, p) → (g -1 , p). Definition. -A point x ∈ X is called G-regular for the rational G-action ρ on X if Breg( ρ) ∩ (G × {p}) is dense in G × {p}.
In other words a point p ∈ X is called G-regular for the rational G-action ρ on X if ρ is biregular in (g, p) for all g in a dense open set of G.

Denote by

X reg ⊂ X the set of G-regular points. Let λ g : G ∼ -→ G be the left multiplication with g ∈ G. For any h ∈ G the diagram G × X ρ G G λ h ×id G × X λ h ×ρ h G × X ρ G G G × X
commutes. This implies the following statement:

Lemma 3.60 ([Kra18]). -If ρ is defined in (g, p) and if ρ h is defined in g • p, then ρ is defined in (hg, p).
If ρ is biregular in (g, p) and if ρ h is biregular in g • p, then ρ is biregular in (hg, p).

Proposition 3.61 ([Kra18]

). -The set X reg of G-regular points is open and dense in X.

If p belongs to X reg and if ρ is biregular in (g, p), then g • p belongs to X reg .

Proof.

-Let G = G 0 ∪ G 1 ∪ . . . ∪ G n be the decomposition into connected components. Then D i = Breg(ρ) ∩ (G i × X
) is open and dense for all i (Lemma 3.57); the same holds for the image D i ⊆ X under the projection onto X. Since

X reg = i D i the set X reg is open and dense in X. If ρ is biregular in (g, p), then ρ -1 = τ • ρ • τ is biregular in (g, g • p). As a consequence ρ is biregular in τ(g, g • p) = (g -1 , g • p). If p is G-regular, then ρ h is biregular in p for all h in a dense open subset G of G.
According to the second assertion of Lemma 3.60 the birational map ρ is biregular in (hg -1 , g • p) for all h ∈ G . Hence g • p belongs to X reg .

A consequence of Proposition 3.61 allows us to only consider the case of a rational G-action such every point is G-regular.

Corollary 3.62 ([Kra18]

). -For the rational G-action on X reg every point is G-regular.

Lemma 3.63 ([Kra18]). -Assume that X = X reg . If ρ g is defined in p, i.e. if p ∈ X Base(ρ g ), then ρ g is biregular in p. Proof. -Suppose that ρ g is defined in p ∈ X. As X = X reg there exists a dense open subset G of G such that for all h ∈ G ρ h is biregular in g • p, ρ hg is biregular in p. Since ρ hg = ρ h • ρ g the map ρ g is biregular in p.
Let us recall that if φ : X Y is a rational map, its graph Γ(φ) is defined by

Γ(φ) = (x, y) ∈ X ×Y | x ∈ X Base(φ) and φ(x) = y .
In particular pr 1 (Γ(φ)) = X Base(φ) and pr 2 (Γ(φ)) = φ(X Base(φ)).

Lemma 3.64 ([Kra18]

). -Let ρ be a rational G-action on a variety X. Suppose that every point of X is G-regular, that is X = X reg . Then for every g ∈ G the graph Γ(ρ g ) of ρ g is closed in X × X.

Proof. -Denote by Γ the closure Γ(ρ g ) of the graph of ρ g in X × X. Let us prove that for any (x 0 , y 0 ) ∈ Γ the rational map ρ g is defined in x 0 . It is equivalent to prove that the morphism pr 1|Γ : Γ → X is biregular in (x 0 , y 0 ). Let h be an element of G such that ρ hg is biregular in x 0 and ρ h is biregular in y 0 . Consider the birational map

φ = (ρ hg , ρ h ) : X × X X × X. If φ is defined in (x, y) ∈ Γ(ρ g ), y = g•x, then φ(x, y) = ((hg)•x, (hg)•x) belongs to the diagonal ∆(X) = (x, x) ∈ X | x ∈ X of X × X. It follows that φ(Γ) ⊆ ∆(X). Since φ is biregular in (x 0 , y 0 ), the restriction ϕ = φ |Γ : Γ ∆(X) of φ to Γ is also biregular in (x 0 , y 0 ). By construction ρ hg • pr 1|Γ = pr 1|∆(X) • ϕ; indeed X × X ρ hg ×ρ h G G X × X Γ ϕ G G pr 1|Γ ? y y ∆(X) pr 1|∆(X) ? y y X ρ hg G G X
But ρ hg is biregular in pr 1|Γ (x 0 , x 0 ), ϕ is biregular in (x 0 , y 0 ) and pr 1|∆(X) is an isomorphism, so pr 1|Γ is biregular in (x 0 , y 0 ).

Lemma 3.65 ([Kra18]

). -Let ρ be a rational action of G on a variety X. Suppose that there

is a dense open subset U of X such that ρ : G × U → G × X, (g, p) → g, ρ(g, p) defines an open immersion. Then the open dense subset Y = g∈G g • U ⊆ X carries a regular G-action. Proof. -Any ρ g induces an isomorphism U ∼ -→ g • U. Therefore, Y = g∈G g • X ⊂ X
is stable under all ρ g . By assumption the induced map on G × U is a morphism, so the induced map on G × g • U is a morphism for all g ∈ G. As a result the induced map G × Y → Y is a morphism.

3.5.2. Construction of a regular model. -

Theorem 3.66 ([Kra18]
). -Let X be a variety with a rational action of G. Suppose that every point of X is G-regular. Then there exists a variety Y with a regular G-action and a G-equivariant open immersion.

Assume now that X is a variety with a rational G-action ρ such that X reg = X. Consider a finite subset S = g 0 = e, g 1 , g 2 , . . . , g m of G. Denote by X (0) , X (1) , . . ., X (m) some copies of X. Consider the disjoint union

X(S) = X (0) ∪ X (1) ∪ . . . ∪ X (m) .
Let us define on X (i) the following relations

∀ i p i ∼ p i ⇐⇒ p i = p i ∀ i, j i = j p i ∼ p j ⇐⇒ ρ g -1
j g i is defined in p i and sends p i to p j This defines an equivalence relation (Lemma 3.63 is needed to prove the symmetry). Consider X(S) = X(S) ∼ the set of equivalence classes endowed with the induced topology.

Lemma 3.67 [START_REF] Kraft | Regularization of rational group actions[END_REF]). -The maps ι i : X (i) → X(S) are open immersions and endow X(S) with the structure of a variety.

Let us fix the open immersion ι 0 : X = X (0) → X(S). Then G acts rationally on X(S) via ρ = ρ S such that ι 0 is G-equivariant. Consider any X (i) as the variety X with the rational G-action

ρ (i) (g, p) = ρ(g i gg -1 i , p); by construction of X(S) the open immersions ι i : X (i) → X(S)
are all G-equivariant.

Lemma 3.68 ([Kra18]

). -Let X (i) be the image of the open immersion ι i : X (i) → X(S). For all i the rational map ρ g i is defined on X (0) . Furthermore ρ g i : X (0) ∼ → X (i) defines an isomorphism.

Proof.

-Consider the open immersion

τ i = ι i • ι -1 0 : X (0) → X(S). Note that im τ i = X (i) . Let us check that τ i (p) = g i p.
It is sufficient to show that it holds on an open dense subset of X (0) . Let U ⊆ X be the open dense set where g i • p is defined. Take p in U. On the one hand by definition

ι 0 (g i • p) = ι i (p); on the other hand ι 0 (g i • p) = g i • ι 0 (p).
As a result g i • ι 0 (p) = ι i (p) and

τ i (p) = ι i (ι -1 0 (p)) = g i • ι 0 (ι -1 0 (p)) = g i • p for any p ∈ ι 0 (U). Proof of Theorem 3.66. -Set D = Breg(ρ) ∩ (G × X). Since X reg = X for any p ∈ X there is an element g in G such that (g, p) ∈ D. As a consequence g∈G g • D = G × X where G acts on G × X by left-multiplication on G. Hence i g i D = G × X for a suitable finite subset S = g 0 = e, g 1 , g 2 , . . . , g m . Recall that X (0) = im(ι 0 ). Let D (0) ⊂ G × X (0) be the image of D. Consider the rational map ρ S : G × X (0) G × X(S).
The map (g, p) → (g, g • p) is the composition of (g, p) → (g, (g -1 i g) • p) and (g, y) → (g, g i • y). The first one is biregular on g i • D (0) and its image is contained in G × X (0) ; the second is biregular on G × X (0) (Lemma 3.68). As G × X (0) = i g i • D (0) the map ρ S is biregular. As a consequence the rational action ρ of G on X(S) has the property that

ρ S : G × X (0) → G × X(S)
defines an open immersion. Lemma 3.65 allows to conclude. ). -Let X, Y , Z be varieties. Assume that Z is affine. Let φ : X ×Y Z be a rational map. Suppose that there exists an open dense subset U of Y such that φ is defined on X × U; there exists a dense subset X of X such that the induced maps φ p : {p} × Y → Z are morphisms for all p ∈ X .

Then φ is a regular morphism.

Consider a rational action ρ : G → Bir(X) of an algebraic group on a variety X. Assume that there is a dense subgroup Γ of G such that ρ(Γ) ⊂ Aut(X).

Let us first prove that the rational G-action on the open dense set X reg ⊆ X is regular. For every p ∈ X reg there is g ∈ Γ such that ρ is biregular in (g, p). By assumption for any h ∈ Γ the map ρ h is biregular on X, hence the map ρ is biregular in (h, p) for any h ∈ Γ (Lemma 3.60). Furthermore h • p belongs to X reg (Proposition 3.61), i.e. X reg is stable under Γ. According to Theorem 3.66 there exists a G-equivariant open immersion

X reg → Y
where Y is a variety with a regular G-action. The complement Y X reg is closed and Γ-stable, so Y X reg is stable under Γ = G.

From the previous point the rational map ρ : G × X X has the following properties:

-there is a dense open set X reg ⊆ X such that ρ is regular on G × X reg ; -for every g ∈ Γ the rational map

ρ g : X → X, p → ρ(g, p)
is a regular isomorphism. Lemma 3.69 implies that ρ is a regular action in case X is affine.

Remark 3.70. -All the statements of this section hold for an algebraically closed field.

CHAPTER 4 GENERATORS AND RELATIONS OF THE CREMONA GROUP

As we already say

σ 2 : (z 0 : z 1 : z 2 ) (z 1 z 2 : z 0 z 2 : z 0 z 1 ).
This result is well-known as the Theorem of Noether and Castelnuovo. Noether was the first mathematician to state this result at the end of the XIXth century. Nevertheless the first exact proof is due to Castelnuovo. Noether's idea was the following. Let us consider a birational self map φ of P 2 C . Take a quadratic birational self map q of P 2 C such that the three base-points of q are three base-point of φ of highest multiplicity. Then deg(φ • q) < deg φ. By induction one gets a birational map of degree 1. But such a quadratic birational map q may not exist. This is for instance the case if one starts with the polynomial automorphism

(z 0 : z 1 : z 2 ) (z 3 1 -z 0 z 2 2 : z 1 z 2 2 : z 3 2 ).
In [START_REF] Alexander | On the factorization of Cremona plane transformations[END_REF] Alexander fixes Noether's proof by introducing the notion of complexity of a map: start with a birational self map φ of the complex projective plane; one can find a quadratic birational self map q of the complex projective plane such that either the complexity of φ • q is strictly less that the complexity of φ; or the complexities of φ • q and φ are equal but #Base(φ • q) < #Base(φ).

Alexander's proof is a proof by induction on these two integers. ). This result is "weaker" nevertheless it has the following nice property:

Theorem 4.3 ([AC02]
). -Let φ be an element of Bir(P 2 C ). There exist j 1 , j 2 , . . ., j k in J and A in PGL(3, C) such that

φ = A • j k • j k-1 • . . . • j 2 • j 1 ; for any 1 ≤ i ≤ n -1 deg(A • j k • j k-1 • . . . • j i+1 • j i ) > deg(A • j k • j k-1 • . . . • j i+2 • j i+1 ).
The first presentation of the plane Cremona group is given by Gizatullin:

Theorem 4.4 ([Giz82]). -The Cremona group Bir(P 2 C ) is generated by the set Q of all quadratic maps.
The relations in Bir(P 2 C ) are consequences of relations of the form q 1 • q 2 • q 3 = id where q 1 , q 2 , q 3 are quadratic birational self maps of P 2 C . In other words we have the following presentation

Bir(P 2 C ) = Q | q 1 • q 2 • q 3 = id ∀ q 1 , q 2 , q 3 ∈ Q such that q 1 • q 2 • q 3 = id in Bir(P 2 C )
Two years later Iskovskikh proved the following statement:

Theorem 4.5 ([Isk83, Isk85]). -The group Bir(P 1 C × P 1 C
) is generated by the group B of birational maps preserving the fibration given by the first projection together with τ : (z 0 , z 1 ) → (z 1 , z 0 ).

Moreover the following relations form a complete system of relations:

relations inside the groups Aut(P 1

C × P 1 C ) and B; τ • (z 0 , z 1 ) → z 0 , z 0 z 1 3 = id; (τ • ((z 0 , z 1 ) → (-z 0 , z 1 -z 0 ))) 3 = id.
In 1994 Iskovskikh, Kabdykairov and Tregub present a list of generators and relations of Bir(P 2 k ) over arbitrary perfect field k (see [START_REF] Iskovskikh | Relations in a twodimensional Cremona group over a perfect field[END_REF]). The group Bir(P 2 C ) hasn't a structure of amalgamated product ( [START_REF] Cornulier | Appendix to normal subgroups in the Cremona group[END_REF]). Nevertheless a presentation of the plane Cremona group in the form of a generalized amalgam was given by Wright: C ) is the amalgamated product of the Jonquières group with the group of automorphisms of the plane along their intersection, divided by the relation σ 2 • τ = τ • σ 2 where σ 2 is the standard involution and τ is the involution (z 0 : z 1 : z 2 ) → (z 1 : z 0 : z 2 ).

As we have seen in Chapter 3 there is an Euclidean topology on the Cremona group ( [START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). With respect to this topology Bir(P 2 C ) is a Hausdorff topological group. Furthermore the restriction of the Euclidean topology to any algebraic subgroup is the classical Euclidean topology. To show that Bir(P 2 C ) is compactly presentable with respect to the Euclidean topology Zimmermann established the following statement:

Theorem 4.8 ([Zim16]). -The group Bir(P 2 C ) is isomorphic to the amalgamated product of Aut(P 2 C ), Aut(F 2 ), Aut(P 1 C × P 1 C ) along their pairwise intersection in Bir(P 2 C ) modulo the relation τ • σ 2 • τ • σ 2 = id
where σ 2 is the standard involution and τ the involution τ : (z 0 :

z 1 : z 2 ) → (z 1 : z 0 : z 2 ).
Urech and Zimmermann got a presentation of the plane Cremona group with respect to the generators given by the Theorem of Noether and Castelnuovo:

Theorem 4.9 ([UZ19]). -The Cremona group Bir(P 2 C ) is isomorphic to σ 2 , PGL(3, C) | (R 1 ), (R 2 ), (R 3 ), (R 4 ), (R 5 ) where (R 1 ) g 1 • g 2 • g -1 3 = id for all g 1 , g 2 , g 3 ∈ PGL(3, C) such that g 1 • g 2 = g 3 ; (R 2 ) σ 2 2 = id (R 3 ) σ 2 • η • (η • σ 2 ) -1 = id for all η in the symmetric group S 3 ⊂ PGL(3, C) of order 6 acting on P 2 C by coordinate permutations (R 4 ) σ 2 • d • σ 2 • d = id for all diagonal automorphisms d in the subgroup D 2 ⊂ PGL(3, C) of diagonal automorphisms; (R 5 ) (σ 2 • h) 3 = id where h : (z 0 : z 1 : z 2 ) → (z 2 -z 0 : z 2 -z 1 : z 2 ) Remarks 4.10. -
The relations (R 2 ), (R 3 ) and (R 4 ) occur in the group Aut(C * × C * ) which is given by the group of monomial maps GL(2, Z) D 2 . (R 5 ) is a relation from the group Aut(P 1

C × P 1 C ) 0 PGL(2, C) × PGL(2, C
) which is considered as a subgroup of Bir(P 2 C ) by conjugation with the birational equivalence

P 1 C × P 1 C P 2 C (u 0 : u 1 ), (v 0 : v 1 ) (u 1 v 0 : u 0 v 1 : u 1 v 1 )
Remark 4.11. -All the results are stated on C but indeed [Cor13, UZ19, Giz82, Isk83, Isk85, Wri92, Bla12] work for the plane Cremona group over an algebraically closed field, [START_REF] Zimmermann | The Cremona group of the plane is compactly presented[END_REF] works for the plane Cremona group over a locally compact local field.

In the first section we recall the proof of Noether and Castelnuovo due to Alexander.

In the second section we give an outline of the proof of the result of [START_REF] Cornulier | Appendix to normal subgroups in the Cremona group[END_REF] that says that the plane Cremona group does not decompose as a non-trivial amalgam. We also recall the proof of Theorem 4.6.

The third section is devoted to generators and relations in the Cremona group. We first give a sketch of the proof of Theorem 4.7. We also give a sketch of the proof of Theorem 4.9. We then explain why there is no Noether and Castelnuovo theorem in higher dimension.

Noether and Castelnuovo theorem

Let us now deal with the proof of Theorem 4.1 given by Alexander [START_REF] Alexander | On the factorization of Cremona plane transformations[END_REF]). Recall the two following formulas proved in §1.3. Consider a birational self map φ of P 2 C of degree ν; denote by p 1 , p 2 , . . ., p k the base-points of φ and by m p i the multiplicity of p i . Then

k ∑ i=0 m p i = 3(ν -1) (4.1.1) k ∑ i=0 m 2 p i = ν 2 -1. (4.1.2)
From (4.1.2) and (4.1.1) one gets

k ∑ i=0 m p i m p i -1 = (ν -1)(ν -2). (4.1.3)
Consider a birational self map of P 2 C of degree ν. If ν = 1, then according to (4.1.1) the map φ is an automorphism of P 2 C . So let us now assume that ν > 1. Let Λ φ be the linear system associated to φ. Denote by p 1 , p 2 , . . ., p k the base-points (in P 2 C or infinitely near) of φ and m p i their multiplicity. Up to reindexation let us assume that

m p 0 ≥ m p 1 ≥ . . . ≥ m p k ≥ 1.
Alexander introduced the notion of complexity: the complexity of Λ φ is the integer 2c = νm p 0 . It is the number of points except p 0 that belong to the intersection of a general line passing through p 0 and a curve of Λ φ .

Remarks 4.12. -One has 2c ≥ 0: the degree of the hypersurfaces of Λ φ is ν, so a point has multiplicity ≤ ν; furthermore 2c ≥ 1; indeed if an homogeneous polynomial of degree ν has a point of multiplicity ν, then the hypersurface given by this polynomial is the union of ν lines.

Set C = p ∈ Base(φ) {p 0 } | m p > c and n = #C.
Bezout theorem implies that the line through p 0 and p 1 intersects any curve of Λ φ in ν points (counted with multiplicity). Furthermore it intersects any curve of Λ φ at p 0 with multiplicity

m p 0 . Consequently m p 1 ≤ ν -m p 0 = 2c and c < m p k ≤ . . . ≤ m p 2 ≤ m p 1 ≤ 2c (4.1.4) Lemma 4.13.
-There are at least three base-points of multiplicity > c = ν-m p 0 2 , i.e. n ≥ 2; hence m p 0 > ν 3 . Furthermore if ν ≥ 3, then p 1 , p 2 , . . ., p n are not aligned.

Proof. -According to (4.1.2) and (4.1.3) one has on the one hand

c k ∑ i=0 m p i (m p i -1) -(c -1) k ∑ i=0 m 2 p i = k ∑ i=0 m p i (cm p i -c -cm p i + m p i ) = k ∑ i=0 m p i (m p i -c)
and on the other hand

(ν -1)(ν -2)c -(ν 2 -1)(c -1) = (ν -1)(νc -2c -νc + ν -c + 1) = (ν -1)(ν -3c + 1). As a result k ∑ i=0 m p i (m p i -c) = (ν -1)(ν -3c + 1) (4.1.5) Since m p n+i ≤ c for any i > 0 one gets n ∑ i=0 m p i (m p i -c) ≥ k ∑ i=0 m p i (m p i -c).
According to (4.1.5)

n ∑ i=0 m p i (m p i -c) ≥ (ν -1)(ν -3c + 1) = ν(ν -3c) + 3c -1. But 3c -1 ≥ 1 2 > 0, so n ∑ i=0 m p i (m p i -c) > ν(ν -3c) = ν(m p 0 -c). Consequently n ∑ i=0 m p i (m p i -c) > ν(m p 0 -c) -m p 0 (m p 0 -c) = (ν -m p 0 )(m p 0 -c) = 2c(m p 0 -c).
As 2c ≥ m p i for any i ≥ 1 (see (4.1.4)) one gets 2c

n ∑ i=1 (m p i -c) > 2c(m p 0 -c) and 2c n ∑ i=1 (m p i -c) > 2c(m p 0 -c)
that is

m p i -c > m p 0 -c (4.1.6) since c > 0. But m p 1 ≤ m p 0 , so n ≥ 2. Therefore, m p 0 + m p 1 + m p 2 > 3 ν-m p 0 2
and m p 0 > ν 3 . Let us assume that n ≥ 3; then (4.1.6) can be rewritten

n ∑ i=1 m p i -nc > m p 0 -c = ν -3c and n ∑ i=1 m p i > ν + (n -3)c ≥ ν.
Definition. -A general quadratic birational self map of Bir(P 2 C ) centered at p, q r is the map, up to linear automorphism, that blows up the three distinct points p, q, r of P 2 C and blows down the strict transform of the lines (pq), (qr) and (pr). These lines are thus sent onto points denoted p , q and r .

The line (p q ) (resp. (q r ), resp. (p r )) corresponds to the exceptional line of the blow up of r (resp. p, resp. q). Lemma 4.14. -Compose φ with a general quadratic birational self map of P 2 C centered at p 0 , q and r where p 0 is the base-point of φ of maximal multiplicity.

The complexity of the new system is equal to the complexity of the old system if and only if p 0 is of maximal multiplicity.

If it is not the case, then the complexity of the new system is strictly less than the complexity of the old one.

Proof. -The complexity of the new system is 2c = νm max where m max denotes the highest multiplicity of the base-points of the new system. Then

2c = ν -m max = 2ν -m p 0 -m q -m r -m max = ν -m p 0 + (ν -m q -m r ) -m max = ν -m p 0 + m p 0 -m max = 2c + m p 0 -m max .
Hence c ≤ c and c = c if and only if m p 0 = m max .

Lemma 4.15. -If there exist two points p i and p j in C = p 1 , p 2 , . . . , p n such that p i and p j are not infinitely near; p i and p 0 are not infinitely near; p j and p 0 are not infinitely near.

Then there exists a general quadratic birational self map of P 2 C such that after composition with φ either the complexity of the system decreases, or #C = n decreases by 2.

Proof. -Suppose that there exist two points p i and p j in C = p 1 , p 2 , . . . , p n such that p i and p j are not infinitely near; p i and p 0 are not infinitely near; p j and p 0 are not infinitely near.

Let us now compose φ with a general quadratic birational self map of P 2 C centered at p 0 , p i and p j . The degree of the new linear system

Λ φ is ν = 2ν -m p 0 -m p i -m p j . Let us remark that ν = 2ν -m p 0 -m p i -m p j = ν + (ν -m p 0 -m p i -m p j ) = ν + (2c -m p i -m p j ) < ν
i.e. the degree has decreased. The new linear system Λ φ has complexity c and we denote by C the set of points of multiplicity > c .

The points p 0 , p i and p j are no more points of indeterminacy; the other base-points and their multiplicity do not change. There are three new base-points which are p 0 , p i and p j . By definition the multiplicity of p 0 (resp. p i , resp. p j ) is equal to the number of intersection points (counted with multiplicity) between the corresponding contracted line and the strict transform of a general curve of the linear system. From Bezout theorem we thus have The complexity of the system does not change, the cardinal of C does not change. There is no point infinitely near p 0 .

     m p 0 = ν -m p i -m p j m p i = ν -m p 0 -m p j m p j = ν -m p 0 -m p i If p 0 is
Proof. -Consider a point q of the complex projective plane such that the lines (p 0 q) and (p k q) contain no base-point; there is no point infinitely near p 0 in the direction of the line (p 0 q); there is no point infinitely near p k in the direction of the line (p k q).

Compose φ with a general quadratic birational map centered at p 0 , p k and q. The degree of the new linear system is

ν = 2ν -m p 0 -m p k = ν + 2c -m p k ≥ ν.
The point p 0 is the point of highest multiplicity:

     m p 0 = ν -m p k ≥ ν -m p 0 = 2c ≥ m p 1 m p k = ν -m p 0 = 2c > c m q = ν -m p 0 -m p k = 2c -m p k < c
hence the complexity remains constant (Lemma 4.14). Note that #C = #C.

The assumptions on q allow to say that a point infinitely near p k (resp. p 0 ) is not transformed in a point infinitely near p 0 . Similarly a point infinitely near p k (resp. p 0 ) is not transformed in a point infinitely near q .

Lemma 4.17. -Assume that all the points of C are above the point of highest multiplicity p 0 . Then one can disperse them with a general quadratic birational self map; in other words there is no base-point of C infinitely near the point p 0 of highest multiplicity of the new system. The cardinal n increases by 2 but the complexity of the system remains constant.

Proof. -Take two points q and r in P 2 C such that the lines (p 0 r), (p 0 q) and (rq) do not contain base-points; are not in the direction of the points infinitely near p 0 .

The degree of the new linear system is ν = 2νm p 0 > ν. Since the curves of the system do not pass through q and r Bezout theorem implies that m p 0 = ν; it is thus the point of highest multiplicity. Furthermore 2c = 2νm p 0ν = 2c.

Any curve of the linear system intersects (p 0 r) and (p 0 q) in νm p 0 = 2c points. As a result m r = m q = 2c > c = c . Moreover r and q belong to C and n = n + 2. The points infinitely near p 0 have been dispersed onto the line (q r ). As there is no basepoint on the line (qr) there is no base-point infinitely near p 0 .

Proof of Theorem 4.1. -Let us consider a birational self map φ of P 2 C of degree ν. Denote by p 0 , p 1 , . . ., p k its base-points and by Λ φ the linear system associated to φ. Let m p i be the multiplicity of p i and assume up to reindexation that

m p 0 ≥ m p 1 ≥ . . . ≥ m p k .
Recall that the complexity of the system Λ φ is c where 2c = νm p 0 , that

C = p ∈ Base(φ) {p 0 } | m p > c
and that n = #C. We will now compose φ with a sequence of general quadratic birational maps in order to decrease the complexity until the complexity equals to 1.

Step 1. -If all points of C are above p 0 , let us apply Lemma 4.17. One gets that p 0 has no more infinitely near base-points and that n = n + 2. Let us now apply Lemma 4.16 until the points of C are all distinct; note that C and n do not change. According to Lemma 4.13 the points of C are not aligned. Let us take two of these points, denoted by p i and p j such that there exist two base-points p k and p with the following property: p k and p do not belong to the lines (p 0 p i ), (p 0 p j ) and (p i p j ). Apply two times Lemma 4.15 to the points p k and p . If the complexity decreases (the first or the second time anyway), then let us start this process again; otherwise the first application of Lemma 4.15 yields to n = n and the second to n = n -2. Furthermore there is no more base-point of C infinitely near p 0 and we go to Step 2.

Step 2. -We distinguish two possibilities:

Step 2i. Either there are two base-points in C that are not infinitely near and one applies Lemma 4.15. If the complexity decreases, come back to Step 1, otherwise come back to Step 2.

Step 2ii. Or let us apply Lemma 4.16, then there are two base-points that are not infinitely near and one can apply Step 2i.

According to Lemma 4.13 if ν > 1, then #C ≥ 3. As a result Step 1 and Step 2 allow to decrease the complexity. When the complexity is 1, the point p 0 has the highest multiplicity and from Lemmas 4.15, 4.16 and 4.17 one gets that #C decreases until 0. In other words our system has at most one base-point. From (4.1.1) and (4.1.2) one gets that ν = 1 and that there is no base-point.

Amalgamated product and

= Aut(P 2 C ) ∩ Aut(A 2 C ) and the group J A 2 C = J ∩ Aut(A 2
C ) along their intersection. On the contrary Bir(P 2 C ) is not the amalgamated product of Aut(P 2 C ) and J .

Indeed there exist elements of Bir(P 2 C ) of finite order which are neither conjugate to an element of Aut(P 2 C ), nor to an element of J (see [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]), contrary to the case of amalgamated products.

More precisely Cornulier proved that the plane Cremona group does not decompose as a non-trivial amalgam ([Cor13]); we will give a sketch of the proof in this section.

A graph Γ consists of two sets X and Y , and two applications

Y → X × X, y → (o(y),t(y)) Y → Y, y → y such that: ∀ y ∈ Y y = y, y = y, o(y) = t(y).
An element of X is a vertex of Γ; an element y ∈ Y is an oriented edge, and y is the reversed edge. The vertex o(y) = t(y) is the origin of y, and the vertex t(y) = o(y) is the terminal vertice. These two vertices are called the extremities of y.

An orientation of a graph Γ is a part Y + of Y such that Y is the disjoint union of Y + and Y + . An oriented graph is defined, up to isomorphism, by the data of two sets X and Y + , and an application Y + → X × X. The set of edges of the corresponding graph is

Y = Y + Y + .
A graph is connected if two vertices are the extremities of at least one path.

Examples 4. -Let n be an integer. Let us consider the oriented graph It has n + 1 vertices 0, 1, . . ., n and the orientation is given by the n egdes

[i, i + 1], 0 ≤ i < n with o([i, i + 1]) = i and t([i, i + 1]) = i + 1.
Let n ≥ 1 be an integer. Consider the oriented graph given by The set of vertices is Z nZ , and the orientation is given by the n edges

[i, i + 1], i ∈ Z nZ , with o([i, i + 1]) = i and t([i, i + 1]) = i + 1.
Definitions. -A path of length n in a graph Γ is a morphism from Ch n to Γ.

A cycle of length n in a graph is a subgraph isomorphic to Cir n .

A tree is a non-empty, connected graph without cycle.

Definition. -A group G is said to have property (FA) if every action of G on a tree has a global fixed point.

Definitions. -A geodesic metric space is a metric space if given any two points there is a path between them whose length equals the distance between the points. A real tree can be defined in the following equivalent ways ([Chi01]): a geodesic metric space which is 0-hyperbolic in the sense of Gromov; a uniquely geodesic metric space for which

[a, c] ⊂ [a, b] ∪ [b, c
] for all a, b and c; a geodesic metric space with no subspace homeomorphic to the circle.

In a real tree a ray is a geodesic embedding of the half line. An end is an equivalence class of rays modulo being at bounded distance.

For a group of isometries of a real tree, to stably fix an end means to pointwise stabilize a ray modulo eventual coincidence (i.e. it fixes the end as well as the corresponding Busemann function (1) ).

Definition. -A group has property (FR) if for every isometric action on a complete real tree every element has a fixed point.

Remark 4.18. -Property (FR) implies property (FA).

Lemma 4.19 ([Cor13]

). -Let G be a group. Property (FR) has the following equivalent characterizations:

for every isometric action of G on a complete real tree every finitely generated subgroup has a fixed point; every isometric action of G on a complete real tree either has a fixed point, or stably fixes a point at infinity. Let us give the main steps of the proof of Theorem 4.21. From now on T is a complete real tree and all actions on T are isometric.

Definition. -A group G decomposes as a non-trivial amalgam if G G 1 * H G 2 with G 1 = H = G 2 .
Step 1. -Let p 0 , p 1 , . . ., p k be points of T and s ≥ 0. Suppose that the following equality holds d(p i , p j ) = s|i -j| for all i, j such that |i -j| ≤ 2. Then it holds for all i and j.

(1) Let (X, d) be a metric space. Given a ray γ the Busemann function

B γ : X → R is defined by B γ (x) = lim t→+∞ d(γ(t), x) -t .
Step 2. -If d ≥ 3, then SL(d, C) has property (FR). In particular if d ≥ 3, then PGL(d, C) has property (FR).

Step 3. -Let us recall that a torus T in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus T n for some integer n). Given a torus T, the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. A Cartan subgroup of an algebraic group is one of the subgroups whose Lie algebra is a Cartan subalgebra. For connected algebraic groups over C a Cartan subgroup is usually defined as the centralizer of a maximal torus.

Let C be the normalizer of the standard Cartan subgroup of PGL(3, C), i.e. the semi-direct product of the diagonal matrices by the Weyl group (of order 6). Set ς : (z 0 , z 1 ) → (1z 0 , 1z 1 ). The group generated by ς and C coincides with PGL(3, C):

C, ς = PGL(3, C).
Step 4. -Let Bir(P 2 C ) act on T so that PGL(3, C) has no fixed point and has a (unique) stably fixed end. Then Bir(P 2 C ) fixes this unique end. Step 1 implies that for all k ≥ 1, the distance between the points p 1 and (σ 2 • ς) k p 1 is exactly sk. This contradicts the fact that

Step 5. -Note that σ 2 = ς • σ 2 • ς • ς • σ 2 -1 . Since Bir(P 2 C ) = σ 2 , PGL(3, C) the groups H 1 = PGL(3, C) and H 2 = σ 2 • PGL(3, C) • σ -1 2 generate
(σ 2 • ς) 3 = id.
4.2.2. It is an amalgamated product of three groups. -In [START_REF] Wright | Two-dimensional Cremona groups acting on simplicial complexes[END_REF] the author shows that Bir(P 2 C ) = Aut C C(z 0 , z 1 ) acts on a two-dimensional simplicial complex C, which has as vertices certain models in the function field C(z 0 , z 1 ) and whose fundamental domain consists of one face F. This yields a structure description of Bir(P 2 C ) as an amalgamation of three subgroups along pairwise intersections. The subgroup Aut(A 2 C ) acts on C by restriction; more precisely the face F has an edge E satisfying the following property: the Aut(A 2 C )-translates of E form a tree T , and the action of Aut(A 2 C ) on T yields the well-known structure theory for Aut(A 2 C ) as an amalgamated product ( [START_REF] Jung | Über ganze birationale Transformationen der Ebene[END_REF]). Let us give some details. Recall that

Aut(P 1 C × P 1 C ) = PGL(2, C) × PGL(2, C) Z 2Z
and

J = PGL(2, C) PGL(2, C(z 0 )).
Proof of Theorem 4.6. -It is based on Theorem 4.5. Denote by G be the group obtained by amalgamating PGL(3, C), Aut(P 1 C × P 1 C ), J along their pairwise intersections in Bir(P 2 C ). Let τ be the involution τ : (z 0 , z 1 ) → (z 1 , z 0 ). Consider the group homomorphism α : G → Bir(P 2 C ) restricting to the identity on

PGL(3, C) ∪ Aut(P 1 C × P 1 C ) ∪ J . As im α contains J and τ ∈ Aut(P 1 C × P 1 C ) Theorem 4.5 implies that α is surjective. Since id, τ ⊂ Aut(P 1 C × P 1
C ) Theorem 4.5 gives a map β from the free product id, τ * J to G. Since Aut(P 1

C × P 1 C ) ⊂ G the equality τ • (ϕ 0 , ϕ 1 ) • τ = (ϕ 1 , ϕ 0 ) ∀ (ϕ 0 , ϕ 1 )
also holds in G. Let us now prove that in G we have (τ • ε) 3 = σ 2 where ε : (z 0 , z 1 ) → z 0 , z 0 z 1 . First note that the equality ε = ρ • σ 2 , where ρ : (z 0 , z 1 ) → 1 z 0 , z 1 z 0 , holds in J and so in G. On the one hand σ 2 and ρ commute in J so in G, on the other hand σ 2 and τ commute in

Aut(P 1 C × P 1 C ) hence in G. Therefore, one has the following equality in G (τ • ε) 3 = (τ • ρ • σ 2 ) 3 = (τ • ρ) 3 • σ 3 2 (4.
× P 1 C ) belong to im β. As G is generated by PGL(3, C) ∪ Aut(P 1 C × P 1 C ) ∪ J , β is surjective. Therefore, α is an isomorphism (α -1 = β).
The amalgamated product group structure of Theorem 4.6 reflects the fact that it acts on a simply connected two-dimensional simplicial complex. This follows from a higher dimensional analogue of Serre's tree theory (see for instance [START_REF] Soulé | Groupes opérant sur un complexe simplicial avec domaine fondamental[END_REF][START_REF] Swan | Generators and relations for certain special linear groups[END_REF]). Let us detail it.

Definitions. -A simplicial complex K is a finite collection of non-empty finite sets such that if X ∈ K and / 0 = Y ⊆ X then Y ∈ K . The union of all members of K is denoted by V (K ). The elements of V (K ) are called the vertices of K . The elements of K are called the simplices of K .

The dimension of a simplex S ∈ K is dim S = |S| -1. The dimension of K is the maximum dimension of any simplex in K .

Admissible models

A model is a reduced, irreducible, separated C-scheme having function field C(z 0 , z 1 ). Consider the set of models S satisfying one of the three properties

S P 2 C , S P 1 C × P 1 C , S P 1
k for some subfield k of C(z 0 , z 1 ) necessarily of pure transcendance degree 1 over C. Such a C-scheme S will be called an admissible model . In the first (resp. second, resp. third) case, we say that S is P 2 C (resp. S is a P 1 C × P 1 C , resp. S is a P 1 k ). The complex C It is constructed using as vertices the set of admissible models. The three models S, V and R, where S is a P 2 C , V is a P 1 C × P 1 C and R is a P 1 k , determine a face when there exist two distinct points p and q on S such that V is the P 1 C × P 1 C ( F 0 ) obtained by blowing up S at p and q, then blowing down the proper transform of the line in S containing p and q; R is the generic P 1 C obtained by blowing up S at p. If S is the standard P 2 C , p = (0 : 1 : 0) and q = (1 : 0 : 0), then V is the standard P 1 C × P 1 C , and R the standard P 1 C(z 0 ) . The standard models form a face called the standard face in C. Fundamental domain Note that from the construction of C the group Bir(P 2 C ) acts on C without inverting any edge or rotating any face. A fondamental domain for the action is given by any one face If as before we choose S to be the standard P 2 C , p = (0 : 1 : 0) et q = (1 : 0 : 0) one gets the standard face. For this choice the centralizer of S, V and R are respectively PGL(3, C),

Aut(P 1 C × P 1 C ), J .
Let us recall that two simplices S 0 and S n are k-connected if there is a sequence of simplices S 0 , S 1 , S 2 , . . ., S n such that any two consecutive ones share a k-face, i.e. they have at least k + 1 vertices in common. The complex K is k-connected if any two simplices in K of dimension ≥ k are k-connected.

Wright establishes the two following results ([Wri92]):

the simplicial complex C is 1-connected; the complex C contains the Aut(A 2 )-tree.

• τ = τ • σ 2 where τ : (z 0 : z 1 : z 2 ) → (z 1 : z 0 : z 2 ).
Blanc's proof is inspired by Iskovskikh's proof but Blanc stays on P 2 C . It is clear that σ 2 •τ = τ • σ 2 , so it suffices to prove that no other relation holds.

Blanc first establishes the following statement:

Lemma 4.24 ([Bla12]
). -Let ϕ be an element of J such that p 1 = (1 : 0 : 0), q ⊂ Base(ϕ)

where q is a proper point of

P 2 C {p 1 }. If ν ∈ Aut(P 2 C ) exchanges p 1 and q, then ψ = ν • ϕ • ν -1 belongs to J , the relation ν • ϕ -1 = ψ -1 • ν is generated by the relation σ 2 • τ = τ • σ 2
in the amalgamated product of J and Aut(P 2 C ). Let φ be an element of Aut(P 2 C ) * Aut(P 2

C )∩J J modulo the relation σ 2 • τ = τ • σ 2 . Write φ as j r • a r • j r-1 • a r-1 • . . . • j 1 • a 1
where j i ∈ J and a i ∈ Aut(P 2 C ) for i = 1, . . ., r. Note that this decomposition is of course not unique.

Let Λ 0 be the linear system of lines of P 2 C . For any i = 1, . . ., r let us denote by Λ i the linear system ( j i • a i • . . . • j 1 • a 1 )(Λ 0 ), and by d i the degree of Λ i . Set

D = max d i | i = 1, . . . , r , n = max i | d i = D , k = n ∑ i=1 (deg j i ) -1 .
Recall that j i belongs to J ⊂ Bir(P 2 C ) and satisfies the following property:

deg j i = deg j i (Λ 0 ) = deg j -1
i (Λ 0 ). In particular deg j i = 1 if and only if j i ∈ Aut(P 2 C ). Let us give an interpretation of k: the number k determines the complexity of the word

j n • a n • j n-1 • a n-1 • . . .• j 1 • a 1 which corresponds to the birational self map j i • a i • . . .• j 1 • a 1 of the highest degree.
Let us now give the strategy of the proof. If D = 1, then each j i is an automorphism of P 2

C and the word φ is equal to an element of Aut(P 2 C ) in the amalgamated product. Since Aut(P 2 C ) → Bir(P 2 C ) this eventuality is clear. Assume now that D > 1, and prove the result by induction on the pairs (D, k) (we consider the lexicographic order).

Fact. -We can suppose that

j n+1 , j n ∈ J Aut(P 2 C ), a n+1 ∈ Aut(P 2 C ) J . Remark.
-The point p = (1 : 0 : 0) is the base-point of the pencil associated to J . As a n+1 ∈ J , one has a n+1 (p) = p.

Properties of the Jonquières maps. -Since j n , j n+1 do not belong to Aut(P 2 C ), then deg

j n > 1, deg j n+1 > 1. Set D L = deg j n+1 , D R = deg j n .
The maps j n+1 and j n preserve the pencil of lines through p. Furthermore p is a base-point of j n+1 (resp. j n ) of multiplicity D L -1 (resp. D R -1). Since j ± n+1 (Λ 0 ) (resp. j ± n (Λ 0 )) is the image of the system Λ 0 it is a system of rational curves with exactly one free intersection point. The system j ± n+1 (Λ 0 ) (resp. j ± n (Λ 0 )) has 2D L -2 (resp. 2D R -2) base-points distinct from p, which all have multiplicity 1.

Set

Ω L = ( j n+1 • a n+1 ) -1 (Λ 0 ) and Ω R = ( j n • a n )(Λ 0 ).
Since the automorphisms a n+1 , a n are changes of coordinates the following properties hold:

deg Ω L = D L and 0 = a -1 n+1 (p) = p is a base-point of Ω L of multiplicity D L -1; deg Ω R = D R and r 0 = p is a base-point of Ω R of multiplicity D R -1.
The author uses these systems to compute the degrees d n+1 , resp. d n-1 of the systems

Λ n+1 = ( j n+1 • a n+1 )(Λ n ), resp. Λ n-1 = (a -1 n • j -1 n )(Λ n ).
Indeed for any i the integer d i coincides with the degree of Λ i which is on the one hand the intersection of Λ i with a general line, on the other hand the free intersection of Λ i with Λ 0 . So d n+1 (resp. d n-1 ) is the free intersection of

Λ n+1 = ( j n+1 •a n+1 )(Λ n ) (resp. Λ n-1 = (a -1 n • j -1 n )(Λ n ))
with Λ 0 but also the free intersection of Λ n with Ω L (resp. Ω R ).

Denote by m(q) the multiplicity of a point q as a base-point of Λ n . Let 1 , . . ., 2D L -2 (resp. r 1 , . . ., r 2D R -2 ) be the base-points of Ω L (resp. Ω R ). Assume that up to reindexation m( i ) ≥ m( i+1 ) (resp. m(r i ) ≥ m(r i+1 )) and if i (resp. r i ) is infinitely near to j (resp. r j ), then i > j. The following equalities hold:

         d n+1 = D L d n -(D L -1)m( 0 ) - 2D L -2 ∑ i=1 m( i ) < d n d n-1 = D R d n -(D R -1)m(r 0 ) - 2D R -2 ∑ i=1 m(r i ) < d n (4.3.1) Inequalities (4.3.1) imply m( 0 ) + m( 1 ) + m( 2 ) > d n m(r 0 ) + m(r 1 ) + m(r 2 ) ≥ d n
First case: m( 0 ) ≥ m( 1 ) and m(r 0 ) ≥ m(r 1 ). -Let q be a point in 1 , 2 , r 1 , r 2 { 0 , r 0 } with the maximal multiplicity m(q) and so that q is a proper point of P 2 C or infinitely near to 0 or r 0 .

Either 1 = r 0 , m(q) ≥ m( 2 ) and m( 0

) + m(r 0 ) + m(q) ≥ m( 0 ) + m( 1 ) + m( 2 ) > d n by (4.3.1). Or 1 = r 0 , m(q) ≥ m( 1 ) ≥ m( 2 ) hence m( 0 ) + m(q) > 2d n 3 . The inequalities m(r 0 ) ≥ m(r 1 ) ≥ m(r 2 ) imply m(r 0 ) ≥ d n
3 and then m( 0 ) + m(r 0 ) + m(q) > d n holds. The inequality m( 0 ) + m(r 0 ) + m(q) > d n implies that 0 , r 0 and q are not aligned and there exists an element θ in J of degree 2 with base points 0 , r 0 , q. Note that

deg θ(Λ n ) = 2d n -m( 0 ) -m(r 0 ) -m(q) < d n .
Let us recall that the automorphism a n+1 of P 2 C sends 0 onto r 0 = p. Take ν ∈ Aut(P 2 C ) ∩ J such that ν fixes r 0 = p and sends a n+1 (r 0 ) onto 0 . Replace a n+1 (resp. j n+1 ) by ν • a n+1 (resp. j n+1 • ν -1 ); we can thus assume that a n+1 exchanges 0 and r 0 . As a consequence according to Lemma 4.24 and modulo the relation

σ 2 • τ = τ • σ 2 j n+1 • a n+1 • j n = j n+1 • a n+1 • θ -1 • θ • j n = ( j n+1 • θ -1 ) • a n+1 • (θ • j n ) where θ = a n+1 • θ • a -1 n+1 ∈ J . Both j n+1 • θ -1 and θ • j n belong to J , but a n+1 belongs to Aut(P 2 C ). Since θ(Λ n ) = (θ • j n )(Λ n-1
) has degree < d n this rewriting decreases the pair (D, k).

Second case: m( 0 ) < m( 1 ) or m(r 0 ) < m(r 1 ). -The author comes back to the first case by changing the writing of φ in the amalgamated product and modulo the relation σ 2 • τ = τ • σ 2 without changing (D, k) but reversing the inequalities.

An other set of generators and relations for Bir(P 2 C

). -The idea of the proof of Theorem 4.9 is the same as in [START_REF] Iskovskikh | Generators and relations in a two-dimensional Cremona group[END_REF][START_REF] Iskovskikh | Proof of a theorem on relations in the two-dimensional Cremona group[END_REF][START_REF] Blanc | Simple relations in the Cremona group[END_REF]. The authors study linear systems of compositions of birational maps of the complex projective plane and use the presentation of Bir(P 2 C ) given in Theorem 4.7. Before giving the proof of Theorem 4.9 let us state the following:

Proposition 4.25 ([UZ19]). -Let φ 1 , φ 2 , . . ., φ n be some elements of PGL(3, C)∩ J . Suppose that φ n • σ 2 • φ n-1 • σ 2 • . . . • σ 2 • φ 1 = id as maps.
Then this expression is generated by relations (R 1 )-(R 5 ).

Proof of Theorem 4.9. -Let G be the group generated by σ 2 and PGL(3, C) divided by the relations

(R 1 )-(R 5 ) G = σ 2 , PGL(3, C) | (R 1 ) -(R 5 ) .
Denote by π : G → Bir(P 2 C ) the canonical homomorphism that sends generators onto generators. Proposition 4.25 asserts that sending an element of J onto its corresponding word in G is well defined. Hence there exists a homomorphism w : J → G such that

π • w = id J .
In particular w is injective.

The universal property of the amalgamated product implies that there exists a unique homomorphism

ϕ : PGL(3, C) * PGL(3,C)∩J J → G such that the following diagram commutes G J H H w G G PGL(3, C) * PGL(3,C)∩J J
* PGL(3,C)∩J J τ • σ 2 • τ • σ 2 .
This yields a homomorphism ϕ : Bir(P 2 C ) → G. More precisely the homomorphisms π : G → Bir(P 2 C ) and ϕ :

Bir(P 2 C ) → G both send generators to generators π(σ 2 ) = σ 2 and π(A) = A ∀ A ∈ PGL(3, C) ϕ(σ 2 ) = σ 2 and ϕ(A) = A ∀ A ∈ PGL(3, C)
The homomorphisms π and ϕ are thus isomorphisms that are inverse to each other.

Let us give some Lemmas and Remarks that allow to give a proof of Proposition 4.25. In [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF] the author gave a general formula for the degree of a composition of two elements of Bir(P 2 C ) but the multiplicities of the base-points of the composition is hard to compute in general. If we impose that one of the two maps has degree 2 then it is a rather straight forward computation ( [START_REF] Alberich-Carramiñana | Geometry of the plane Cremona maps[END_REF]). Denote by m p (φ) the multiplicity of φ at the point p. For any φ ∈ J of degree d one has

m (1:0:0) (φ) = d -1, m p (φ) = 1 ∀ p ∈ Base(φ) {(1 : 0 : 0)}, so according to [AC02] one has: Lemma 4.26 ([UZ19]
). -Let φ, resp. ψ be a Jonquières map of degree 2, resp. d. Let p 1 , p 2 be the base-points of φ different from (1 : 0 : 0) and q 1 , q 2 be the base-points of φ -1 different from (1 : 0 : 0) such that the pencil of lines through p i is sent by φ onto the pencil of lines through q i .

Then

deg(ψ • φ) = d + 1 -m q 1 (ψ) -m q 2 (ψ), m (1:0:0) (ψ • φ) = d -m q 1 (ψ) -m q 2 (ψ) = deg(ψ • φ) -1, m p i (ψ • φ) = 1 -m q j (ψ) if i = j.

Remark 4.27 ([UZ19]

). -These equalities can be translate as follows when Λ ψ denotes the linear system of ψ:

deg(ψ • φ) = deg(φ -1 (Λ ψ )) = d + 1 -m q 1 (Λ ψ ) -m q 2 (Λ ψ ), m (1:0:0) φ -1 (Λ ψ ) = d -m q 1 (Λ ψ ) -m q 2 (Λ ψ ) = deg φ -1 (Λ ψ ) -1, m p i φ -1 (Λ ψ ) = 1 -m q j (Λ ψ ) i = j.
But the multiplicity of Λ ψ in a point different from (1 : 0 : 0) is 0 or 1 so

   eiter deg φ -1 (Λ ψ ) = deg(Λ ψ ) + 1 and m q 1 (Λ ψ ) = m q 2 (Λ ψ ) = 0 or deg φ -1 (Λ ψ ) = deg(Λ ψ ) and m q 1 (Λ ψ ) + m q 2 (Λ ψ ) = 1 or deg φ -1 (Λ ψ ) = deg(Λ ψ ) -1 and m q 1 (Λ ψ ) = m q 2 (Λ ψ ) = 1
Furthermore Bezout theorem implies that (1 : 0 : 0) and any other base-points of ψ are not collinear; indeed (1 : 0 : 0) is a base-point of multiplicity d -1, all other base-points of multiplicity 1 (since ψ belongs to J ) and a general member of Λ ψ intersects a line in d points counted with multiplicity.

Lemma 4.28

([UZ19]). -Let φ be an element of PGL(3, C) ∩ J . Suppose that σ 2 • φ • σ 2 is linear.
Then σ 2 • φ • σ 2 is generated by the relations (R 1 ), (R 3 ) and (R 4 ).

Proof.

-By Lemma 4.26 to say that σ 2 • φ • σ 2 is linear means that Base(σ 2 • φ) = Base(σ 2 ) = (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) .

(σ 2 • ψ i • σ 2 • ψ i-1 • . . . • σ 2 • ψ 1 ) -1
does not have any infinitely near base-points for all

1 ≤ i ≤ n.
Idea of the Proof of Proposition 4.25. -Let us introduce similar notations as in the proof of Theorem 4.7. Let Λ 0 be the complete linear system of lines in P 2 C and for 1 ≤ i ≤ j let Λ i be the following linear system

Λ i := σ 2 • ϕ i-1 • σ 2 • . . . • σ 2 • ϕ 1 (Λ 0 ). Set δ i := deg Λ i , D i := max δ i | i = 1, 2, . . . , j , n := max i | δ i = D .
Consider the lexicographic order. Let us prove the result by induction on pairs of positive integers (D, n).

If D = 1, then j = 1, and there is nothing to prove. Assume now that D > 1. We can suppose that for 1 ≤ i ≤ j the map

φ i • σ 2 • φ i-1 • σ 2 • . . . • σ 2 • φ 1 -1
does not have any infinitely near base-points (Lemma 4.30). Furthermore we can do this without increasing the pair (D, n). Hence any Λ i , 1 ≤ i ≤ j, does not have any infinitely near base-points.

The maps φ i are Jonquières ones, so fix (1 : 0 : 0). The maps σ 2 • φ i and σ 2 always have (1 : 0 : 0) as common base-points. In particular deg(σ 2 • φ i • σ 2 ) ≤ 3 for any 1 ≤ i ≤ j (Lemma 4.26). Let us now deal with the three distinct cases: deg Second case: deg(σ 2 • φ n • σ 2 ) = 2. The maps σ 2 and φ n • σ 2 have exactly two common base-points, one of them being (1 : 0 : 0). One can assume that the other one is (0 : 1 : 0). More precisely there are two coordinate permutations t 1 and t 2 in S 3 ∩ J such that t 1 • φ n • t 2 fixes (1 : 0 : 0) and (0 : 1 : 0), that is

(σ 2 •φ n •σ 2 ) = 1, deg(σ 2 •φ n •σ 2 ) = 2, deg(σ 2 • φ n • σ 2 ) = 3.
t 1 • φ n • t 2 : (z 0 : z 1 : z 2 ) → (a 1 z 0 + a 2 z 2 : b 1 z 1 + b 2 z 2 : cz 2 ) for some a 1 , a 2 , b 1 , b 2 , c in C. Using (R 1 ) and (R 3 ) we get φ j • σ 2 • . . . • φ n+1 • σ 2 • t -1 1 • t 1 • φ n • t 2 • t -1 2 • σ 2 • . . . • φ 1 = φ j • σ 2 • . . . • σ 2 • (φ n+1 • t -1 1 ) • σ 2 • (t 1 • σ 2 • t 2 ) • σ 2 • (t -1 2 • φ n-1 ) • σ 2 • . . . • φ 1
The pair (D, n) is unchanged. Let us thus assume that t 1 = t 2 = id and

φ n : (z 0 : z 1 : z 2 ) → (a 1 z 0 + a 2 z 2 : b 1 z 1 + b 2 z 2 : cz 2 ).
Recall that by assumption for any 1 ≤ i ≤ n the maps

φ i • σ 2 • φ i-1 • σ 2 • . . . • σ 2 • φ 1 have
no infinitely other base-points. As a result Λ n has no infinitely near base-points.

Claim 4.31 ([UZ19]

). -The product a 2 b 2 is non-zero.

Proof. -Assume by contradiction that a 2 b 2 = 0. Then q := φ -1 n (0 : 0 : 1) is a base-point of σ 2 • φ n that lies on a line contracted by (σ 2 • φ n-1 ) -1 . By Remark 4.27 one has

D -1 = δ n+1 = D + 1 -m (0:1:0) (Λ n ) -m q (Λ n ). In particular m q (Λ n ) = 2 -m (0:1:0) (Λ n ) = 1. As q ∈ Base(σ 2 ) one has: q ∈ Base (σ 2 • φ n-1 ) -1 . Its proper image by (σ 2 • φ n-1 ) -1 is thus a base-point of Λ n-1 . But a 2 b 2 = 0;
as a result q is an infinitely near point: contradiction.

If a 2 b 2 is non-zero, then no three of the base-points of σ 2 and σ 2 • φ n are collinear. According to Lemma 4.30 there exist ψ and ϕ in PGL(3, C) such that the word σ 2 •φ n •σ 2 can be replaced by the word ψ • σ 2 • ϕ using (R 1 ), (R 3 ), (R 4 ) and (R 5 ). We thus get a new pair (D , n ) where D ≤ D;

moreover if D = D , then n < n. Third case: deg(σ 2 • φ n • σ 2 ) = 3. See [UZ19].

Let us give an application of this new presentation ([UZ19]). In

C • (A 0 , A 1 , A 2 ) = CA 0 t C, CA 1 t C, CA 2 t C . Lemma 4.32 ([UZ19]
). -The variety S(2, n) is a rational variety, and

dim S(2, n) = (n + 1)(n + 2) 2 -1.
Remark 4.33. -The variety S(2, n) has thus the same dimension as the space of plane curves of degree n.

An element A = (A 0 , A 1 , A 2 ) of PGL(3, C) induces an automorphism on (Sym(n, C)) 3 by φ(A 0 , A 1 , A 2 ) := φ 0 (A 0 , A 1 , A 2 ), φ 1 (A 0 , A 1 , A 2 ), φ 2 (A 0 , A 1 , A 2 ) .
This automorphism commutes with the action of GL(n, C); we thus obtain a regular action of PGL(3, C) on S(2, n). Theorem 4.9 allows to give a short proof of the following statement:

Proposition 4.34 ([Giz99]
). -The regular action of PGL(3, C) extends to a rational action of Bir(P 2 C ). Proof. -Define the birational action of σ 2 on S(2, n) by

(A 0 , A 1 , A 2 ) (A -1 0 , A -1 1 , A -1
2 ). According to Theorem 4.9 to see that this indeed defines a rational action of Bir(P 2 C ) on S(2, n) it is sufficient to see that (R 1 )-(R 5 ) are satisfied which is the case. 4.3.3. Why no Noether and Castelnuovo theorem in higher dimension ? -Let us give an idea of the proof of the fact that there is no Noether and Castelnuovo theorem in higher dimension: C . First introduce some notations: let

P ∈ C[z 0 , z 1 , . . . , z n ] d , Q ∈ C[z 0 , z 1 , . . . , z n ] and R 0 , R 1 , . . ., R n-1 ∈ C[z 0 , z 1 , . . . , z n ] d-
be some homogeneous polynomials of degree d, resp. , resp. d -. Consider ψ P,Q,R and ψ R the rational maps given by ψ P,Q,R : (z 0 : z 1 : . . . : z n ) (QR 0 : QR 1 : . . . : QR n-1 : P), ψ R : (z 0 : z 1 : . . . : z n ) (R 0 : R 1 : . . . : R n-1 ).

Lemma 4.36 ([Pan99]

). -Let d and be some integers such that d

≤ + 1 ≤ 2. Take Q ∈ C[z 0 , z 1 , . . . , z n ] and P ∈ C[z 0 , z 1 , . . . , z n ] d without common factors. Let R 1 , R 2 , . . ., R n be some elements of C[z 0 , z 1 , . . . , z n-1 ] d-. Assume that P = z n P d-1 + P d Q = z n Q -1 + Q with P d-1 , P d , Q -1 , Q ∈ C[z 0 , z 1 , . . . , z n-1 ] of degree d -1, resp. d, resp. -1, resp. and such that (P d-1 , Q -1 ) = (0, 0). The map ψ P,Q,R is birational if and only if ψ R is.
This statement allows to prove that given a hypersurface of P n C one can construct a birational self map of P n C that blows down this hypersurface: Lemma 4.37 [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]). -Let n ≥ 3. Let S be an hypersurface of P n C of degree ≥ 1 having a point p of multiplicity ≥ -1.

Then there exists a birational self map of P n C of degree d ≥ + 1 that blows down S onto a point.

Proof. -Let us assume without loss of generality that p = (0 : 0 : . . . : 0 : 1). Suppose that S is given by (Q = 0). Take a generic plane passing through p given by (H = 0). Choose P = z n P d-1 + P d such that P d-1 ∈ C[z 0 , z 1 , . . . , z n-1 ] of degree d -1 and = 0;

P d ∈ C[z 0 , z 1 , . . . , z n-1 ] of degree d; pgcd(P, HQ) = 1.
Set Q = H d--1 q and R i = z i . The statement then follows from Lemma 4.36.

Proof of Theorem 4.35. -Consider the family of hypersurfaces given by Q(z 1 , z 2 , z 3 ) = 0

where

(Q = 0) defines a smooth curve C Q of degree on z 0 = z 4 = z 5 = . . . = z n = 0 . Note that (Q = 0) is birationally equivalent to P n-2 C × C Q . Furthermore (Q = 0
) and (Q = 0) are birationally equivalent if and only if C Q and C Q are isomorphic. Take = 2; the set of isomorphism classes of smooth cubics is a 1-parameter family. For any C Q there exists a birational self map of P n C that blows down C Q onto a point (Lemma 4.37). As a result any set of group generators of Bir(P n C ), n ≥ 3, has to contain uncountably many elements of Bir(P n C ) PGL(n + 1, C). As we have seen one consequence of Noether and Castelnuovo theorem is that the Jonquières group and Aut(P 2 C ) = PGL(3, C) generate Bir(P 2 C ). This statement does also not hold in higher dimension ( [BLZar]): let n ≥ 3, the n-dimensional Cremona group is not generated by Aut(P n C ) and by Jonquières elements, i.e. elements that preserve a family of lines through a given point, which form a subgroup PGL(2, C(z 2 , z 3 , . . . , z n )) Bir(P n-1 C ) ⊆ Bir(P n C ). A more precise statement has been established in dimension 3 in [BYar]: the 3-dimensional Cremona group is not generated by birational maps preserving a linear fibration

P 3 C P 2 C .
CHAPTER 5

ALGEBRAIC PROPERTIES OF THE CREMONA GROUP

The group Bir(P 2 C ) has many properties of linear groups, so we wonder if Bir(P 2 C ) has a faithful linear representation; in the first section we show that the answer is no ( [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF][START_REF] Cornulier | Nonlinearity of some subgroups of the planar Cremona group[END_REF]). Still in the first section we give the proof of the following property: the plane Cremona group contains non-linear finitely generated subgroups ( [Cor]).

In the second section we give the proof of the facts that the normal subgroup generated by σ 2 in Bir(P 2 C ) is Bir(P 2 C ). the normal subgroup, generated by a non-trivial element of PGL(3, C) = Aut(P 2 C ) in Bir(P 2 C ) is Bir(P 2 C ). As a consequence Bir(P 2 C ) is perfect ([CD13]), that is [Bir(P 2 C ), Bir(P 2 C )] = Bir(P 2 C ). We finish this chapter by the description of the endomorphisms of the plane Cremona group; as a corollary we get the Theorem 5.1 ([D 07a]). -The plane Cremona group is hopfian, i.e. any surjective endomorphism of Bir(P 2 C ) is an automorphism. We use for that the classification of the representations of SL(3, Z) in Bir(P 2 C ), we thus recall and establish it in the third section:

Theorem 5.2 ([D 06a]).
-Let Γ be a finite index subgroup of SL(3, Z). Let υ be an injective morphism from Γ to Bir(P 2 C ). Then, up to birational conjugacy, either υ is the canonical embedding, or υ is the involution A → ( t A) -1 .

As a result we obtain the:

. From [A, B] = C we get [A |E α , B |E α ] = C |E α ; but C |E α = αid |E α hence [A |E α , B |E α ] = αid |E α , that is (B -1 AB) |E α = αA |E α . Note that (B -1 AB) |E α and A |E α are conju- gate thus (B -1 AB) |E α and A |E α
have the same eigenvalues. Furthermore these eigenvalues are non-zero. If λ is an eigenvalue of A |E α , then αλ, α 2 λ, . . ., α p-1 λ are also eigenvalues of A |E α . As p is prime and α distinct from 1, the numbers α, α 2 , . . ., α p-1 are distinct, dim E α ≥ p, and n ≥ p.

Proof of Proposition 5.4. -Assume by contradiction that there exists an injective morphism ζ from Bir(P 2 C ) into GL(n, k). For any prime p let us consider in the affine chart z 2 = 1 the group generated by the maps (z 0 , z 1 ) → (e -2iπ/p z 0 , z 1 ), (z 0 , z 1 ) (z 0 , z 0 z 1 ), (z 0 , z 1 ) → (z 0 , e -2iπ/p z 1 ).

The images of these three elements of Bir(P 2 C ) satisfy the assumptions of Birkhoff Lemma; therefore, p ≤ n for any prime p: contradiction.

In [Cor] Cornulier gives an example of a non-linear finitely generated subgroup of the plane Cremona group. The existence of such subgroup is not new, for instance it follows from an unpublished construction of Cantat. The example in [Cor] has the additional feature of being 3-solvable. To prove its non-linearity Cornulier proves that it contains nilpotent subgroups of arbitrary large nilpotency length.

Let G be a group. Recall that [g, h] = g • h • g -1 • h -1 denotes the commutator of g and h.

If H 1 and H 2 are two subgroups of G, then [H 1 , H 2 ] is the subgroup of G generated by the elements of the form [g, h] with g ∈ H 1 and h ∈ H 2 . We defined the derived series of G by setting G (0) = G and for all n ≥ 0

G (n+1) = [G (n) , G (n) ].
The soluble length (G) of G is defined by

(G) = min k ∈ N ∪ {0} | G (k) = {id} with the convention: min / 0 = ∞. We say that G is solvable if (G) < ∞.
The descending central series of a group G is defined by C 0 G = G and for all n ≥ 0

C n+1 G = [G,C n G].
The group G is nilpotent if there exists j ≥ 0 such that C j G = {id}. If j is the minimum non-negative number with such a property, we say that G is of nilpotent class j.

Take f in C(z 0 ) and g in C(z 0 ) * ; define α f and µ g by

α f : (z 0 , z 1 ) z 0 , z 1 + f (z 0 ) , µ g : (z 0 , z 1 ) z 0 , z 1 g(z 0 ) .
Note that

α f + f = α f • α f µ gg = µ g • µ g µ g • α f • µ -1 g = α f g (5.1.1)
Take t ∈ C and consider s t : (z 0 , z 1 ) → (z 0 + t, z 1 ). The following equalities hold

s t • α f (z 0 ) • s -1 t = α f (z 0 -t) , s t • µ g(z 0 ) • s -1 t = µ g(z 0 -t) (5.1.2)
Let Γ n be the subgroup of Bir(P 2 C ) defined for any n ≥ 0 by

Γ n = s 1 , α z n 0 .
Remark that Γ n is indeed a subgroup of the Jonquières group. It satisfies the following properties:

Lemma 5.7 ( [Cor]). -The nilpotency length of Γ n is exactly n + 1, and Γ n is torsion free.

Proof. -Let A n be the abelian subgroup of the Jonquières group consisting of all α P where P ranges over polynomials of degree at most n. The group A n is normalized by s 1 , and [s 1 , A n ] ⊂ A n-1 for n ≥ 1 while A 0 = {id}. Therefore, the largest group generated by s 1 and A n is nilpotent of class at most n + 1, and so is Γ n .

Consider now the n-iterated group commutator given by

[s 1 , [s 1 , . . . , [s 1 , α z n 0 ] . . .] It coincides with α ∆ n z n
0 where ∆ is the discrete differential operator ∆P(z 0 ) = -P(z 0 ) + P(z 0 -1). Remark that ∆ n z n 0 = 0 and Γ n is not n-nilpotent. Clearly Γ n is torsion-free.

The group

G = s 1 , α 1 , µ z 0 ⊂ Bir(P 2
Q ) satisfies the following properties: Proposition 5.8 ( [Cor]). -The group G ⊂ Bir(P 2 Q ) is solvable of length 3, and is not linear over any field.

A consequence of this statement is Proposition 5.4.

Proof. -Relations (5.1.1) and (5.1.2) imply that s 1 , α f , µ g | f ∈ C(z 0 ), g ∈ C(z 0 ) * is solvable of length at most three. The subgroup

s 1 , α f , µ g | f ∈ C(z 0 ), g = ∏ n∈Z (z 0 -n) k n , k n finitely supported contains Γ n ,
and is torsion free.

As

µ n z 0 • α 1 • µ -n z 0 = α z n 0 , the group Γ n is contained in G for all n.
But Γ n is nilpotent of length exactly n + 1, hence G has no linear representation over any field.

The Cremona group is perfect

In this section let us prove the following statement Theorem 5.9 ( [START_REF] Cerveau | Transformations birationnelles de petit degré[END_REF]). -The plane Cremona group is perfect, i.e. the commutator subgroup of Bir(P 2 C ) is Bir(P 2 C ):

Bir(P 2 C ), Bir(P 2 C ) = Bir(P 2 C ).
Let G be a group, and let g be an element of G. We denote by g G the normal subgroup of G generated by g:

g G = h • g • h -1 , h • g -1 • h -1 | h ∈ G . Since PGL(3, C) is simple then A PGL(3,C) = PGL(3, C) (5.2.1)
for any non-trivial element A of PGL(3, C). Consider now a birational self map φ of Bir(P 2 C ). The Noether and Castelnuovo Theorem implies that

φ = (A 1 ) • σ 2 • A 2 • σ 2 • A 3 • . . . • A n • (σ 2 ) (5.2.2)
with A i ∈ PGL(3, C). The relation (5.2.1) implies that

(z 0 , z 1 ) → (-z 0 , -z 1 ) PGL(3,C) = PGL(3, C);
thus any A i in (5.2.2) can be written

B 1 • (z 0 , z 1 ) → (-z 0 , -z 1 ) • B -1 1 • B 2 • (z 0 , z 1 ) → (-z 0 , -z 1 ) • B -1 2 • . . . • B n • (z 0 , z 1 ) → (-z 0 , -z 1 ) • B -1 n with B i ∈ PGL(3, C).
The involutions (z 0 , z 1 ) → (-z 0 , -z 1 ) and σ 2 being conjugate via Consider now a non-trivial automorphism A of P 2 C . As A PGL(3,C) = PGL(3, C) (see (5.2.1)) the involution (z 0 , z 1 ) → (-z 0 , -z 1 ) can be written as a composition of PGL(3, C)-conjugates of A. Since (z 0 , z 1 ) → (-z 0 , -z 1 ) and σ 2 are conjugate via (z 0 , z 1 ) → According to (5.2.2) and Proposition 5.11 one has Corollary 5.12. -Any birational self map of P 2 C can be written as the composition of Bir(P 2 C )-conjugates of the translation (z 0 , z 1 ) → (z 0 , z 1 + 1).

(z 0 , z 1 ) → z 0 +1 z 0 -1 , z 1 +1 z 1 -1 ∈ PGL(2, C) × PGL(2,
z 0 +1 z 0 -1 , z 1 +1 z 1 -1 ∈ PGL(2, C) × PGL(2, C) one gets σ 2 = ϕ 1 • A • ϕ -1 1 • ϕ 2 • A • ϕ -1 2 • . . . • ϕ n • A • ϕ -1 n with ϕ i ∈ Bir(P 2 C ).
But the translation (z 0 , z 1 ) → (z 0 , z 1 + 1) is a commutator

(z 0 , z 1 ) → (z 0 , z 1 + 1) = (z 0 , z 1 ) → (z 0 , 3z 1 ), (z 0 , z 1 ) → z 0 , z 1 + 1 2
and Corollary 5.12 thus implies Theorem 5.9.

5.3. Representations of SL(n, Z) into Bir(P 2 C ) for n ≥ 3 We will now give a sketch of the proofs of Theorem 5.2 and Corollary 5.3. Let us introduce some notations. Given A ∈ Aut(P 2 C ) = PGL(3, C) we denote by t A the linear transpose of A. The involution

A → A ∨ = ( t A) -1
determines an exterior and algebraic automorphism of the group Aut(P 2 C ) (see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]). Let us recall some properties about the groups SL(n, Z) (see for instance [START_REF] Steinberg | Some consequences of the elementary relations in SL n[END_REF]). For any integer q let us introduce the morphism Θ q : SL(n, Z) → SL n, Z qZ induced by the reduction modulo q morphism Z → Z qZ . Denote by Γ(n, q) the kernel of Θ q and by Γ(n, q) the reciprocical image of the subgroup of diagonal matrices of SL n, Z qZ by Θ q . The Γ(n, q) are normal subgroups called congruence subgroups.

Theorem 5.13 ([Ste85]

). -Let n ≥ 3 be an integer. Let Γ be a subgroup of SL(n, Z).

If Γ has finite index, there exists an integer q such that the following inclusions hold

Γ(n, q) ⊂ Γ ⊂ Γ(n, q).
If Γ has infinite index, then Γ is finite.

Take 1 ≤ i, j ≤ n, i = j. Let us denote by δ i j the n×n Kronecker matrix and set e i j = id+δ i, j .

Proposition 5.14 [START_REF] Steinberg | Some consequences of the elementary relations in SL n[END_REF]). -The group SL(3, Z) has the following presentation

e i j | [e i j , e k ] =    id if i = and j = k e i if i = and j = k e -1 k j if i = and j = k
, (e 12 e -1 21 e 12 ) 4 = id .

Remark 5.15.

-The e q i j 's generate Γ(3, q) and satisfy relations similar to those verified by the e i j 's except (e 12 e -1 21 e 12 ) 4 = id. The e q i j 's are called the standard generators of Γ(3, q). Definition. -Let k be an integer. A k-Heisenberg group is a group with the following presentation

H k = f , g, h | [ f , g] = h k , [ f , h] = [g, h] = id .
We will say that f , g and h are the standard generators of H k .

Remarks 5.16. -

The subgroup of H k generated by f , g and h k is a subgroup of index k.

The groups Γ(3, q) contain a lot of k-Heisenberg groups; for instance if 1 ≤ i = j = ≤ 3, then e q i j , e q i , e q j is a q-Heisenberg group of Γ(3, q). Let G be a finitely generated group, let a 1 , a 2 , . . . , a n be a generating set of G, and let g be an element of G. The length ||g|| of g is the smallest integer k for which there exists a sequence (s 1 , s 2 , . . . , s k ) with s i ∈ a 1 , a 2 , . . . , a n , a -1 1 , a -1 2 , . . . , a -1 n for any 1 ≤ i ≤ k, such that g = s 1 s 2 . . . s k .

We say that lim k→+∞ ||g k || k is the stable length of g. A distorted element of G is an element of infinite order of G whose stable length is zero.

Lemma 5.17

([D 06a]). -Let H k = f , g, h | [ f , g] = h k , [ f , h] = [g, h] = id be a k-Heisenberg group.
The element h k is distorted.

In particular the standard generators of Γ(3, q) are distorted.

Proof.

-Since [ f , g] = [g, h] = id on has [ f , g m ] = h m for any integer , m. In particular

h k 2 = [ f , g ]. As a result ||h k 2 || ≤ 4 .
Each standard generator of Γ(3, q) satisfies e q 2 i j = [e q i , e q j ].

Lemma 5.18 ([D 07b]

). -Let G be a finitely generated group. Let υ be a morphism from G to Bir(P 2 C ). Any distorted element g of G satisfies λ(υ(g)) = 1, i.e. υ(g) is an elliptic map or a parabolic one.

Proof. -Let a 1 , a 2 , . . . , a n be a generating set of G. The inequalities

λ(υ(g)) k ≤ deg(υ(g) k ) ≤ max i deg(υ(a i )) ||g k || imply the following ones 0 ≤ log λ(υ(g)) ≤ ||g k || k log max i deg(υ(a i )) .
But since g is distorted lim

k→+∞ ||g k || k = 0 and log λ(υ(g)) = 0.
Remark 5.19. -We follow the proof of [D 06a]; nevertheless it is possible to "simplify it" by using the following result: any distorted element of Bir(P 2 C ) is algebraic ([BF19, CdC20]). According to Corollary 3.34 we thus have: any distorted element of Bir(P 2 C ) is elliptic.

Definition. -Let φ 1 , φ 2 , . . ., φ k be some birational self maps of a rational surface S. Assume that φ 1 , φ 2 , . . ., φ k are virtually isotopic to the identity. We say that φ 1 , φ 2 , . . ., φ k are simultaneously virtually isotopic to the identity if there exists a surface S, a birational map ψ : S S such that for any 1 ≤ i ≤ k the map ψ -1 • φ i • ψ belongs to Aut( S) and ψ -1 • φ i • ψ belongs to Aut( S) 0 for some integer .

Proposition 5.20 ([D 06a]

). -Let υ be a representation from

H k = f , g, h | [ f , g] = h k , [ f , h] = [g, h] = id into Bir(P 2 C ).
Assume that any standard generator υ( f ), υ(g) and υ(h) of υ(H k ) is virtually isotopic to the identity. Then υ( f ), υ(g) and υ(h) are simultaneously virtually isotopic to the identity.

Proof. -According to Proposition 2.12 the maps υ( f ) and υ(g) are simultaneously virtually isotopic to the identity. Since g and h commute, Exc(υ(g)) and Ind(υ(g)) are invariant by υ(h). The relation [ f , g] = h k implies that both Exc(υ(g)) and Ind(υ(g)) are invariant by υ( f ). A reasoning analogous to that of the proof of Proposition 2.12 and [DF01, Lemma 4.1] allows us to establish the statement.

The second assertion of Remarks 5.16 leads us to study the representations of Heisenberg k-groups into automorphisms groups of minimal rational surfaces. Let us deal with it.

REPRESENTATIONS OF SL(n, Z) INTO Bir(P 2

C ) FOR n ≥ 3 105

Lemma 5.21 ([D 06a]). -Let υ be a morphism from H k into Aut(P 1 C × P 1 C ). The morphism υ is not an embedding.

Proof. -We can assume that υ( f ), υ(g) and υ(h) fixe the two standard fibrations (if it is not the case we can consider H 2k instead of H k ); in other words we can assume that im υ is contained in PGL(2, C) × PGL(2, C). Denote by pr i , i ∈ {1, 2}, the i-th projection. Note that pr i (υ(H 2k )) is a solvable subgroup of PGL(2, C). Furthermore pr i (υ(h k )) is a commutator. Hence pr i (υ(h k )) is conjugate to the translation z → z + β i . Let us prove by contradiction that β i = 0; assume β i = 0. Then both pr i (υ( f )) and pr i (υ(g)) are also some translation since they commute with pr i (υ(h k )). But then pr i (υ(h k )) = [pr i (υ( f )), pr i (υ(g))] = id: contradiction with β i = 0. As a result β i = 0 and υ is not an embedding.

Lemma 5.22 ([D 06a]

). -Let υ be a morphism from H k into Aut(F n ) with n ≥ 1.

Then up to birational conjugacy υ(H k ) is a subgroup of (z 0 , z 1 ) → (αz 0 + P(z 1 ),

βz 1 + γ) | α, β ∈ C * , γ ∈ C, P ∈ C[z 1 ] .
Moreover up to birational conjugacy υ(h 2k ) : (z 0 , z 1 ) → (z 0 + P(z 1 ), z 1 )

for some P ∈ C[z 1 ].
(z 0 , z 1 ) → (z 0 + ζz 1 , z 1 + β) υ(g) : (z 0 , z 1 ) → (z 0 + γz 1 , z 1 + δ) υ(h k ) : (z 0 , z 1 ) → (z 0 + k, z 1 )
where ζ, δ, β γ denote complex numbers such that ζδβγ = k.

Proof. -The Zariski closure υ(H k ) of υ(H k ) is an algebraic unipotent subgroup of PGL(3, C). By assumption υ is an embedding, so the Lie algebra of υ(H k ) is isomorphic to

h =      0 ζ β 0 0 γ 0 0 0   | ζ, β, γ ∈ C    .
Let pr be the canonical projection from SL(3, C) into PGL(3, C). The Lie algebra of pr -1 (υ(H k )) coincides with h up to conjugacy. Let us recall that the exponential map sends h in the group H of upper triangular matrices and that H is a connected algebraic group. As a consequence pr -1 (υ(H k )) 0 = H. Any element of pr -1 (υ(H k )) acts by conjugation on H, so belongs to H, j • id | j 3 = 1 . As pr(j • id) = id, the restriction pr |H of pr to H is surjective on υ(H k ). It is also injective. Hence it is an isomorphism. Therefore, υ can be lifted to a representation υ from H k into H. The map υ(h k ) can be written as a commutator; it is thus unipotent. The relations satisfied by the generators imply that up to conjugacy in SL(3, C)

υ( f ) : (z 0 , z 1 ) → (z 0 + ζz 1 , z 1 + β) υ(g) : (z 0 , z 1 ) → (z 0 + γz 1 , z 1 + δ) υ(h k ) : (z 0 , z 1 ) → (z 0 + k, z 1 )
with ζδβγ = k.

Let ρ be an embedding of Γ(3, q) into Bir(P 2 C ). According to Lemma 5.17 and Lemma 5.18 for any standard generator e i j of SL(3, Z) one has λ(ρ(e i j )) = 1. Theorem 2.9 implies that (i) either one of the ρ(e q i j ) preserves a unique fibration that is rational or elliptic, (ii) or any standard generator of Γ(3, q) is virtually isotopic to the identity.

Let us first assume that (i) holds.

Lemma 5.24 ([D 06a]

). -Let Γ be a Kazhdan group that is finitely generated. Let ρ be a morphism from Γ into PGL(2, C(z 1 )) (resp. PGL(2, C)). Then ρ has finite image.

Proof. -Denote by γ i the generators of Γ and by

a i (z 1 ) b i (z 1 ) c i (z 1 ) d i (z 1 ) their image by ρ. A finitely generated Q-group is isomorphic to a subfield of C. Hence Q(a i (z 0 ), b i (z 0 ), c i (z 0 ), d i (z 0 ))
is isomorphic to a subfield of C and one can assume that im ρ ⊂ PGL(2, C) = Isom(H 3 ). As Γ is Kazhdan any continuous action of Γ by isometries of a real or complex hyperbolic space has a fixed point. The image of ρ is thus up to conjugacy a subgroup of SO(3, R); according to [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger)[END_REF] the image of ρ is thus finite.

Proposition 5.25 ([D 06a]

). -Let ρ be a morphism from Γ(3, q) to Bir(P 2 C ). If one ρ(e q i j ) preserves a unique fibration, then im ρ is finite.

Proof. -Let us assume without loss of generality that ρ(e q 12 ) preserves a unique fibration F . The relations satisfied by the e q i j imply that F is invariant by any ρ(e q 2 i j ). Hence for any ρ(e q 2 i j ) there exist

F : P 2 C → Aut(P 1 C ) that defines F , and
h i j ∈ PGL(2, C) such that F • ρ(e q 2 i j ) = h i j • F. Let υ be the morphism defined by υ : Γ(3, q 2 ) → PGL(2, C), e q 2
i j → h i j . The group Γ(3, q 2 ) is a Kazhdan group, so Γ = ker υ is of finite index (Lemma 5.24); as a consequence Γ is a Kazhdan group.

REPRESENTATIONS OF SL(n, Z) INTO Bir(P 2

C ) FOR n ≥ 3 107

Remark that F can not be elliptic; indeed the group of birational maps that preserve fiberwise an elliptic fibration is metabelian and a subgroup of Γ(3, q 2 ) of finite index can not be metabelian.

Let us assume that F is a rational fibration. One can assume that F = (z 1 = constant). The group of birational maps of the complex projective plane that preserves F is identified with PGL(2, C(z 1 )) PGL(2, C) hence ρ |Γ : Γ → PGL(2, C(z 1 )) has finite image (Lemma 5.24).

Consider now the case (ii), i.e. assume that any standard generator of Γ(3, q) is virtually isotopic to the identity.

Remark 5.26. -Two irreducible homologous curves of negative self-intersection coincide. As a consequence an automorphism ϕ of a surface S isotopic to the identity fixes any curve of negative self-intersection. Furthermore for any sequence of blow-downs ψ from S to a minimal model S of S the map ψ • ϕ • ψ -1 is an automorphism of S isotopic to the identity.

According to Remark 5.26, Proposition 5.20, Lemma 5.17 and Lemma 5. ) belong, for some integer n, to

PGL(3, C) = Aut(P 2 C ), then ρ(Γ(3, q 2 n 2 )) is a subgroup of PGL(3, C) = Aut(P 2 C ).
To establish this statement we will need the two following results; the first one was obtained by Cantat and Lamy when they study the embeddings of lattices from simple Lie groups into the group of polynomial automorphisms Aut(A 2 C ) whereas the second one is a technical one.

Theorem 5.29 ([CL06]

). -Let G be a simple real Lie group. Let Γ be a lattice of G. If there exists an embedding of Γ into Aut(A 2 C ), then G is isomorphic to either PSO(1, n) or PSU(1, n) for some integer n. ). Finally any ρ(e

q 2 n 2 i j ) is a polynomial automorphism of A 2
C and ρ is not an embedding (Theorem 5.29). Assume that βδ = 0. Since ζδβγ = q 2 n 2 one has (β, δ) = (0, 0).

Suppose that β = 0. The conjugacy by

(z 0 , z 1 ) → z 0 + γ 2 z 1 - γ 2δ z 2 1 ,
(z 0 , z 1 ) → z 0 1 + ζz 1 + b z 1 1 + ζz 1 , z 1 1 + ζz 1 .
Thanks to [ρ(e Assume that δ = 0; using a similar reasoning we get a contradiction.

Proof of Theorem 5.2. -Any ρ(e i j ) is virtually isotopic to the identity (Lemma 5.18 and Proposition 5.25). The maps ρ(e n 12 ), ρ(e n 13 ) and ρ(e n 23 ) are, for some integer n, conjugate to automorphisms of a minimal rational surface (Proposition 5.20 and Remark 5.16). Up to conjugacy one can assume that ρ(Γ(3, n 2 )) ⊂ PGL(3, C) (Lemmas 5.21, 5.27 and 5.28). The restriction ρ |Γ(3,n 2 ) of ρ to Γ(3, n 2 ) can be extended to an endomorphism of PGL(3, C) (see [START_REF] Steinberg | Some consequences of the elementary relations in SL n[END_REF]). But PGL(3, C) is simple, so this extension is both injective and surjective. The automorphisms of PGL(3, C) are obtained from inner automorphisms, automorphisms of the field C and the involution u → u ∨ (see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF] Chapter IV]). But automorphisms of the field C do not act on Γ(3, n 2 ); hence up to linear conjugacy ρ |Γ(3,n 2 ) coincides with the identity or the involution u → u ∨ .

Let φ be an element of ρ(SL(3, Z)) ρ(Γ(3, n 2 )) that blows down at least one curve C . The group Γ(3, n 2 ) is a normal subgroup of Γ. As a consequence C is invariant by ρ(Γ(3, n 2 )), and so by ρ(Γ(3, n 2 )) = PGL(3, C) which is impossible. Finally φ does not blow down any curve, and ρ(SL(3, Z)) ⊂ PGL(3, C).

Proof of Corollary 5.3. -Let Γ be a subgroup of finite index of SL(4, Z), and let ρ be a morphism from Γ into the plane Cremona group. We will prove that im ρ is finite. To simplify let us suppose that Γ = SL(4, Z). Denote by e i j the standard generators of SL(4, Z). The morphism ρ induces a faithful representation ρ from SL(3, Z) into Bir(P 2 C ):

SL(4, Z) ⊃ SL(3, Z) 0 0 1 ρ → Bir(P 2 C )
According to Theorem 5.2 the map ρ is, up to birational conjugacy, the identity or the involution u → u ∨ . Let us first assume that up to birational conjugacy ρ = id. Assume that Exc(ρ(e 34 )) = / 0. Since [e 34 , e 31 ] = [e 34 , e 32 ] = id the map ρ(e 34 ) commutes with

(z 0 , z 1 , z 2 ) → (z 0 , z 1 , az 0 + bz 1 + z 2 )
where a, b ∈ C and Exc(ρ(e 34 )) is invariant by (z 0 , z 1 , z 2 ) → (z 0 , z 1 , az 0 +bz 1 +z 2 ). Moreover e 34 commutes with e 12 and e 21 , in other words e 34 commutes with the following copy of SL(2, Z)

SL(4, Z) ⊃   SL(2, Z) 0 0 0 1 0 0 0 1  
The action of SL(2, Z) on C 2 has no invariant curve, so Exc(ρ(e 34 )) is contained in the line at infinity. But the image of this line by (z 0 , z 1 , z 2 ) → (z 0 , z 1 , az 0 + bz 1 + z 2 ) intersects C 2 : contradiction. Hence Exc(ρ(e 34 )) = / 0 and ρ(e 34 ) belongs to PGL(3, C). Similarly we get that ρ(e 43 ) belongs to PGL(3, C). The relations satisfied by the standard generators thus imply that ρ(SL(4, Z)) is contained in PGL(3, C). As a consequence im ρ is finite.

A similar idea allows to conclude when ρ is, up to conjugacy, the involution u → u ∨ . Let n ≥ 4 be an integer. Consider a subgroup of finite index Γ of SL(n, Z). Let ρ be a morphism from Γ to Bir(P 2 C ). According to Theorem 5.13 the group Γ contains a congruence subgroup Γ(n, q). The morphism ρ induces a representation ρ from Γ(4, q) to Bir(P 2 C ). As we just see the kernel of this representation is infinite so does ker ρ.

The group Bir(P 2 C

) is hopfian Let V be a projective variety defined over a field k ⊂ C. The group Aut k (C) of automorphisms of the field extension C k acts on V (C), and on Bir(V ) as follows

κ ψ(p) = (κ • ψ • κ -1 )(p) (5.4.1)
for any κ ∈ Aut k (C), any ψ ∈ Bir(V ), and any point p ∈ V (C) for which both sides of (5.4.1) are well defined. As a consequence Aut k (C) acts by automorphisms on Bir(V ). If κ : C → C is a field morphism, then this construction gives an injective morphism Aut(P n C ) → Aut(P n C ), g → κ g.

THE GROUP Bir(P 2 C ) IS HOPFIAN 111

Write C as the algebraic closure of a purely transcendental extension Q(x i , i ∈ I) of Q; if f : I → I is an injective map, then there exists a field morphism

κ : C → C, x i → x f (i) .
Such a morphism is surjective if and only if f is onto. The group Aut(Bir(P 2 C )) has been described in [D 06b] and [D 06a] via two different methods:

Theorem 5.31 ([D 06b, D 06a]). -Let ϕ be an element of Aut(Bir(P 2 C )). Then there exist a birational self map ψ of P 2

C and an automorphism κ of the field C such that

ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ Bir(P 2
C ) The proof of [D 06b] will be deal with in §7.2. The proof of [D 06a] can in fact be used to describe the endomorphisms of the plane Cremona group: Theorem 5.32 ([D 07a]). -Let ϕ be a non-trivial endomorphism of Bir(P 2 C ). Then there exist ψ in Bir(P 2 C ) and an immersion κ of the field C such that

ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ Bir(P 2 C ) Let us work in the affine chart z 2 = 1. The group of translations is T = (z 0 , z 1 ) → (z 0 + α, z 1 + β) | α, β ∈ C .

Lemma 5.33 ([D 07a]

). -Let ϕ be a birational self map of P 2 C . Assume that ϕ commutes with both (z 0 , z 1 ) → (z 0 + 1, z 1 ) and (z 0 , z 1 ) → (z 0 , z 1 + 1).

Then ϕ belongs to T.

Proof. -Let ϕ = (ϕ 0 , ϕ 1 ) be an element of Bir(P 2 C ) that commutes with both (z 0 , z 1 ) → (z 0 + 1, z 1 ) and (z 0 , z 1 ) → (z 0 , z 1 + 1). In particular

ϕ 0 (z 0 + 1, z 1 ) = ϕ 0 (z 0 , z 1 ) + 1 ϕ 1 (z 0 + 1, z 1 ) = ϕ 1 (z 0 , z 1 ) From ϕ 1 (z 0 + 1, z 1 ) = ϕ 1 (z 0 , z 1 ) we get that ϕ 1 = ϕ 1 (z 1 ). The equality ϕ 0 (z 0 + 1, z 1 ) = ϕ 0 (z 0 , z 1 ) + 1 implies ∂ϕ 0 ∂z 0 (z 0 + 1, z 1 ) = ∂ϕ 0 ∂z 0 (z 0 , z 1 );
as a consequence ∂ϕ 0 ∂z 0 = a(z 1 ) and ϕ 0 = a(z 1 )z 0 + b(z 1 ) for some a, b in C(z 1 ). Then ϕ 0 (z 0 + 1, z 1 ) = ϕ 0 (z 0 , z 1 ) + 1 yields a(z 1 ) = 1. In other words ϕ : (z 0 , z 1 ) (z 0 + b(z 1 ), ϕ 1 (z 1 )).

Let us now write that ϕ • (z 0 , z 1 + 1) : (z 0 , z 1 ) (z 0 , z 1 + 1) • ϕ; we get that ϕ : (z 0 , z 1 ) (ϕ 0 (z 0 ), z 1 + c(z 0 )).

Finally ϕ : (z 0 , z 1 ) (z 0 + b(z 1 ), ϕ 1 (z 1 )) and ϕ : (z 0 , z 1 ) (ϕ 0 (z 0 ), z 1 + c(z 0 )) imply that ϕ belongs to T.

Proof of

: (z 0 , z 1 ) → z 0 z 0 -1 , z 0 -z 1 z 0 -1 .
According to [START_REF] Kh | Defining relations for the Cremona group of the plane[END_REF] one has ( • σ 2 ) 3 = id. As a result ϕ(( • σ 2 ) 3 ) = id. Since ϕ( ) = (recall that belongs to PGL(3, C)) one gets that ϕ(σ 2 ) = id. As the plane Cremona group is generated by PGL(3, C) and σ 2 one gets that ϕ = id.

Assume now that ϕ |PGL(3,C) is injective. According to Theorem 5.2 the restriction ϕ |SL(3,Z) of ϕ to SL(3, Z) is, up to inner conjugacy, the canonical embedding or A → A ∨ . Suppose first that ϕ |SL(3,Z) is the canonical embedding. Denote by U the group of unipotent upper triangular matrices. Set

f β = ϕ(z 0 + β, z 1 ), g α = ϕ(z 0 + α, z 1 ), h γ = ϕ(z 0 , z 1 + γ).
Since f β and h γ commute to both (z 0 , z 1 ) → (z 0 + 1, z 1 ) and (z 0 , z 1 ) → (z 0 , z 1 + 1) one gets from Lemma 5.33 that

f β : (z 0 , z 1 ) → (z 0 + λ(β), z 1 + ζ(β)) h γ : (z 0 , z 1 ) → (z 0 + η(γ), z 1 + µ(γ))
where λ, ζ, η and µ are additive morphisms from C to C. As g γ commutes with (z 0 , z 1 ) → (z 0 + z 1 , z 1 ) and (z 0 , z 1 ) → (z 0 + 1, z 1 ) there exists a α in C(y) such that g γ : (z 0 , z 1 ) → (z 0 + a α (z 1 ), z 1 ).

The equality

(z 0 + αz 1 , z 1 ) • (z 0 , z 1 + γ) • (z 0 + αz 1 , z 1 ) -1 • (z 0 , z 1 + γ) -1 = (z 0 + αz 1 , z 1 ) implies that g α • h α = f αγ • h γ • g α for any α, γ in C. As a consequence f β : (z 0 , z 1 ) → (z 0 + λ(β), z 1 ) g α : (z 0 , z 1 ) → (z 0 + θ(α)z 1 + ζ(α), z 1 )
and θ(α)µ(α) = λ(αγ). From (z 0 , z 1 ) → (z 0 + α, z 1 ) , (z 0 , z 1 ) → (z 0 , z 1 + βz 0 ) = (z 0 , z 1 ) → (z 0 , z 1α)

THE GROUP Bir(P 2 C ) IS HOPFIAN 113

one gets h γ : (z 0 , z 1 ) → (z 0 , z 1 + µ(γ)). In other words for any α, β ∈ C one has

ϕ(z 0 + α, z 1 + β) = f α • h β = (z 0 , z 1 ) → (z 0 + λ(α), z 1 + µ(β)).
Therefore, ϕ(T) ⊂ T and ϕ(U) ⊂ U. To finish let us assume for instance that ϕ |PGL(3,C) comes, up to inner conjugacy, from the composition of A → A ∨ and an embedding of the field

C into itself. Set (η 1 , η 2 ) = ϕ (z 0 , z 1 ) → z 0 , 1 z 1 . From (z 0 , z 1 ) z 0 , 1 z 1 •((z 0 , z 1 ) → (αz 0 , βz 1 ))• (z 0 , z 1 ) z 0 , 1 z 1 = (z 0 , z 1 ) → αz 0 , z 1 β one gets η 1 λ(α -1 )z 0 , λ(β -1 )z 1 = λ(α -1 )η 1 (z 0 , z 1 ) η 2 λ(α -1 )z 0 , λ(β -1 )z 1 = λ(β)η 2 (z 0 , z 1 ) Hence ϕ (z 0 , z 1 ) → z 0 , 1 z 1 = (z 0 , z 1 ) → ±z 0 , ± 1 z 1 . But (z 0 , z 1 ) → (z 1 , z 0 ) • (z 0 , z 1 ) → z 0 , 1 z 1 2 = σ 2 , so ϕ(σ 2 ) = ±σ 2 . Furthermore ϕ( ) = (z 0 , z 1 ) → (-z 0 -z 1 -1, z 1 ) as ϕ |SL(3,Z) coincides with A → A ∨ . Then the second component of ϕ • σ 2 3 is ± 1 z 1 : contradiction with ϕ • σ 2 3 = id.
If ϕ |PGL(3,C) comes, up to inner conjugacy, from an embedding of C similar computations imply that ϕ(σ 2 ) = σ 2 and one concludes with Noether and Castelnuovo theorem.

CHAPTER 6 FINITE SUBGROUPS OF THE CREMONA GROUP

The classification of finite subgroups of Bir(P 1 C ) = PGL(2, C) is well known and goes back to Klein. It consists of cyclic, dihedral, tetrahedral, octahedral and icosahedral groups. Groups of the same type and same order constitute a unique conjugacy class in Bir(P 1 C ). What about the two-dimensional case, i.e. what about the finite subgroups of Bir(P 2 C ) ? The story starts a long time ago with Bertini ([Ber77]) who classified conjugacy classes of subgroups of order 2 in Bir(P 2 C ). Already the answer is drastically different from the onedimensional case. The set of conjugacy classes is parameterized (see Theorem 6.3) by a disconnected algebraic variety whose connected components are respectively isomorphic to either the moduli spaces of hyperelliptic curves of genus g, or the moduli space of canonical curves of genus 3, or the moduli space of canonical curves of genus 4 with vanishing theta characteristic.

Bertini's proof is considered to be incomplete; a complete and short proof was published only a few years ago by Bayle and Beauville ([BB00]).

In 1894 Castelnuovo proved that any element of Bir(P 2 C ) of finite order leaves invariant either a net of lines, or a pencil of lines, or a linear system of cubic curves with n ≤ 8 basepoints ( [START_REF] Castelnuovo | Le trasformationi generatrici del gruppo cremoniano nel piano[END_REF]). Kantor announced a similar result for arbitrary finite subgroups of Bir(P 2 C ); his proof relies on a classification of possible groups in each case ( [START_REF] Kantor | Theorie der endlichen Gruppen von eindeutigen Transformationen in der ebene[END_REF]). Unfortunately Kantor's classification, even with some corrections made by Wiman ([Wim96]), is incomplete in the following sense:

given some abstract finite group, it is not possible using their list to say whether this group is isomorphic to a subgroup of Bir(P 2 C ); the possible conjugation between the groups of the list is not considered.

The Russian school has made great progress since the 1960's: Manin and Iskovskikh classified the minimal G-surfaces into automorphisms of del Pezzo surfaces and of conic bundles ([Man67, Isk79]). Many years after people come back to this problem. As we already mention Bayle and Beauville classified groups of order 2. It is the first example of a precise description of conjugacy classes; it is shown that the non-rational curves fixed by the groups determine the conjugacy classes. Groups of prime order were also studied ([BB04, dF04, Zha01]). Zhang applies Bayle and Beauville strategy to the case of birational automorphisms of prime order p ≥ 3. It turns out that nonlinear automorphisms occur only for p = 3 and p = 5; the author describes them explicitly. The techniques of [START_REF] Bayle | Birational involutions of P 2[END_REF] are also generalized by de Fernex to cyclic subgroups of prime order ([dF04]). The list is as precise as one can wish, except for two classes of groups of order 5: the question of their conjugacy is not answered. Beauville and Blanc completed this classification ([BB04]); they prove in particular that a birational self map of the complex projective plane of prime order is not conjugate to a linear automorphism if and only if it fixes some non-rational curve. Beauville classified p-elementary groups ([Bea07]). Blanc classified all finite cyclic groups ([Bla07a]), and all finite abelian groups ( [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF]). The goal of [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF] is to update the list of Kantor and Wiman. The authors used the modern theory of G-surfaces, the theory of elementary links, and the conjugacy classes of Weyl groups.

In the first section we recall the definitions of Geiser involutions, Bertini involutions and Jonquières involutions. We give a sketch of the proof of the classification of birational involutions of the complex projective plane due to Bayle and Beauville.

In the second section we deal with finite abelian subgroups of the plane Cremona group. Results due to Dolgachev and Iskovskikh are recalled.

In the last section we state some results of Blanc about finite cyclic subgroups of Bir(P 2 C ), isomorphism classes of finite abelian subgroups of Bir(P 2 C ) but also a generalization of a theorem of Castelnuovo which states that an element of finite order which fixes a curve of geometric genus > 1 has order 2, 3 or 4.

6.1. Classification of subgroups of order 2 of Bir(P 2 C ) 6.1.1. Geiser involutions. -Let p 1 , p 2 , . . ., p 7 be seven points of the complex projective plane in general position. Denote by L the linear system of cubics through the p i 's. The linear system L of cubic curves through the p i 's is two-dimensional. Take a general point p, and consider the pencil of curves from L passing through p. A general pencil of cubic curves has nine base-points; let us define I G (p) as the ninth base-point of the pencil. The map I G is a

Geiser involution ([Gei67]

). The algebraic degree of a Geiser involution is equal to 8.

One can also see a Geiser involution as follows. The linear system L defines a rational map of degree 2, ψ :

P 2 C |L| * P 2 C .
The points p and I G (p) lie in the same fibre. As a consequence I G is a birational deck map of this cover. If we blow up p 1 , p 2 , . . ., p 7 we get a del Pezzo surface S of degree 2 and a regular map of degree 2 from S to P 2 C . Furthermore the Geiser involution becomes an automorphism of S.

Note that the fixed points of I G lie on the ramification curve of ψ. It is a curve of degree 6 with double points p 1 , p 2 , . . ., p 7 and is birationally isomorphic to a canonical curve of genus 3.

A third way to see Geiser involutions is the following. Let S be a del Pezzo surface of degree 2. The linear system | -K S | defines a double covering S → P 2 C , branched along a smooth quartic curve ([DPT80]). The involution ι which exchanges the two sheets of this covering is called a Geiser involution; it satisfies

Pic(S) ι ⊗ Q Pic(P 2 C ) ⊗ Q = Q.
The exceptional locus of a Geiser involution is the union of seven cubics passing through the seven points of indeterminacy of I G and singular at one of these seven points.

Bertini involutions. -Let us fix in P 2

C eight points p 1 , p 2 , . . ., p 8 in general position. Consider the pencil of cubic curves through these points. It has a ninth base-point p 9 . For any general point p there is a unique cubic curve C (p) of the pencil passing through p. Take p 9 as the zero of the group law of the cubic C (p); define I B (p) as the negative -p with respect to the group law. The map I B is a birational involution called Bertini involution ( [START_REF] Bertini | Ricerche sulle trasformazioni univoche involutorie nel piano. Annali di Mat[END_REF]).

The algebraic degree of a Bertini involution is equal to 17. The fixed points of a Bertini involution lie on a canonical curve of genus 4 with vanishing theta characteristic isomorphic to a nonsingular intersection of a cubic surface and a quadratic cone in P 3 C . Another way to see a Bertini involution is the following. Consider a del Pezzo surface S of degree 1. The map S → P 3 C defined by the linear system | -2K S | induces a degree 2 morphism of S onto a quadratic cone Q ⊂ P 3 C , branched along the vertex of Q and a smooth genus 4 curve ( [START_REF] Demazure | Séminaire sur les Singularités des Surfaces[END_REF]). The corresponding involution, the Bertini involution, satisfies rk Pic(S) I B = 1. 6.1.3. Jonquières involutions. -Let C be an irreducible curve of degree ν ≥ 3. Assume that C has a unique singular point p and that p is an ordinary multiple point with multiplicity ν -2. To (C , p) we associate a birational involution I J that fixes pointwise C and preserves lines through p. Let m be a generic point of P 2 C C . Let r m , q m and p be the intersections of the line (mp) and C . The point I J (m) is the point such that the cross ratio of m, I J (m), q m and r m is equal to -1. The map I J is a Jonquières involution of degree ν centered at p; it preserves C . More precisely its fixed points are the curve C of genus ν -2 as soon as ν ≥ 3.

If ν = 2, then C is a smooth conic ; the same construction can be done by choosing a point p that does not lie on C .

Lemma 6.1 ([DI09]

). -Let G be a finite subgroup of Bir(P 2 C ). Let C 1 , C 2 , . . ., C k be nonrational irreducible curves on P 2 C such that each of them contains an open subset C 0 i whose points are fixed under all g ∈ G.

Then the set of birational isomorphism classes of the curves C i is an invariant of the conjugacy class of G in Bir(P 2 C ). Proof. -Assume that G = ψ • H • ψ -1 for some subgroup H of Bir(P 2 C ) and some birational self map ψ of the complex projective plane. Replacing C 0 i by a smaller open subset if needed we assume that ψ -1 (C 0 i ) is defined and consists of fixed points of H. As C i is not rational, ψ -1 (C 0 i ) is not a point. Its Zariski closure is thus a rational irreducible curve C i birationally isomorphic to C i that contains an open subset of fixed points of H. Corollary 6.2. -Jonquières involutions of degree ≥ 3 are not conjugate to each other, not conjugate to projective involutions, not conjugate to Bertini involutions, not conjugate to Geiser involutions.

Bertini involutions are not conjugate to Geiser involutions, not conjugate to projective involutions.

Geiser involutions are not conjugate to projective involutions.

Proof. -The statement follows from Lemma 6.1 and the above properties: a connected component of the fixed locus of a projective map is a line or a point; the fixed points of a Geiser involution lie on a curve birationally isomorphic to a canonical curve of genus 3; the fixed points of a Bertini involution lie on a canonical curve of genus 4 with vanishing theta characteristic; the set of fixed points of a Jonquières involution of degree ν ≥ 3 outside the base locus is an hyperelliptic curve of degree ν -2.

We can thus introduce the following definition.

Definition. -An involution is of Jonquières type if it is birationally conjugate to a Jonquières involution.

An involution is of Bertini type if it is birationally conjugate to a Bertini involution. An involution is of Geiser type if it is birationally conjugate to a Geiser involution.

The classification of subgroups of Bir(P 2 C ) of order 2 is given by the following statement:

Theorem 6.3 ([BB00]
). -A non-trivial birational involution of the complex projective plane is conjugate to one and only one of the following:

a Jonquières involution of a given degree ν ≥ 2; a Geiser involution; a Bertini involution.

More precisely the parameterization of each conjugacy class is known. Before stating it let us give some definitions.

Remarks 6.4. -Let S, S be two rational surfaces and ι ∈ Bir(S), ι ∈ Bir(S ) be two involutions. They are birationally equivalent if there exists a birational map ϕ : S S such that ϕ • ι = ι • ϕ. Note that in particular two involutions of Bir(P 2 C ) are equivalent if and only if they are conjugate in Bir(P 2 C ). Assume that ι fixes a curve C. Then ι = ϕ • ι • ϕ -1 fixes the proper transform of C under ϕ which is a curve birational to C except possibly if C is rational; indeed, if C is rational it may be contracted to a point. The normalized fixed curve of ι is the union of the normalizations of the non-rational curves fixed by ι. This is an invariant of the birational equivalence class of ι.

Proposition 6.5 ([BB00]

). -The map which associates to a birational involution of P 2 C its normalized fixed curve establishes a one-to-one correspondence between conjugacy classes of Jonquières involutions of degree ν and isomorphism classes of hyperelliptic curves of genus ν -2 (ν ≥ 3); conjugacy classes of Geiser involutions and isomorphism classes of non-hyperelliptic curves of genus 3; conjugacy classes of Bertini involutions and isomorphism classes of non-hyperelliptic curves of genus 4 whose canonical model lies on a singular quadric.

Jonquières involutions of degree 2 form one conjugacy class.

The approach of Bayle and Beauville is different from the approach of Castelnuovo. It is based on the following observation: any birational involution of P 2 C is conjugate, via an appropriate birational isomorphism S ∼ P 2 C to a biregular involution ι of a rational surface S. Therefore, the authors are reduced to the birational classification of the pairs (S, ι). In [START_REF] Ju | Rational surfaces over perfect fields[END_REF] Manin classified the pairs (S, G) where S is a surface and G a finite group. This question has been simplified by the introduction of Mori theory. This theory allows Bayle and Beauville to show that the minimal pairs (S, ι) fall into two categories, those which admit a ι-invariant base-point free pencil of rational curves, and those with rk Pic(S) ι = 1. The first case leads to the so-called Jonquières involutions whereas the second one leads to the Geiser and Bertini involutions.

Let us now give some details. By a surface we mean a smooth, projective, connected surface over C. We consider pairs (S, ι) where S is a rational surface and ι a non-trivial biregular involution of S. Recall that the pair (S, ι) is minimal if any birational morphism ψ : S → S such that there exists a biregular involution ι of S with ψ • ι = ι • ψ is an isomorphism. Lemma 6.6 ([BB00]). -The pair (S, ι) is minimal if and only if for any exceptional curve (1) E on S the following hold:

ι(E) = E E ∩ ι(E) = / 0.
Proof. -Suppose that (S, ι) is not minimal. Then there exist a pair (S , ι ) and a birational morphism ψ : S → S such that ψ • ι = ι • ψ and ψ contracts some exceptional curve E. Then ψ contracts the divisor E + ι(E). Therefore, (E + ι(E)) 2 ≤ 0, and so

E • ι(E) ≤ 0, i.e. ι(E) = E or E ∩ ι(E) = / 0.
Conversely assume that there exists an exceptional curve E on S such that ι(E) = E (resp. E ∩ ι(E) = / 0). Let S be the surface obtained by blowing down E (resp. E ∪ ι(E)). Then ι induces an involution ι of S so that (S, ι) is not minimal.

The only piece of Mori theory used by Bayle and Beauville is the following one: Lemma 6.7 ([BB00]). -Let (S, ι) be a minimal pair with rk Pic(S) ι > 1. Then S admits a base-point free pencil stable under ι.

It allows them to establish the: Theorem 6.8 ([BB00]). -Let (S, ι) be a minimal pair. One of the following holds:

(1) there exists a smooth P 1 C -fibration f : S → P 1 C and a non-trivial involution

I of P 1 C such that f • ι = I • f ;
(2) there exists a fibration f : S → P 1 C such that f • ι = f , the smooth fibres of f are rational curves on which ι induces a non-trivial involution, any singular fibre is the union of two rational curves exchanged by ι, meeting at one point;

(3) S is isomorphic to

P 2 C ; (4) (S, ι) is isomorphic to P 1 C × P 1 C with the involution (z 0 , z 1 ) → (z 1 , z 0 ); (
5) S is a del Pezzo surface of degree 2 and ι is the Geiser involution; (6) S is a del Pezzo surface of degree 1 and ι is the Bertini involution.

Proof. -Assume rk Pic(S) ι = 1. As Pic(S) ι contains an ample class, -K S is ample, i.e. S is a del Pezzo surface. If rk Pic(S) = 1, then one obtains case (3).

(1) Recall that an exceptional curve E on a surface S is a smooth rational curve with E 2 = -1.

CLASSIFICATION OF SUBGROUPS OF ORDER 2 OF Bir(P 2

C ) 121

If rk Pic(S) > 1, then -ι is the orthogonal reflection with respect to K ⊥ S . Such a reflection is of the form

x → x -2 (α • x) (α • α) α with (α • α) ∈ {1, 2} and K S proportional to α. If K S is divisible, then S is isomorphic to P 1 C × P 1
C and since ι must act non-trivially on Pic(S) we get case (4). The only remaining eventualities are K 2 S ∈ {1, 2}. The Geiser and Bertini involutions have the required properties ( §6.1.1, §6.1.2). An automorphism ϕ of S acting trivially on Pic(S) is the identity; indeed S is the blow up of P 2 C at 9d points in general position, ϕ induces an automorphism of P 2 C which must fix these points. Hence Geiser and Bertini involutions are the only ones to have the required properties. Suppose now that rk Pic(S) ι > 1. According to Lemma 6.7 the surface S admits a ιinvariant pencil |F| of rational curves. This defines a fibration f : S → P 1 C with fibre F, and an involution

I of P 1 C such that f • ι = I • f .
If f is smooth, then this gives (1) or a particular case of (2). If f is not smooth, let F 0 be a singular fibre of f . It contains an exceptional divisor

E. Since (S, ι) is minimal, then ι(E) = E and E • ι(E) ≥ 1. As a result (E + ι(E)) 2 ≥ 0, so F 0 = E + ι(E) and E • ι(E) = 1. Set p = E ∩ ι(E).
The involution induced by ι on T p S exchanges the directions of E and ι(E); it thus has eigenvalues 1 and -1. As a consequence ι fixes a curve passing through p; this curve must be horizontal and I trivial.

Furthermore the fixed curve of ι being smooth, the involution induced by ι on a smooth fibre cannot be trivial. We get case (2).

Bayle and Beauville precised which pairs in the list of Theorem 6.8 are indeed minimal ([BB00, Proposition 1.7]).

Let us now give the link between biregular involutions of rational surfaces and birational involutions of the complex projective plane: Lemma 6.9 ([BB00]). -Let ι be a birational involution of a surface S 1 . There exists a birational morphism ϕ : S → S 1 and a biregular involution I of S such that ϕ • I = ι • ϕ.

To prove it we need some results, let us state and prove them. Theorem 6.10 (see for instance [START_REF] Beauville | Complex algebraic surfaces[END_REF], Theorem II.7). -Let S be a surface, and let X be a projective variety. Let φ : S X be a rational map. Then there exist a surface S , a morphism η : S → S which is the composition of a finite number of blow-ups,

a morphism ψ : S → X such that S ψ 1 1 η Ð Ð S φ G G X commutes.
Proof. -As X lies in some projective space we may assume that X = P m C . Furthermore we can suppose that φ(S) lies in no hypersurface of P m C . As a result φ corresponds to a linear system P ⊂ |D| of dimension m on S without fixed component.

If P has no base-point, then φ is a morphism and there is nothing to do.

Assume that P has at least one base-point p. Consider the blow up ε : Bl p S → S at p. Set S 1 = Bl p S. The exceptional curve E occurs in the fixed part of the linear system ε * P ⊂ |ε * D| with some multiplicity k ≥ 1; that is, the system P 1 = |ε * P -kE| ⊂ |ε * D -kE| has no fixed component. It thus defines a rational map

φ 1 = φ • ε : S 1 P m C .
If φ 1 is a morphism, then the result is proved. If not, we repeat the "same step". We get by induction a sequence ε n : S n → S n-1 of blow ups and a linear system

P n ⊂ |D n | = |ε * n D n-1 -k n E n | on S n with no fixed part. On the one hand D 2 n = D 2 n-1 -k 2 n < D 2 n-1
; on the other hand P n has no fixed part, so D 2 n ≥ 0 for any n. Consequently the process must end. More precisely after a finite number of blow ups we obtain a system P n with no base-points which defines a morphism ψ : S n → P m C as required. Lemma 6.11 (see for instance [START_REF] Beauville | Complex algebraic surfaces[END_REF]). -Let S be an irreducible surface. Let S be a smooth surface. Let φ : S → S be a birational morphism. Assume that the rational map φ -1 is not defined at a point p ∈ S .

Then φ -1 (p) is a curve on S.

Proof. -We assume that S is affine so that there is an embedding j : S → A n C . The rational map

j • φ -1 : S A n C
is defined by rational functions g 1 , g 2 , . . . g n . One of them, say for instance g 1 is undefined at p, that is g 1 ∈ O S ,p . Set g 1 = u v with u, v ∈ O S ,p , u and v coprime and v(p) = 0. Consider the curve D on S given by φ * v = 0. On S ⊂ A n C denote by z 0 the first coordinate function. We have

φ * u = z 0 φ * v on S. Hence φ * u = φ * v = 0 on D. Consequently D = φ -1 (Z) where Z = u = v = 0 ⊂ S .

CLASSIFICATION OF SUBGROUPS OF ORDER 2 OF Bir(P 2 C ) 123

By assumption u and v are coprime, so Z is finite. Shrinking S if necessary we can assume that Z = {p}. Finally D = φ -1 (p).

Lemma 6.12 (see for instance [START_REF] Beauville | Complex algebraic surfaces[END_REF]). -Let S, S be two surfaces. Let φ : S S be a birational map such that φ -1 is not defined at p ∈ S .

Then there exists a curve C on S such that φ(C) = {p}.

Proof.

-The map φ corresponds to a morphism ψ : U → S for some subset U of S. Denote by

Γ = u, ψ(u) | u ∈ U ⊂ U × S
the graph of ψ. Let Γ be the closure of Γ in S × S ; it is an irreducible surface, possibly with singularities. The projections pr 1 : Γ → S, pr 2 : Γ → S are birational morphisms and the diagram

Γ pr 1 Ð Ð pr 2 0 0 S φ G G S is commutative.
By assumption φ -1 is not defined at p ∈ S , so does pr -1 2 . There is an irreducible curve C on Γ such that pr 2 (C ) = {p} (Lemma 6.11). As Γ ⊂ S × S the image pr 1 (C ) of C by pr 1 is a curve C in S such that φ(C) = {p}. Proposition 6.13 (see for instance [START_REF] Lamy | Une preuve géométrique du théorème de Jung[END_REF]). -Let X and S be two surfaces. Let φ : X → S be a birational morphism of surfaces. Suppose that the rational map φ -1 is not defined at a point p of S.

Then

Bl p S ε 3 3 X ψ a a φ G G S
where ψ : X → Bl p S is a birational map and ε : Bl p S → S is the blow up at p.

Proof.

-Set ψ = ε -1 • φ. Suppose that ψ is not a morphism, and let m be a point of X such that ψ is not defined at m. On the one hand φ(m) = p and φ is not locally invertible at m; on the other hand there exists a curve in Bl p S blown down onto m by ψ -1 (Lemma 6.12). This curve has to be the exceptional divisor E associated to ε. Let r and q be two distinct points of E at which ψ -1 is well defined; consider C, C two germs of smooth curves transverse to E at r and q respectively. Then ε(C) and ε(C ) are two germs of smooth curves transverse at p, which are images by φ of two germs of curves at m. The differential of φ at m has thus rank 2: contradiction with the fact that φ is not invertible at m.

Proof of Lemma 6.9. -There exists a birational morphism ϕ : S → S 1 such that the rational map ψ = ι • ϕ is everywhere defined (Theorem 6.10). Furthermore ϕ can be written as

ϕ = ε n-1 • ε n-2 • . . . • ε 1
where ε i : S i+1 → S i , 1 ≤ i ≤ n -1, is obtained by blowing up a point p i ∈ S i and S = S n . The map ι is not defined at p 1 , so ψ -1 = ϕ -1 • ι is not defined at p 1 . Proposition 6.13 implies that ψ factors as

S 2 ε 1 1 1 S ψ G G g 1 d d S 1
Proceeding by induction we see that ψ factors as ϕ • I where I is a birational morphism. Since ϕ • I 2 = ϕ, the map I is an involution.

In other words Lemma 6.9 says that any birational involution of a surface is birationally equivalent to a biregular involution ι : S → S; furthermore (S, ι) can be assumed to be minimal. Therefore, the classification of conjugacy classes of involutions in Bir(P 2 C ) is equivalent to the classification of minimal pairs (S, ι) up to birational equivalence. Remark 6.14. -Recall that the P 1 C -bundles over P 1 C are of the form

F n = P P 1 C O P 1 C ⊕ O P 1 C (n)
for some integers n ≥ 0 (see §3.3.2).

For n ≥ 1 the fibration f :

F n → P 1 C
has a unique section of self-intersection -n. Consider a fibre F of f , and a point p of F. Assume that ι is a birational involution of F n regular in a neighborhood of F and fixing p.

After the elementary transformation at p we get a birational involution of F n+1 regular in a neighborhood of the new fibre.

Proof of Theorem 6.3. -The unicity assertion follows from Remark 6.4. Using Lemma 6.9 we will prove that the involutions of Theorem 6.8 are birationally equivalent to one of Theorem 6.3.

Cases (5) and (6) give by definition the Geiser and Bertini involutions.

An involution of type (4) is birationally equivalent to a Jonquières involution of degree 2. Indeed let Q be a smooth conic in P 2 C , and let p ∈ P 2 C Q be a point. Consider the birational involution ι of P 2 C that maps a point x to its harmonic conjugate on the line (px) through p and x with respect to the two points of (px) ∩ Q. This involution is not defined at the following three points: p and the two points q and r where the tangent line to Q passes through p. Set S = Bl p,q,r P 2 C . The involution ι extends to a biregular involution I of S, the Jonquières involution of degree 2.

In case (3) take a point p ∈ P 2 C such that ι(p) = p. Let us blow up p, ι(p) and then blow down the proper transform of the line (pι(p)) which is a ι-invariant exceptional curve. We get a pair (T, ι ) with T P 1 C × P 1 C by stereographic projection and rk Pic(T ) ι = 1: we are thus in case (4), so in the case of a Jonquières involution of degree 2.

Let us now deal with case (1). The surface S is isomorphic to F n for some n ≥ 0. The involution ι has two invariant fibres, any of them containing at least two fixed points. One of these points does not belong to s n (section of self-intersection -n on F n ), hence after a (finite) sequence of elementary transformations we get n = 1. Let us thus focus on the case n = 1. Let F 1 be the surface obtained by blowing up a point p ∈ P 2 C . Projecting from p defines a P 1 -bundle f : F 1 → P 1 C . Any biregular involution ι of F 1 preserves this fibration hence defines a pair (F 1 , ι) of case (1) or (2). The involution ι preserves the unique exceptional curve E 1 of F 1 ; the pair (F 1 , ι) is thus not minimal: ι induces a biregular involution of P 2 C . We finally get a Jonquières involution of degree 2 as we just see.

We now consider case (2). Let us distinguish two possibilities: denote by F 1 , F 2 , . . ., F s the singular fibres of f and by p i , 1 ≤ i ≤ s, the singular point of F i . The fixed locus of ι is a smooth curve C passing through p 1 , p 2 , . . ., p s . The degree 2 covering C → P 1 C induced by f is ramified at p 1 , p 2 , . . ., p s .

(2a) Either f is smooth, s = 0 and C is the union of two sections of f which do not intersect; (2b) or f is not smooth, C is a hyperelliptic curve of genus g ≥ 0 and s = 2g + 2.

First assume that we are in case (2a). After elementary transformations we can suppose that S = F 1 . The fixed locus of ι is the union of E 1 and a section which does not meet E 1 . Blowing down E 1 one gets case (4).

Finally let us look at case (2b) for g ≥ 0. Let us blow down one of the components in each singular fibre. We thus have a birational involution on a surface F n , the fixed curve C embedded into F n . After elementary transformations at general points of C one gets a birational involution on a surface F 1 , the fixed curve C embedded into F 1 . The genus formula implies that E 1 •C = g. Suppose that C is tangent to E 1 at some point q ∈ F 1 . After an elementary transformation at q then an elementary transformation at some general point of C the order of contact of C and E 1 at q decreases by 1. Proceeding in this way we arrive at the following situation: E 1 and C meet transversally at g distinct points. Let blow down E 1 to a point p of P 2 C ; the curve C maps to a plane curve C of degree g+2 with an ordinary multiple point of multiplicity g at p and no other singularity. This yields to a birational involution of P 2 C which preserves the lines through p and admits C as fixed curve, i.e. a Jonquières involution with center p and fixed curve C.

Finite abelian subgroups of the Cremona group

Dolgachev and Iskovskikh used a modern approach to the problem initiated in the works of Manin and Iskovskikh who gave a clear understanding of the conjugacy problem via the concept of a G-surface ( [START_REF] Ju | Rational surfaces over perfect fields[END_REF][START_REF] Iskovskih | Minimal models of rational surfaces over arbitrary fields[END_REF]). Let G be a finite group. A G-surface is a pair (S, ψ) where S is a nonsingular projective surface and ψ is an isomorphism from G to Aut(S). A morphism of the pairs (S, ψ) → (S , ψ ) is defined to be a morphism of surfaces φ : S → S such that

ψ (G) = φ • ψ(G) • φ -1 .
In particular let us note that two subgroups of Aut(S) define isomorphic G-surfaces if and only if they are conjugate inside Aut(S).

Let (S, ψ) be a rational G-surface. Take a birational map φ : S P 2 C . For any g ∈ G the map φ • g • φ -1 belongs to Bir(P 2 C ). This yields to an injective homomorphism ι φ : G → Bir(P 2 C ). In other words a birational isomorphism class of G-surfaces defines a conjugacy class of subgroups of Bir(P 2 C ) isomorphic to G. The following result shows that any conjugacy class is obtained in this way: Lemma 6.16 ( [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]). -Let G be a finite subgroup of Bir(P 2 C ). There exist a rational Gsurface (S, ψ) and a birational map φ : Hence one has:

S P 2 C such that G = φ • ψ(G) • φ -1 .
Theorem 6.17

([DI09]

). -There is a natural bijective correspondence between birational isomorphism classes of rational G-surfaces and conjugate classes of subgroups of Bir(P 2 C ) isomorphic to G.

Therefore, the goal of Dolgachev and Iskovskikh is to classify G-surfaces up to birational isomorphism of G-surfaces.

There is a G-equivariant analogue of minimal surfaces:

Definition. -A minimal G-surface is a G-surface (S, ψ) such that any birational morphism of G-surfaces (S, ψ) → (S , ψ ) is an isomorphism.

Note that it is enough to classify minimal rational G-surfaces up to birational isomorphism of G-surfaces. The authors can rely on the following fundamental result: Theorem 6.18. -Let S be a minimal rational G-surface. Then either S admits a structure of a conic bundle with Pic(S) G Z 2 ; or S is isomorphic to a del Pezzo surface with Pic(S) G Z.

An analogous result from the classical literature is showed by using the method of the termination of adjoints, first introduced for linear system of plane curves in the work of Castelnuovo. This method is applied to find a G-invariant linear system of curves in the plane in [START_REF] Kantor | Theorie der endlichen Gruppen von eindeutigen Transformationen in der ebene[END_REF];

Kantor essentially stated the result above but without the concept of minimality. A first modern proof can be found in [START_REF] Ju | Rational surfaces over perfect fields[END_REF] and [START_REF] Iskovskih | Minimal models of rational surfaces over arbitrary fields[END_REF]. Nowadays Theorem 6.18 follows from a G-equivariant version of Mori theory ([dF04]).

As a result to complete the classification Dolgachev and Iskovskikh need (i) to classify all finite groups G that may occur in a minimal G-pair;

(ii) to determine when two minimal G-surfaces are birationally isomorphic.

To achieve (i) the authors computed the full automorphisms group of a conic bundle surface on a del Pezzo surface and then made a list of all finite subgroups acting minimally on the surface.

To achieve (ii) the authors used the ideas of Mori theory to decompose a birational map of rational G-surfaces into elementary links.

Finite cyclic subgroups of Bir(P 2

C ) In [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF] the author gave the list of finite cyclic subgroups of the plane Cremona group, up to conjugation. The curves fixed by one element of the group, and the action of the whole group on these curves, are often sufficient to distinguish the conjugacy classes. It was done in [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF] in many cases, but some remain unsolved. In [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF] the author completed this classification with the case of abelian non-cyclic groups.

Its classification implies several results we will now mention.

Theorem 6.19 ([Bla07a]

). -For any integer n ≥ 1 there are infinitely many conjugacy classes of elements of Bir(P 2 C ) of order 2n, that are non-conjugate to a linear automorphism. If n > 15, a birational map of P 2 C of order 2n is a n-th root of a Jonquières involution and preserves a pencil of rational curves.

If an element of Bir(P 2

C ) is of finite odd order and is not conjugate to a linear automorphism of P 2 C , then its order is 3, 5, 9 or 15. In particular any birational map of P 2 C of odd order > 15 is conjugate to a linear automorphism of the plane.

Then Blanc generalized a theorem of Castelnuovo which states that an element of finite order which fixes a curve of geometric genus > 1 has order 2, 3 or 4 (see [START_REF] Castelnuovo | Le trasformationi generatrici del gruppo cremoniano nel piano[END_REF]): Theorem 6.20 [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF]). -Let G be a finite abelian group which fixes some curve of positive geometric genus.

Then G is cyclic, of order 2, 3, 4, 5 or 6, and all these cases occur.

If the curve has geometric genus > 1, then G is of order 2 or 3.

Theorem 6.21 ([Bla07a]

). -Let G be a finite abelian subgroup of Bir(P 2 C ).

The following assertions are equivalent:

any g ∈ G id does not fix a curve of positive geometric genus; the group G is birationally conjugate to a subgroup of Aut(P 2 C ), or to a subgroup of Aut(P 1 C × P 1 C ), or to the group isomorphic to Z 2Z × Z 4Z generated by the two following elements

(z 0 : z 1 : z 2 ) → z 1 z 2 : z 0 z 1 : -z 0 z 2 , (z 0 : z 1 : z 2 ) → z 1 z 2 (z 1 -z 2 ) : z 0 z 2 (z 1 + z 2 ) : z 0 z 1 (z 1 + z 2 ) .
Furthermore this last group is conjugate neither to a subgroup of Aut(P 2 C ), nor to a subgroup of Aut(P 1

C × P 1 C ).
In [Bea07] Beauville gave the isomorphism classes of p-elementary subgroups of the plane Cremona group. Blanc generalized it as follows: Theorem 6.22 [START_REF] Blanc | Finite abelian subgroups of the Cremona group of the plane[END_REF]). -The isomorphism classes of finite abelian subgroups of the plane Cremona group are the following:

Z mZ × Z nZ for any integers m, n ≥ 1, Z 2nZ × Z 2Z 2 for any integer n ≥ 1, Z 4Z 2 × Z 2Z , Z 3Z 3 , Z 2Z 4 .
In [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF] the author finished the classification of cyclic subgroups of finite order of the Cremona group, up to conjugation. He gave natural parameterizations of conjugacy classes, related to fixed curves of positive genus. The classification of finite cyclic subgroups that are not of Jonquières type was almost achieved in [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]. Let us explain what we mean by "almost": a list of representative elements is available; explicit forms are given; the dimension of the varieties which parameterize the conjugacy classes are provided.

What is missing ? A finer geometric description of the algebraic variety parameterizing conjugacy classes according to [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF].

The case of groups conjugate to subgroups of Aut(P 2 C ) was studied in [START_REF] Beauville | On Cremona transformations of prime order[END_REF]: there is exactly one conjugacy class for each order n, representated by (z 0 : z 1 : z 2 ) → (z 0 : z 1 : e 2iπ/n z 2 ) .

Blanc completed the classification of cyclic subgroups of Bir(P 2 C ) of finite order ([Bla11a]). For groups of Jonquières type he applied cohomology group theory and algebraic tools to the group J and got:

Theorem 6.23 ([Bla11a]). -
For any positive integer m, there exists a unique conjugacy class of linearisable elements of order n, represented by the automorphism

(z 0 : z 1 : z 2 ) → (z 0 : z 1 : e 2iπ/n z 2 ).
Any non-linearisable Jonquières element of finite order of Bir(P 2 C ) has order 2n, for some positive integer n, and is conjugate to an element φ, such that φ and φ n are of the following form

φ : (z 0 , z 1 ) e 2iπ/n z 0 , a(z 0 )z 1 + (-1) δ p(z n 0 )b(z 0 ) b(z 0 )z 1 + (-1) δ a(z 0 ) φ n : (z 0 , z 1 ) z 0 , p(z n 0 ) z 1
where a, b belongs to C(z 0 ), δ to 0, ±1 , and p ∈ C[z 0 ] is a polynomial with simple roots.

The curve Γ of equation z 2 1 = p(z n 0 ), pointwise fixed by φ n , is hyperelliptic, of positive geometric genus, and admits a (2 : 1)-map φ 2 1 : Γ → P 1 C . The action of φ on Γ has order n, and is not a root of the involution associated to any φ 2 1 . Furthermore the above association yields a parameterization of the conjugacy classes of non-linearisable Jonquières elements of order 2n of Bir(P 2 C ) by isomorphism classes of pairs (Γ, ψ), where Γ is a smooth hyperelliptic curve of positive genus, ψ ∈ Aut(Γ) is an automorphism of order n, which preserves the fibres of the φ 2 1 and is not a root of the involution associated to the φ 2 1 .

The analogous result for finite Jonquières cyclic groups holds, and follows directly from this statement.

Note that if the curve Γ has geometric genus ≥ 2, the φ 2 1 is unique, otherwise it is not. Blanc also dealt with cyclic subgroups of Bir(P 2 C ) that are not of Jonquières type. Using the classification of [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF] and some classical tools on surfaces and curves he provided the parameterization of the 29 families of such groups.

The classification is divided in two parts: find representative families and prove that each group is conjugate to one of these; parameterize the conjugacy classes in each families by algebraic varieties. For cyclic groups of prime order the varieties parameterizing the conjugacy classes are the moduli spaces of the non-rational curves fixed by the groups. Blanc needs to generalize it, by looking for the non-rational curves fixed by the non-trivial elements of the group. Let us give the definition of this invariant which provides a simple way to decide whether two cyclic groups are conjugate. Recall that a birational map of the complex projective plane fixes a curve if it restricts to the identity on the curve.

Definition. -Let φ be a non-trivial element of Bir(P 2 C ) of finite order. If no curve of positive geometric genus is (pointwise) fixed by φ, then NFC(φ) = / 0; otherwise φ fixes exactly one curve of positive genus ([BB00, dF04]), and NFC(φ) is then the isomorphism class of the normalization of this curve.

Two involutions φ, ψ of Bir(P 2 C ) are conjugate if and only if NFC(φ) = NFC(ψ) (see §6.1). If φ, ψ are elements of Bir(P 2 C ) of the same prime order, then φ and ψ are conjugate if and only if NFC(φ) = NFC(ψ) (see [START_REF] Beauville | On Cremona transformations of prime order[END_REF][START_REF] De Fernex | On planar Cremona maps of prime order[END_REF]). This is no longer the case for cyclic groups of composite order as observed in [START_REF] Beauville | On Cremona transformations of prime order[END_REF]: the automorphism φ of the cubic surface Let us now give a simple way to decide whether two cyclic subgroups of finite order of Bir(P 2 C ) are conjugate: Theorem 6.24 [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]). -Let G and H be two cyclic subgroups of Bir(P 2 C ) of the same finite order. Then G and H are conjugate in Bir(P 2 C ) if and only if NFCA(φ) = NFCA(ψ) for some generators φ of G and ψ of H.

z 3 0 + z 3 1 + z 3 2 + z 3 3 = 0 in

CHAPTER 7 UNCOUNTABLE SUBGROUPS OF THE CREMONA GROUP

All the results of this Chapter have been proved without the construction of the action of the isometric action of Bir(P 2 C ) on the hyperbolic space H ∞ and we keep this point of view. Different ideas and tools are used in any section: foliations and group theory are the main ingredients.

The study of the automorphis groups starts a long time ago. For instance for classical groups let us see [START_REF] Dieudonné | La géométrie des groupes classiques[END_REF]. Consider the automorphism group of the complex projective space P n C ; it is PGL(n + 1, C). The automorphism group of PGL(n + 1, C) is generated by the inner automorphisms, the involution M → M ∨ and the action of the field automorphisms of C. In 1963 Whittaker showed that any isomorphism between homeomorphism groups of connex topological varieties is induced by an homeomorphism between the varieties themselves ( [START_REF] Whittaker | On isomorphic groups and homeomorphic spaces[END_REF]). In 1982 Filipkiewicz proved a similar statement for differentiable varieties.

Theorem 7.1 ( [START_REF] Filipkiewicz | Isomorphisms between diffeomorphism groups[END_REF]). -Let V , W be two connected varieties of class C k , resp. C j . Let Diff k (V ) be the group of C k -diffeomorphisms of V . Let φ : Diff k (V ) → Diff j (V ) be an isomorphism group. Then k = j and there exists a C k -difffeomorphism ψ : V → W such that

φ(ϕ) = ψ • ϕ • ψ -1 ∀ ϕ ∈ Bir(P 2 C
). The description of uncountable maximal abelian subgroups of the plane Cremona group allows to characterize the automorphisms group of Bir(P 2 C ): Theorem 7.2 ([D 06b]). -Let ϕ be an automorphism of Bir(P 2 C ). There exist a birational self map ψ of the complex projective plane and an automorphism κ of the field C such that

ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ Bir(P 2 C ).
In other words the non-inner automorphism group of Bir(P 2 C ) can be identified with the automorphisms of the field C.

In the first section we study uncountable maximal abelian subgroups of Bir(P 2 C ); let G be such a group. We give an outline of the proofs of the following results: any element of G preserves at least one singular holomorphic foliation; either no element of G is torsion-free, or G leaves invariant a holomorphic foliation; if G is torsion-free, then G is conjugate to a subgroup of the Jonquières group.

In the second section we describe the automorphism group of Bir(P 2 C ). A study of the torsion-free maximal abelian subgroups of the Jonquières group shows that the group

J a = (z 0 , z 1 ) (z 0 + a(z 1 ), z 1 ) | a ∈ C(z 1 )
is invariant by any automorphism of Bir(P 2 C ). Some work on special subgroups of J a achieves the description of Aut(Bir(P 2 C )).

In a session problems during the International Congress of Mathematicians Mumford proposed the following ([Mum76]):

"Let G = Aut C C(z 0 , z 1 ) be the Cremona group (...) the problem is to topologize G and associate to it a Lie algebra consisting, roughly, of those meromorphic vector fields D on P 2 C which "integrate" into an analytic family of Cremona transformations."

In the third section we deal with a contribution in that direction: the description of 1parameter subgroups of quadratic birational self maps of P 2 C . In [START_REF] Ghys | Sur les groupes engendrés par des difféomorphismes proches de l'identité[END_REF] Ghys showed that any nilpotent subgroup of Diff ω (S 2 ) is metabelian; as a consequence he got that if Γ is a subgroup of finite index of SL(n, Z), n ≥ 4, then any morphism from Γ into Diff ω (S 2 ) has finite image. In the same spirit the nilpotent subgroups of the plane Cremona group are described in the fourth section: if Γ is a strongly nilpotent group of length > 1, then either G is metabelian up to finite index, or G is a torsion group. As a consequence as soon as n ≥ 5 no subgroup of SL(n, Z) of finite index embeds into Bir(P 2 C ). The description of centralizers of discrete dynamical systems is an important problem in real/complex dynamics. Julia ([Jul22, Jul68]) then Ritt ([Rit23]) show that the set

Cent(φ) = ψ : P 1 C → P 1 C | ψ • φ = φ • ψ of
rational functions that commute to a rational function φ coincide in general (1) with φ n 0 | n ∈ N where φ 0 is an element of Cent(φ). In the 60's Smale considered generic diffeomorphisms φ of compact manifolds and asked if its centralizer coincides with φ n | n ∈ Z . Many mathematicians have considered this question (for instance [START_REF] Bonatti | The C 1 generic diffeomorphism has trivial centralizer[END_REF][START_REF] Palis | Rigidity of the centralizers of diffeomorphisms and structural stability of suspended foliations[END_REF][START_REF] Palis | Centralizers of Anosov diffeomorphisms on tori[END_REF][START_REF] Palis | Rigidity of centralizers of diffeomorphisms[END_REF]). The fifth section deals with centralizers of elliptic birational maps, Jonquières twists and Halphen twists.

(1) except monomial maps z → z k , Tchebychev polynomials, Lattès examples ...

UNCOUNTABLE MAXIMAL ABELIAN SUBGROUPS OF Bir(P 2

C ) 135 7.1. Uncountable maximal abelian subgroups of Bir(P 2 C )

Let S be a complex compact surface. A foliation F on S is given by a family (χ i ) i of holomorphic vector fields with isolated zero defined on some open cover (U i ) i of S. The vector fields χ i have to satisfy the following conditions: there exist g i j ∈ O * (U i ∩ U j ) such that χ i = g i j χ j on U i ∩ U j . Let us remark that a non-trivial meromorphic vector field on S defines such a foliation.

Lemma 7.3 ([D 06b]

). -Let G be an uncountable abelian subgroup of Bir(P 2 C ). There exists a rational vector field χ such that

ϕ * χ = χ ∀ ϕ ∈ G.
In particular G preserves a foliation.

Proof. -Since G is uncountable, there exists an integer d such that

G d = G ∩ Bir d (P 2 C ) is uncountable. Hence the Zariski closure G d of G d in Bir ≤d (P 2 C ) is an algebraic set of dimen- sion ≥ 1. Consider a curve in G d , i.e. a map η : D → G d , t → η(t).
Remark that elements of G d are rational maps that commute. Let us define the rational vector field χ at any m ∈ P 2 C Ind(η(0) -1 ) by

χ(m) = ∂η(s) ∂s s=0 η(0) -1 (m) .
Let ϕ be an element of G d . If we differentiate the equality

ϕη(s)ϕ -1 (m) = η(s)(m)
with respect to s, m being fixed, one gets: ϕ * χ = χ. In other words χ is invariant by the elements of G d , and so by any element of G.

As a result for any uncountable abelian subgroup G of Bir(P 2 C ), there exists a foliation on ). If S is a projective surface endowed with a foliation F , we denote by Bir(S, F ) (resp. Aut(S, F )) the group of birational maps (resp. holomorphic maps) of S preserving the foliation F . In general Bir(S, F ) coincides with Aut(S, F ) and is finite. In [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF] the authors dealt with the opposite case and got a classification.

Theorem 7.4 ([CF03]

). -Let F be a foliation on S such that Aut(X, ϕ * F ) Bir(X, ϕ * F ) for any birational map ϕ : X S. Then, up to conjugacy, there exists an element of infinite order in Bir(S, F ) and either F is a rational fibration, or up to a finite cover there exist some integers p, q, r, s such that

Bir(P 2 C , F ) = (z 0 , z 1 ) (z p 0 z q 1 , z r 0 z s 1 ), (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C * .
Before stating the opposite case Aut(S, F ) infinite, let us give some definitions. Let Λ be a lattice in C 2 ; it induces a complex torus T = C 2 Λ of dimension 2. For instance the product of an elliptic curve by itself is a complex torus. An affine map ψ that preserves Λ induces an automorphism of the torus T. If the linear part of ψ is of infinite order, then either the linear part of ψ is hyperbolic and ψ induces an Anosov automorphism that preserves two linear foliations; or the linear part of ψ is unipotent and ψ preserves an elliptic fibration.

Sometimes there is a finite automorphism group of T normalized by ψ. Denote by T G the desingularization of T G . The automorphism induced by ψ on T G preserves the foliations induced the stable and unstable foliations preserved by ψ when ψ is hyperbolic; an elliptic fibration when the linear part of ψ is unipotent.

If G = id, (z 0 , z 1 ) → (-z 0 , -z 1 ) we say that T G is a Kummer surface; otherwise T G is a generalized Kummer surface.

Theorem 7.5 ([CF03]

). -Let F be a singular holomorphic foliation on a projective surface S. Assume that Aut(S, F ) is infinite. Then Aut(S, F ) contains at least one element ϕ of infinite order and one of the following holds:

F is invariant by an holomorphic vector field; F is an elliptic fibration; the surface S is a generalized Kummer surface, ϕ can be lifted to an Anosov automorphism ϕ of the torus and F is the projection on S of the unstable or stable foliation of ϕ.

Remarks 7.6. -

The foliations invariant by an holomorphic vector field are described in [CF03, Proposition 3.8]. The last two cases are mutually exclusive.

Using these two statements one can prove the following one: Theorem 7.7 ([D 06b]). -Let G be an uncountable maximal abelian subgroup of Bir(P 2 C ). Then: either G has an element of finite order; or G is up to conjugacy a subgroup of the Jonquières group.

Idea of the proof. -Assume first that Aut(X, φ * F ) Bir(X, φ * F ) for any birational map φ : X S. Then according to Theorem 7.4 either G preserves a rational fibration, and then G is up to birational conjugacy contained in the Jonquières group; or G is up to conjugacy and finite cover a subgroup of

(z 0 , z 1 ) (z p 0 z q 1 , z r 0 z s 1 ), (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C * , α = α p β q , β = α r β s . If G is conjugate to the diagonal group D = (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C * ,
then G contains elements of finite order. Otherwise since G is uncountable it can not be reduced to

(z 0 , z 1 ) (z p 0 z q 1 , z r 0 z s 1 )
. Therefore, there exists a non-trivial element (z 0 , z 1 ) → (λz 0 , µz 1 ) in G such that λ = λ p µ q and µ = λ r µ s . For any the map (z 0 , z 1 ) → (λ z 0 , µ z 1 ) satisfies these equalities, so belongs to G. Consider such that λ = i; then µ = e iπ 1-p 2q is also a root of unity and (z 0 , z 1 ) → (λ z 0 , µ z 1 ) is thus an element of finite order of G. More precisely G contains periodic elements of any order. Suppose now that there exist a surface S and a birational map ψ : S P 2 C such that Aut(S, ψ * F ) = Bir(S, ψ * F ). According to Theorem 7.5 either ψ * F is invariant by an holomorphic vector field on S; or ψ * F is an elliptic fibration.

Since G is uncountable the last eventuality can not occur [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]). Let us thus assume that ψ * F is invariant by an holomorphic vector field on S. According to [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF] one can assume up to conjugacy that G is a subgroup of Aut( S) where S is a minimal model of S. But minimal rational surfaces are P 2 C , P 1 C × P 1 C and the Hirzebruch surfaces F n , n ≥ 2, and their automorphisms groups are known (see Chapter 3).

The description of the uncountable maximal abelian subgroups of minimal rational surfaces gives:

Proposition 7.8 ([D 06b]).
-Let S be a minimal rational surface. Let G be an uncountable abelian subgroup of Aut(S) maximal in Bir(S). Then: either G contains an element of finite order, or G coincides with (z 0 , z 1 ) → (z 0 + P(z 1 ), z 1 )

| P ∈ C[z 1 ], deg P ≤ n , or G = (z 0 , z 1 ) → (z 0 + α, z 1 + β) | α, β ∈ C .
A study of the uncountable maximal abelian subgroups of the Jonquières group allows to refine Theorem 7.7 as follows:

Theorem 7.9 ([D 06b]). -Let G be an uncountable maximal abelian subgroup of the plane Cremona group. Then up to conjugacy: either G contains an element of finite order, or G = (z 0 , z 1 )

(z 0 + a(z 1 ), z 1 ) | a ∈ C(z 1 ) , or G = (z 0 , z 1 ) → (z 0 + α, z 1 + β) | α, β ∈ C , or any subgroup of Bir(P 2
C ) acting by conjugacy on G is, up to finite index, solvable.

Description of the automorphisms group of the Cremona group

Let us give an idea of the proof of Theorem 7.2. The description of uncountable maximal abelian subgroups of Bir(P 2 C ) yields to Corollary 7.10

([D 06b]). -Let ϕ be an automorphism of Bir(P 2 C ). Set J a = (z 0 , z 1 ) (z 0 + a(z 1 ), z 1 ) | a ∈ C(z 1 ) .
Up to birational conjugacy ϕ(J a ) = J a and (z 0 , z 1 ) → (z 0 + 1, z 1 ) is invariant by ϕ.

Let us consider

T 1 = (z 0 , z 1 ) → (z 0 + α, z 1 ) | α ∈ C , T 2 = (z 0 , z 1 ) → (z 0 , z 1 + β) | β ∈ C , and
D 1 = (z 0 , z 1 ) → (αz 0 , z 1 ) | α ∈ C * , D 2 = (z 0 , z 1 ) → (z 0 , βz 1 ) | α ∈ C . Proposition 7.11 ([D 06b]).
-Let ϕ be an automorphism of Bir(P 2 C ). Assume that ϕ(J a ) = J a and (z 0 , z 1 ) → (z 0 + 1, z 1 ) is invariant by ϕ. Then up to birational conjugacy:

ϕ(J a ) = J a , (z 0 , z 1 ) → (z 0 + 1, z 1 ) is invariant by ϕ, ϕ(T 1 ) = T 1 and ϕ(T 2 ) = T 2 , ϕ(D 1 ) = D 1 and ϕ(D 2 ) = D 2 .
As a consequence an automorphism of Bir(P 2 C ) induces two automorphisms of the group Aff(C) of affine maps of the complex line.

Lemma 7.12. -Let ϕ be an automorphism of Aff(C). Then ϕ is the composition of an inner automorphism and an automorphism of the field C.

Sketch of the proof. -The maximal abelian subgroups of Aff(C) are the group of translations

T = z → z + β | β ∈ C
and the groups of affine maps that preserve a point

D z 0 = z → α(z -z 0 ) + z 0 |α ∈ C * .
Since T does not contain element of finite order, ϕ sends T onto T. In other words there exists an additive bijection κ 2 : C → C such that ϕ(z+β) = z+κ 2 (β). Up to conjugacy by an element of T one can assume that ϕ(D 0 ) = D 0 . In other words there exists a multiplicative bijection κ 1 : C * → C * such that ϕ(αz) = κ 1 (α)z. On the one hand

ϕ z → αz + α = ϕ z → z + α • ϕ z → αz = z → κ 1 (α)z + κ 2 (α)
and on the other hand

ϕ z → αz + α = ϕ z → αz • ϕ z → z + 1 = z → κ 1 (α)z + κ 1 (α)κ 2 (1) .

Hence for any α the equality

z → κ 1 (α)z + κ 2 (α) = z → κ 1 (α)z + κ 1 (α)κ 2 (1) holds. Since µ = κ 2 (1) is non-zero, κ 2 is
additive and multiplicative. As a result κ 2 is an isomorphism of the field C and

ϕ z → αz + β) = z → κ 1 αz + κ 2 β = z → κ 1 αz + µ κ 1 β = z → κ 1 αz + κ -1 1 µβ = z → κ 1 ( κ -1 1 µz) • (αz + β) • ( κ 1 µz) = z → κ 1 ( κ 1 µz) -1 • (αz + β) • ( κ 1 µz) .
Sketch of the proof of Theorem 7.2. -Proposition 7.11 and Lemma 7.12 imply that for any α, β in C * , for any γ, δ in C one has

ϕ (z 0 , z 1 ) → (αz 0 + γ, βz 1 + δ) = (z 0 , z 1 ) → ( κ 1 αz 0 + µ κ 1 γ, κ 2 βz 1 + η κ 2 δ)
where η, µ are two non-zero complex numbers and κ 1 , κ 2 two automorphisms of the field C. Since (z 0 , z 1 ) → (z 0 + z 1 , z 1 ) and (z 0 , z 1 ) → (αz 0 , αz 1 ) commute their image by ϕ also, and so κ 1 = κ 2 . As a consequence up to conjugacy by an inner automorphism and an automorphism of the field C, the groups

T = (z 0 , z 1 ) → (z 0 + α, z 1 + β) | α, β ∈ C and D = (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C *
are pointwise invariant. Then one can check that the involutions (z 0 , z 1 ) → z 0 , 1 z 1 and (z 0 , z 1 ) → (z 1 , z 0 ) are invariant by ϕ. But the group generated by T, D, (z 0 , z 1 ) → z 0 , 1

z 1 and (z 0 , z 1 ) → (z 1 , z 0 ) contains PGL(3, C). Furthermore σ 2 = (z 0 , z 1 ) → z 0 , 1 z 1 • ((z 0 , z 1 ) → (z 1 , z 0 )) 2 hence ϕ(σ 2 ) = σ 2 .
We conclude thanks to the Noether and Castelnuovo Theorem.

Corollary 7.13 ([D 06b]). -An isomorphism of the semi-group of rational self maps of P 2 C is inner up to the action of an automorphism of the field C.

In the spirit of the result of Filipkiewicz (Theorem 7.1) one has: Corollary 7.14 ([D 06b]). -Let S be a complex projective surface. Let ϕ be an isomorphism between Bir(S) and Bir(P 2 C ). There exist a birational map ψ : S P 2 C and an automorphism of the field C such that Note that there is no description of Aut(Bir(P n C )) for n ≥ 3. Nevertheless there are two results in that direction: the first one is Theorem 7.17 ([D 15b]). -Let ϕ be an automorphism of Bir(P n C ); there exist an automorphism κ of the field C, and a birational self map

ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ Bir(S).
ψ of P n C such that ϕ(φ) = κ (ψ • φ • ψ -1 ) ∀ φ ∈ G(n, C) = σ n , PGL(n + 1, C) .
the second one is

Theorem 7.18 ([Can14]
). -Let V be a smooth connected complex projective variety of dimension n. Let r be a positive integer and let ρ : Aut(P r C ) → Bir(V ) be an injective morphism of groups. Then n ≤ r.

Furthermore if n = r, there exist a field morphism κ : C → C and a birational map

ψ : V P n C such that either ψ • ρ(A) • ψ -1 = κ A for all A ∈ Aut(P n C ), or ψ • ρ(A) • ψ -1 = ( κ A) ∨ for all A ∈ Aut(P n C ). In particular V is rational. Moreover κ is an automorphism of C if ρ is an isomorphism.
Before giving an idea of the proof of this last result let us state two corollaries of it. The first shows that the Cremona groups Bir(P The second characterizes rational varieties V by the structure of Bir(V ), as an abstract group: [START_REF] Cantat | Morphisms between Cremona groups, and characterization of rational varieties[END_REF]). -Let V be an irreducible complex projective variety of dimension n. The following properties are equivalent:

Corollary 7.20 ([
V is rational, Bir(V ) is isomorphic to Bir(P n C ) as an abstract group, there is a non-trivial morphism from PGL(n + 1, C) to Bir(V ).

The strategy that leads to the proof of Theorem 7.18 is similar to the proof of Theorem 7.2 but requires several new ideas:

Weil's regularization Theorem (Theorem 3.56), that transforms a group of birational maps of V with uniformly bounded degrees into a group of automorphisms of a new variety by a birational change of variables; Epstein and Thurston work on nilpotent Lie subalgebras in the Lie algebra of smooth vector fields of a compact manifold ([ET79]).

One-parameter subgroups of Bir

(P 2 C ) 7.3.1. Description of 1-parameter subgroups of quadratic birational maps of P 2 C . -A germ of flow in Bir ≤2 (P 2 C ) is a germ of holomorphic application t → φ t ∈ Bir ≤2 (P 2 C ) such that φ t+s = φ t • φ s φ 0 = id
Since a germ of flow can be generalized we speak about flow. The set of lines blown down by the flow φ t is a germ of analytic sets in the Grassmaniann of lines in P 2 C , i.e. in the dual space (P 2 C ) ∨ . Similarly the set of indeterminacy points of the φ t is a germ of analytic sets of P 2 C .

We call family of contracted curves a continuous map (indeed an analytic one) defined over a germ of closed sector ∆ of vertex 0 in C

D : ∆ → (P 2 C ) ∨
such that for any t ∈ ∆ the lines D t coincide with a line D(t) blown down by φ t .

Similarly a family of indeterminacy points is a continuous map t → p t defined on a sector ∆ such that any p t is an indeterminacy point of φ t . Let φ t be a flow. Let D t (resp. p t ) be a family of curves blown down by φ t (resp. a family of indeterminacy points of φ t ). If D t (resp. p t ) is independent of t, the family is unmobile, otherwise it is mobile.

A rational vector field χ on P 2 C is rationally integrable if its flow is a flow of birational maps. A germ of flow in Bir 2 (P 2 C ) is the flow of a rationally integrable vector field χ = ∂φ t ∂t t=0

called infinitesimal generator of φ t . To this vector field is associated a foliation whose leaves are "grosso modo" the trajectories of χ. Recall that a fibration by lines L of P 2

C is given by λ 1 + µ 2 = 0 where 1 , 2 are linear forms that are not proportional. The base-point is the intersection point p of all these lines. We also say that L is a pencil of lines through p, or L is a foliation by lines singular at p. Recall that a birational self map of P 2 C that preserves a rational fibration belongs up to birational conjugacy to J .

Let φ t be a germ of flow in Bir 2 (P 2 C ). Then the following properties hold: assume that φ t blows down a mobile line, then φ t preserves a fibration by lines, more precisely the family of contracted lines belongs to a fibration invariant by any element of the flow ([CD13, Proposition 2.5, Remark 2.6]); there is at most one unmobile line blown down by φ t (see [CD13, Lemma 2.10]); if φ t blows down a unique line that is moreover unmobile, then there exists an invariant affine chart C 2 such that φ t|C 2 : C 2 → C 2 is polynomial for any t (see [CD13, Proposition 2.12]); assume that there exists an invariant affine chart C 2 such that φ t|C 2 : C 2 → C 2 is polynomial for any t. Then φ t preserves a pencil of lines. Furthermore either φ t is affine, or there exists a normal form for φ t up to linear conjugacy ([CD13, Proposition 2.15]).

Combining all these properties one can state the following result:

Theorem 7.21 ([CD13]). -A germ of flow in Bir 2 (P 2
C ) preserves a fibration by lines. Let φ t be a quadratic birational flow, and let χ be its infinitesimal generator. A strong symmetry Y of χ is a rationally integrable vector field of flow ψ s such that φ t and ψ s commute, i.e. [χ,Y ] = 0,

ψ s ∈ Bir 2 (P 2
C ) for all s, χ and Y are not C-colinear.

Let φ t be a flow in Bir 2 (P 2 C ), and let χ (resp. F χ ) be the associated vector field (resp. foliation). We denote by φ t Z ⊂ Bir 2 (P 2 C ) the Zariski closure of φ t in Bir 2 (P 2 C ). Let G(χ) be the maximal algebraic abelian subgroup of Bir 2 (P 2 C ) that contains φ t Z .

Theorem 7.22 ([CD13]

). -Let φ t be a germ of flow in Bir 2 (P 2 C ), and let χ be its infinitesimal generator.

If dim G(χ) = 1, then F χ is a rational fibration. If dim G(χ) ≥ 2, then F χ has a strong symmetry.
In both cases F χ is defined by a rational closed 1-form.

Proof. -Let us prove the first assertion. If dim G(χ) = 1, then φ t Z is the component of G(χ) that contains the identity. This group viewed as a Lie group is isomorphic to C, or C * , or C Λ . According to Theorem 7.21 the group φ t Z preserves a fibration by lines; let us assume that this fibration is given by z 1 = constant. One yields a morphism

π : φ t Z → PGL(2, C)
that describes the action of φ t on the fibers.

If π is trivial (i.e. if the fibration is preserved fiberwise), then F χ = z 1 = constant and the result holds. Otherwise φ t Z is not isomorphic to C Λ because there is no C Λ among the subgroup of PGL(2, C). Hence the topological closure of φ t Z in P 17

C

Rat 2 is a rational curve. But according to Darboux a foliation of P 2 C whose the closure of all leaves are algebraic curves has a non-constant rational first integral ( [START_REF] Jouanolou | Équations de Pfaff algébriques[END_REF]). In our case the curves are rational, so F χ is a rational fibration.

Let us now prove the second assertion. Assume dim G(χ) ≥ 2. One can find a germ of 1-parameter group ψ s in G(χ) not contained in φ t . Let Y be the infinitesimal generator of ψ s . The vector fields χ and Y commute and are not C-colinear. Let us consider ω a rational 1-form that define F χ , i.e. i χ ω = 0. If χ and Y are generically independent, then Ω = ω i Y ω is closed and define F χ . If χ and Y are not generically independent, then Y = f χ with f rational and non-constant. Since [χ,Y ] = 0 one has χ( f ) = 0. As a result d f defines F χ and is closed.

Remark 7.23. -The last two statements can be generalized as follows:

Theorem 7.24 ([CD13]). -Let φ t be a germ of flow in Bir n (P 2 C ), and let χ be its infinitesimal generator. Denote by G(χ) the abelian maximal algebraic group contained in Bir n (P 2 C ) and that contains φ t Z . Then if dim G(χ) = 1, then F χ is either a rational fibration or an elliptic fibration;

if dim G(χ) ≥ 2, then χ has a strong symmetry.

In both cases F χ is defined by a closed rational 1-form.

Theorem 7.25 ([CD13]

). -Any germ of birational flow in Bir n (P 2 C ) preserves a rational fibration.

7.3.2.

A few words about the classification of germs of quadratic birational flows. -Let φ t be a germ of flow in Bir 2 (P 2 C ); then φ t preserves a fibration by lines ([CD13, Theorem 2.16]). In other words up to linear conjugacy

φ t : (z 0 , z 1 ) A(z 1 ,t)z 0 + B(z 1 ,t) C(z 1 ,t)z 0 + D(z 1 ,t) , ν(z 1 ,t)
with ν(z 1 ,t) = z 1 , or z 1 + t, or e βt z 1 ; A, B, C, D are polynomials in z 1 and deg

z 1 A ≤ 1, deg z 1 B ≤ 2, deg z 1 C = 0, deg z 1 D ≤ 1, B(z 1 , 0) = C(z 1 , 0) = 0 and A(z 1 , 0) = D(z 1 , 0).
The infinitesimal generator χ = ∂φ t ∂t t=0

of φ t can be written

αz 2 0 + (z 1 )z 0 + P(z 1 ) az 1 + b ∂ ∂z 0 + ε(z 1 ) ∂ ∂z 1 with α, a, b ∈ C, , P ∈ C[z 1 ], deg = 1, deg P = 2
and up to linear conjugacy and scalar multiplication ε ∈ {0, 1, z 1 }.

The above vector fields are classified up to automorphisms of P 2 C and renormalization in [CD13, Chapter 2, §2]; such vector fields are detected via the following methods: compute explicitely the flow by integration; or degenerate χ on another vector field χ 0 that is not rationally integrable; or show that a birational model of F χ has an isolated degenerate resonnant singular point (one and only one non-zero eigenvalue), and so F χ has no rational first integral. Then prove that there is no strong symmetry hence χ is not rationally integrable (Theorem 7.22).

Nilpotent subgroups of the Cremona group

In [D 07b] are described the nilpotent subgroups of the plane Cremona group: Theorem 7.26 ([D 07b]). -Let N be a nilpotent subgroup of Bir(P 2 C ). Assume that, up to finite index, N is not abelian. Then either N is a torsion group; or N is metabelian up to finite index, i.e. [N, N] is abelian up to finite index.

Examples 5. -Let α and β be two non zero complex numbers; the group

(z 0 , z 1 ) → (z 0 + αβ, z 1 ), (z 0 , z 1 ) → (z 0 + αz 1 , z 1 ), (z 0 , z 1 ) → (z 0 , z 1 + β)
is a non-abelian, non-finite and nilpotent subgroup of Bir(P 2 C ). If a belongs to C(z 1 ), then

(z 0 , z 1 ) → (z 0 + 1, z 1 ), (z 0 , z 1 ) → (z 0 + z 1 , z 1 ), (z 0 , z 1 ) → (z 0 + a(z 1 ), z 1 -1) ,
is a non-abelian, non-finite and nilpotent subgroup of Bir(P 2 C ). Corollary 7.27

([D 07b]). -Let G be a group. Assume that G contains a subgroup N such that N is of nilpotent class > 1, N has no torsion, N is not metabelian up to finite index.
Then there is no faithfull representation of G into Bir(P 2 C ). Remark 7.28. -Let G be a nilpotent group of nilpotent class n. Take f in G, g in

C (n-2) G and consider h = [ f , g] ∈ C (n-1) G. Since G is of nilpotent class n, then [ f , h] = [g, h] = id.
In other words any nilpotent group contains a distorted element.

According to Remark 7.28 and Lemma 5.18 one has: Proposition 7.29. -Let N be a nilpotent subgroup of the plane Cremona group. It contains a distorted element which is elliptic or parabolic.

Idea of the proof of Theorem 7.26. -Take G ⊂ Bir(P 2 C ) a nilpotent subgroup of class k which is not up to finite index of nilpotent class k -1. Denote by Σ G the set of finitely generated nilpotent subgroups of G that are, up to finite index, not abelian. Then either any element of Σ G is finite and G is a torsion group; or Σ G contains a non-finite element H.

Claim 7.30 ([D 06a]

). -The group H preserves a fibration F that is rational or elliptic.

Any element of C (k-1) H preserves fiberwise F . Let φ be in C (k-1) H. As [φ, G] = id, then a) either φ preserves fiberwise two distinct fibrations; b) of G preserves fiberwise F . If a) holds, then φ is of finite order; if it is the case for any φ ∈ C (k-1) H, then H is, up to finite index, of nilpotent class k -1: contradiction. If b) holds, then G is, up to finite index, metabelian. Let us detail why when F is rational. In that case G is, up to conjugacy, a subgroup of the Jonquières group J . Let pr 2 be the projection J → PGL(2, C). A non-finite nilpotent subgroup of PGL(2, k), where k = C or C(z 1 ), is up to finite index abelian. The group pr 2 (G) is thus, up to finite index, abelian. Consequently we can assume that pr 2 (C (i) G) = {id} for 1 ≤ i ≤ k. In particular C (1) G is a nilpotent subgroup of PGL(2, C(z 1 )) and as a result is, up to finite index, abelian.

Idea of the proof of the Claim 7.30. -Let us recall that H is a non-finite nilpotent subgroup of Bir(P 2 C ) with the following properties:

H is finitely generated, H is nilpotent of class k > 0, H is not, up to finite index, of nilpotent class k -1.
Assume C (k-1) H is not a torsion group. Then H preserves a fibration that is rational or elliptic. According to Lemma 5.18 a non-trivial element of C (k-1) G either preserves a unique fibration F that is rational or elliptic, or is an elliptic birational map. We have the following alternative: a) either C (k-1) G contains an element h that preserves a unique fibration F , b) or any element of C (k-1) G {id} is elliptic.

Let us look at these eventualities:

a) Since [h, G] = {id} any element of G preserves F .
b) The group C (k-1) G is finitely generated and abelian. Let a 1 , a 2 , . . . , a n be a generating set of C (k-1) G. The a i 's are elliptic maps, so there exist a surface S i , a birational map η i :

S i P 2 C and an integer k i > 0 such that η -1 i • a k i i • η i belongs to the neutral component Aut(S i ) 0 of Aut(S i ).
In particular the a i 's fix any curve of negative selfintersection, we can thus assume that S i is a minimal rational surface. A priori all the S i are distinct. Nevertheless according to Proposition 2.12 there exist a minimal rational surface S, a birational map η : S P 2 C and an integer k > 0 such that for any 1 ≤ i ≤ n the map η -1 • a k i • η belongs to the neutral component Aut(S) 0 of Aut(S). Minimal rational surfaces are P 2 C , P 1 C × P 1 C and Hirzebruch surfaces F n , n ≥ 2. Using on the one hand the description of the automorphisms groups of minimal rational surfaces (see Chapter 3), and on the other hand the fact that if K is an algebraic Lie subgroup of GL(n, C), then the semi-simple and nilpotent parts of any element of K belong to K, we prove that G is, up to finite index and up to conjugacy, contained in the Jonquières group J (see [D 07b]).

It remains to consider the case "C (k-1) G is a torsion group"; the ideas are similar (see [D 07b, Proposition 4.5]).

7.5. Centralizers in Bir(P 2 C ) 7.5.1. Centralizers of elliptic birational maps. -We will focus on the case of birational self maps of P 2 C of infinite order. Note for instance that for birational self map of P 2 C of finite order the situation is wild: consider for instance a birational involution φ of P 2 C . If φ is conjugate to an automorphism of P 2 C , then the centralizer of φ in Bir(P 2 C ) is uncountable but if φ is conjugate to a Bertini (or a Geiser) involution, then the centralizer is finite ( [START_REF] Blanc | On birational transformations of pairs in the complex plane[END_REF]).

According to [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF] an elliptic birational self map of P 2 C of infinite order is conjugate to an automorphism of P 2

C which restricts to one of the following automorphisms on some open subset isomorphic to C 2 : (z 0 , z 1 ) → (αz 0 , βz 1 ) where α, β belong to C * and where the kernel of the group homomorphism

Z 2 → C 2 (i, j) → α i β j
is generated by (k, 0) for some k ∈ Z;

(z 0 , z 1 ) → (αz 0 , z 1 + 1) where α ∈ C * .

We can describe the centralizers of such maps; let us start with the centralizer of (z 0 , z 1 ) → (αz 0 , βz 1 ) where α, β belong to C * and where the kernel of the group homomorphism

Z 2 → C 2 (i, j) → α i β j
is generated by (k, 0) for some k ∈ Z. Recall that PGL(2, C) is the group of automorphisms of P 1 C or equivalently the group of Möbius transformations

z 0 az 0 + b cz 0 + d A direct computation implies the following: for any α ∈ C * η ∈ PGL(2, C) | η(αz 0 ) = αη(z 0 ) =    PGL(2, C) if α = 1 z 0 γz ±1 0 | γ ∈ C * if α = -1 z 0 → γz 0 | γ ∈ C * if α 2 = 1
Lemma 7.31. -Let us consider φ : (z 0 , z 1 ) → (αz 0 , βz 1 ) where α, β belongs to C * and where the kernel of the group homomorphism

Z 2 → C 2 (i, j) → α i β j is generated by (k, 0) for some k ∈ Z. The centralizer of φ in Bir(P 2 C ) is (z 0 , z 1 ) (η(z 0 ), z 1 a(z k 0 )) | a ∈ C(z 0 ), η ∈ PGL(2, C), η(αz 0 ) = αη(z 0 ) .
Proof. -Let ψ : (z 0 , z 1 ) (ψ 0 (z 0 , z 1 ), ψ 1 (z 0 , z 1 )) be a birational self map of P 2 C that commutes with φ. Then ψ 0 (αz 0 , βz 1 ) = αψ 0 (z 0 , z 1 ) (7.5.1) and ψ 1 (αz 0 , βz 1 ) = βψ 1 (z 0 , z 1 ) (7.5.2) hold. Denote by φ * the linear automorphism of the C-vector space C[z 0 , z 1 ] given by φ * : ϕ(z 0 , z 1 ) → ϕ(αz 0 , βz 1 ).

Let us write ψ i as P i Q i for i = 0, 1 where P i , Q i are polynomials without common factor. Note that P 0 , P 1 , Q 0 , Q 1 are eigenvectors of φ * , i.e. any of the P i , Q i is a product of a monomial in z 0 , z 1 with an element of C[z k 0 ]. Using (7.5.1) and (7.5.2) we get that ψ 0 (z 0 , z 1 ) = z 0 a 0 (z k 0 ) ψ 1 (z 0 , z 1 ) = z 1 a 1 (z k 0 ) But ψ is birational, so ψ 0 belongs to PGL(2, C). Furthemore ψ 0 satisfies ψ 0 (αz 0 ) = αψ 0 (z 0 ).

Let us now deal with the other possibility:

Lemma 7.32. -Let φ be the automorphism of P 2 C given by φ : (z 0 , z 1 ) → (αz 0 , z 1 + β)

where α ∈ C * , β ∈ C. The centralizer of φ in Bir(P 2 C ) is (z 0 , z 1 ) (η(z 0 ), z 1 + a(z 0 )) | η ∈ PGL(2, C), η(αz 0 ) = αη(z 0 ), a ∈ C(z 0 ), a(αz 0 ) = a(z 0 )
Proof.

-After conjugacy by (z 0 , z 1 ) → (z 0 , βz 1 ) we can assume that β = 1. If ψ : (z 0 , z 1 ) (ψ 0 (z 0 , z 1 ), ψ 1 (z 0 , z 1 )) is a birational map that commutes with φ, then ψ 0 (αz 0 , z 1 + 1) = αψ 0 (z 0 , z 1 ) (7.5.3) and ψ 1 (αz 0 , z 1 + 1) = ψ 1 (z 0 , z 1 ) + 1 (7.5.4) From (7.5.3) and [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF] we get that ψ 0 only depends on z 0 . Hence ψ 0 belongs to PGL(2, C) and commutes with z 0 → αz 0 . From (7.5.4) we get

∂ψ 1 ∂z 1 (αz 0 , z 1 + 1) = ∂ψ 1 ∂z 1 (z 0 , z 1 ) ∂ψ 1 ∂z 0 (αz 0 , z 1 + 1) = 1 α ∂ψ 1 ∂z 0 (z 0 , z 1 )
which again means that both ∂ψ 1 ∂z 0 and ∂ψ 1 ∂z 1 only depend on z 0 . Therefore, ψ 1 :

(z 0 , z 1 ) → γz 1 + b(z 0 ) with γ ∈ C * and b ∈ C(z 0 ). Then (7.5.4) can be rewritten b(αz 0 ) = b(z 0 ) + 1 -γ which implies that ∂b ∂z 0 (αz 0 ) = 1 α ∂b ∂z 0 (z 0 )
and that z 0 Geometrically it corresponds to look at the action of φ ∈ J on the basis of the invariant fibration z 1 = cst. The kernel of pr 2 , i.e. the elements of J which preserve the fibration z 1 = cst fiberwise, is a normal subgroup J 0 PGL(2, C(z 1 )) of J . Up to a birational conjugacy an element φ of J 0 is of one of the following form ([D 06b])

∂b ∂z 0 (z 0 ) is invariant under z 0 → αz 0 . If α is not a root of unity, then ∂b ∂z 0 = δ z 0 for some δ ∈ C. As b is rational, δ is zero and b is constant. As a consequence b(αz 0 ) = b(z 0 )+1-γ implies γ = 1, that is ψ 1 : (z 0 , z 1 ) z 0 +β. Assume that α is a primitive k-th root of unity. The map ψ : (z 0 , z 1 ) (η(z 0 ), γz 1 + b(z 0 )) commutes with φ k : (z 0 , z 1 ) (z 0 , z 1 + k) if and only if γ(z 1 + k) + b(z 0 ) = γz 1 + b(z 0 ) + k, i.e.
(z 0 , z 1 ) (z 0 + a(z 1 ), z 1 ), (z 0 , z 1 ) (b(z 1 )z 0 , z 1 ), (z 0 , z 1 ) c(z 1 )z 0 + F(z 1 ) z 0 + c(z 1 ) , z 1 with a ∈ C(z 1 ), b ∈ C(z 1 ) * , c ∈ C(z 1 ), F ∈ C[z 1 ] and F not a square. Still according to [D 06b]
the non-finite maximal abelian subgroups of J 0 are

J a = (z 0 , z 1 ) (z 0 + a(z 1 )) | a ∈ C(z 1 ) J m = (z 0 , z 1 ) (b(z 1 )z 0 , z 1 ) | a ∈ C(z 1 ) J F = (z 0 , z 1 ) c(z 1 )z 0 + F(z 1 ) z 0 + c(z 1 ) , z 1 | a ∈ C(z 1 )
where F denotes an element of C[z 1 ] which is not a square. Note that we can assume up to conjugacy that F is a polynomial with roots of multiplicity 1.

If φ belongs to J 0 , let us denote by Ab(φ) the non-finite maximal abelian subgroup of J 0 that contains φ. Up to conjugacy either (z 0 , z 1 ) (z 0 + a(z 1 ), z 1 ) and Ab(φ) = J a ;

or (z 0 , z 1 ) (b(z 1 )z 0 , z 1 ) and Ab(φ) = J m ;

or (z 0 , z 1 )

c(z 1 )z 0 +F(z 1 )
z 0 +c(z 1 ) , z 1 and Ab(φ) = J F . Proposition 7.33 [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -Let φ be an element of J 0 that is a Jonquières twist. Then the centralizer of φ in Bir(P 2 C ) is contained in J .

Proof. -Consider a birational self map ϕ : (z 0 , z 1 ) (ϕ 0 (z 0 , z 1 ), ϕ 1 (z 0 , z 1 )) of P 2 C that commutes with φ. If ϕ does not belong to J , then ϕ 1 = cst is a fibration invariant by φ distinct from z 1 = cst. Then φ is of finite order (Lemma 8.17): contradiction with the fact that φ is a Jonquières twist.

7.5.2.1. Centralizers of elements of J a . -Note that elements of J a are not Jonquières twists but elliptic maps. Hence their centralizers are described in §7.5.1. Let us give some details. Let φ : (z 0 , z 1 ) (z 0 + a(z 1 ), z 1 ) be a non-trivial element of J a (i.e. a ≡ 0). Up to conjugacy by (z 0 , z 1 ) (a(z 1 )z 0 , z 1 ) one can assume that a ≡ 1. The centralizer of (z 0 , z 1 )

(z 0 + 1, z 1 )
is isomorphic to J a PGL(2, C) (see §7.5.1). Hence

Corollary 7.34. -The centralizer of a non-trivial element of J a is isomorphic to J a PGL(2, C).

7.5.2.2. centralizers of twists of J m . -An element φ of J m is a Jonquières twist if and only if up to birational conjugacy (z 0 , z 1 ) (a(z 1 )z 0 , z 1 )

with a ∈ C(z 1 ) C * .
Remark that if a belongs to C * , then (z 0 , z 1 ) (az 0 , z 1 ) is an elliptic map whose centralizer is described in §7.5.1. Assume now that φ ∈ J m is a Jonquières twist. Let a ∈ C(z 1 ) C * .

Denote by Stab

(a) = ν ∈ PGL(2, C) | a(ν(z 1 )) = a(z 1 ) ±1
the subgroup of PGL(2, C) and by

stab(a) = ν ∈ PGL(2, C) | a(ν(z 1 )) = a(z 1 )
the normal subgroup of Stab(a). Consider also

stab(a) = (z 0 , ν(z 1 )) | ν ∈ stab(a)
and Stab(a) the group generated by stab(a) and the elements

(z 0 , z 1 ) 1 z 0 , ν(z 1 )
with ν ∈ Stab(a) stab(a).

Proposition 7.35 ([CD12b]

). -Let φ be a Jonquières twist in J m . The centralizer of φ in

Bir(P 2 C ) is J m Stab(a); in particular it is a finite extension of Ab(φ) = J m .
Remark 7.36. -One can write φ as (z 0 , z 1 ) (a(z 1 )z 0 , z 1 ) with a ∈ C(z 1 ) C * . For generic a the group Stab(a) is trivial, so for generic φ ∈ J m the centralizer of φ in Bir(P 2 C ) coincides with J m = Ab(φ).

Proof. -Write φ as (z 0 , z 1 ) (a(z 1 )z 0 , z 1 ) with a ∈ C(z 1 ) C * . If ψ commutes with φ, then ψ preserves the fibration z 1 = cst (Proposition 7.33), i.e.

ψ : (z 0 , z 1 ) A(z 1 )z 0 + B(z 1 ) C(z 1 )z 0 + D(z 1 ) , ν(z 1 ) with A B C D ∈ PGL(2, C(z 1 )
) and ν ∈ PGL(2, C). Since ψ and φ commute, the following

hold A(z 1 )C(z 1 ) 1 -a(ν(z 1 )) = 0 B(z 1 )D(z 1 ) 1 -a(ν(z 1 )) = 0 Therefore, AC ≡ 0 and BD ≡ 0, i.e. B = C = 0 or A = D = 0. Assume first that B = C = 0, i.e. that ψ : (z 0 , z 1 ) (A(z 1 )z 0 , ν(z 1 )).
The condition φ • ψ = ψ • φ implies a(ν(z 1 )) = a(z 1 ). As stab(a) is contained in the centralizer of φ in Bir(P 2 C ) the map φ belongs to J m stab(a).

Suppose now that A = D = 0, i.e. that ψ : (z 0 , z 1 )

B(z 1 ) z 0 , ν(z 1 ) . The equality ψ • ϕ = ϕ • ψ implies that a(ν(z 1 )) = a(z 1 ) -1 . But Stab(a) is contained in the centralizer of φ in Bir(P 2 C ), so ψ belongs to J m Stab(a).
7.5.2.3. Centralizers of elements of J F . -Let φ be a twist in J F . Let us write φ as

(z 0 , z 1 ) c(z 1 )z 0 + F(z 1 ) z 0 + c(z 1 )
, z 1 with c ∈ C(z 1 ) * and F ∈ C[z 1 ] whose roots have multiplicity 1. The curve C of fixed points of φ is given by z 2 0 = F(z 1 ). Since F has simple roots one has

   C is rational when 1 ≤ deg F ≤ 2; the genus of C is 1 when 3 ≤ deg F ≤ 4; the genus of C is ≥ 2 when deg F ≥ 5.
Assume first that the genus of C is positive.

Lemma 7.37

([CD12b]). -Let φ : (z 0 , z 1 ) c(z 1 )z 0 + F(z 1 ) z 0 + c(z 1 ) , z 1 c ∈ C(z 1 ) * , F ∈ C[z 1 ]
be a twist in J F . The curve z 2 0 = F(z 1 ) and the fibers z 1 =cst are invariant and there is no other invariant curves.

Proof. -The map φ has two fixed points on a generic fiber which correspond to the intersection of the fiber with the curve z 2 0 = F(z 1 ). Assume by contradiction that there is an other invariant curve C . The curve C intersects a generic fiber in a finite number of points that are invariant by φ. But a Möbius transformation that preserves more than three points is periodic: contradiction with the fact that φ is a Jonquières twist, so of infinite order.

Proposition 7.38

([CD12b]). -Let φ : (z 0 , z 1 ) c(z 1 )z 0 + F(z 1 ) z 0 + c(z 1 ) , z 1 c ∈ C(z 1 ) * , F ∈ C[z 1 ]
be a twist in J F . Assume that F has only simple roots and deg F ≥ 3, i.e. the curve

z 2 0 = F(z 1 ) has genus ≥ 1. Then the centralizer of φ in Bir(P 2 C ) is a finite extension of Ab(φ) = J F .
Proof. -Take α ∈ C such that F(α) = 0. The restriction φ |z 1 =α of φ on the fiber z 1 = α has two fixed points: (± F(α), α). Note that the centralizer Cent(φ) of φ in Bir(P 2 C ) is contained in J (Proposition 7.33). We will focus on elements ψ of Cent(φ) that preserve the fibration z 1 =cst fiberwise, i.e. on the kernel of pr 2|Cent(φ) : Cent(φ) → PGL(2, C).

Remark that any ψ ∈ Cent(φ) preserves C and that the automorphism ψ |C of C preserves (± F(α), α) . Hence either ψ |C = id, that is ψ ∈ J F , or ψ |C is the involution (z 0 , z 1 ) → (-z 0 , z 1 ) of C . Note that the restriction of τ : (z 0 , z 1 )

-F(z 1 ) z 0 , z 1 to C is τ |C : (z 0 , z 1 )
(-z 0 , z 1 ). Therefore, any birational self map of P 2 C that preserves both C and the fibration z 1 =cst fiberwise belongs either to J F or to τ • J F . But

τ • φ • τ -1 = τ • φ • τ = φ -1 ,
(C ) = {id, τ |C }).
Assume that C is rational.

Lemma 7.39 ([CD12b]

). -Let φ ∈ J F be a Jonquières twist such that the curve C of fixed points of φ is rational. Any element that commutes with φ belongs to J and preserves C .

Proof. -The curve of fixed points of φ is given by z 2 0 = F(z 1 ). Let ψ be a birational self map of P 2 C such that φ • ψ = ψ • φ. According to Proposition 7.33 the map ψ preserves the fibration z 1 =cst. Either ψ contracts C or ψ preserves C . But C is transverse to the fibration z 1 =cst, so ψ can not contract C . As a result ϕ is an element of J that preserves C .

Note that the case deg F ≥ 3 has already been studied, so let us assume that deg

F ≤ 2. Remark that if φ : (z 0 , z 1 ) c(z 1 )z 0 + z 1 z 0 + c(z 1 ) , z 1 and if ϕ : (z 0 , z 1 ) z 0 γz 1 + δ , αz 1 + β γz 1 + δ then ϕ -1 • φ • ϕ is of the following type (z 0 , z 1 ) c(z 1 )z 0 + (αz 1 + β)(γz 1 + δ) z 0 + c(z 1 ) , z 1 .
In other words thanks to

(z 0 , z 1 ) c(z 1 )z 0 + z 1 z 0 + c(z 1 ) , z 1
we obtain all polynomials (αz 1 + β)(γz 1 + δ) of degree 2 with simple roots. So one can suppose that deg F = 1. Note that if deg F = 1, i.e. F(z 1 ) = αz 1 + β, then up to conjugacy by (z 0 , z 1 ) → z 0 , z 1 -β α one can assume that F : z 1 → z 1 .

Lemma 7.40 [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -Consider the birational self map of P 2 C given by φ :

(z 0 , z 1 ) c(z 1 )z 0 + z 1 z 0 + c(z 1 ) , z 1 with c ∈ C(z 1 ) * .
If ψ is a birational self map of P 2 C that commutes with φ, then either pr 2 (ψ) = α z 1 with α ∈ C * ; or pr 2 (ψ) = ζz 1 with ζ root of unity. Furthermore pr 2 (ψ) belongs to the finite group stab 4c 2 (z 1 ) c 2 (z 1 )-z 1 .

For any α non-zero consider the dihedral group

D ∞ (α) = z 1 → α z 1 , z 1 → ζz 1 | ζ root of unity
Note that all the D ∞ (α) are conjugate to D ∞ (1).

Proposition 7.41 [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -Let φ ∈ J F be a Jonquières twist such that the fixed curve of φ is rational. Up to conjugacy we can assume that φ : (z 0 , z 1 ) c(z 1 )z 0 + z 1 z 0 + c(z 1 ) , z 1 with c ∈ C(z 1 ) C. The centralizer of φ in Bir(P 2 C ) is

J z 1 stab 4c 2 (z 1 ) c 2 (z 1 ) -z 1 ∩ D ∞ (α)
for some α ∈ C * .

Proof. -Denote by Cent(φ) the centralizer of φ in Bir(P 2 C ), and by C the fixed curve of φ.

Let us first assume that any element of Cent(φ) preserves the fibration z 1 =cst fiberwise. Then Cent(φ) = J z 1 .

Assume now that there exists an element ψ in Cent(φ) that does not preserve the fibration z 1 =cst fiberwise. According to Lemma 7.40 either pr 2 (ψ) = ζz 1 with ζ root of unity, or pr 2 (ψ) = α z 1 with α in C * . If pr 2 (ψ) = ζz 1 with ζ root of unity, then 4c 2 (ζz 1 ) c 2 (ζz 1 )ζz 1 = 4c 2 (z 1 ) c 2 (z 1 )z 1 i.e. c 2 (ζz 1 ) = ζc 2 (z 1 ). There exists υ such that υ 2 = ζ and c(υ 2 z 1 ) = υc(z 1 ). Note that ϕ : (z 0 , z 1 ) → (υz 0 , υ 2 z 1 ) belongs to Cent(φ). Remark that pr 2 (ψ • ϕ -1 ) = id, so ψ • ϕ -1 belongs to J z 1 .

If pr 2 (ψ) = α z 1 , then 4c 2 α z 1 c 2 α z 1 -z 1 = 4c 2 (z 1 ) c 2 (z 1 ) -z 1 i.e. c 2 α z 1 = α z 2 1 c 2 (z 1 ). There exists β in C such that β 2 = α and c β 2 z 1 = β z 1 c(z 1 ). Remark that the map (z 0 , z 1 ) → βz 0 z 1 , β 2 z 1 commutes with φ. The map ψ • ϕ -1
belongs to Cent(φ) and preserves the fibration z 1 =cst fiberwise; hence ψ • ϕ -1 belongs to J z 1 .

We thus have established: [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -The centralizer of a Jonquières twist φ that preserves fiberwise the fibration in the plane Cremona group is a finite extension of Ab(φ).

Proposition 7.42 ([
7.5.2.4. Centralizers of elements of J J 0 . -The description of the centralizers of elements of J 0 (Proposition 7.42) allows to describe, up to finite index, the centralizer of elements of J .

Generically these maps have a trivial centralizer ([CD12b]

). A consequence of the study of the centralizers of elements of J is: [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -The centralizer of a Jonquières twist is virtually solvable.

Corollary 7.43 ([
Zhao has refined this statement:

Proposition 7.44 ([Zha19]
). -The centralizer of a Jonquières twist whose action on the basis of the rational fibration is of infinite order is virtually abelian.

7.5.3. What about the others ? -7.5.3.1. -Let φ be an Halphen twist. Up to birational conjugacy one can assume that φ is an element of a rational surface S with an elliptic fibration and that this fibration is φ-invariant ( §2.3). Furthermore we can assume that there is no smooth curve of self-intersection -1 in the fibers, i.e. that the fibration is minimal, and so that φ is an automorphism. The elliptic fibration is the unique φ-invariant fibration ( [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). As a result the fibration is invariant by all elements that commute with φ, and the centralizer of φ is contained in Aut(S).

Since the fibration is minimal, the surface S is obtained by blowing up the complex projective plane in the nine base-points of an Halphen pencil and the rank of its Néron-Severi group is equal to 10. The group Aut(S) can be embedded in the endomorphisms of H 2 (S, Z) for the intersection form and preserves the class [K S ] of the canonical divisor, that is the class of the fibration. The dimension of the orthogonal hyperplane to [K S ] is 9, and the restriction of the intersection form on its hyperplane is semi-negative: its kernel coincides with Z[K S ]. As a consequence Aut(S) contains an abelian group of finite index with rank ≤ 8. We can thus state:

Proposition 7.45 [START_REF] Gizatullin | Rational G-surfaces[END_REF]). -Let φ be an Halphen twist. The centralizer of φ in Bir(P 2 C ) contains a subgroup of finite index which is abelian, free and of rank ≤ 8. 7.5.3.2. -We finish the description of the centralizers of birational maps with the case of loxodromic maps in §8.1.2.

CHAPTER 8 CONSEQUENCES OF THE ACTION OF THE CREMONA GROUP ON AN INFINITE DIMENSIONAL HYPERBOLIC SPACE

As we will see in this chapter one of the main techniques to better understand infinite subgroups of Bir(P 2 C ) is the construction of the action by isometries of the plane Cremona group on an infinite dimensional hyperbolic space detailed in Chapter 2 and the use of results from hyperbolic geometry and group theory.

In the first section we recall results of Demazure and Beauville that suggest that the plane Cremona group behaves like a rank 2 group. We give an outline of the proof of the description of the centralizer of a loxodromic element of Bir(P 2 C ). On the one hand it finishes the description of the centralizer of the elements of Bir(P 2 C ), on the other hand it suggests that Bir(P 2 C ) behaves as a group of rank 1. We end this section by recalling the description of the morphisms from a countable group with Kazhdan property (T ) into Bir(P 2 C ) which also insinuates that Bir(P 2 C ) behaves as a group of rank 1. In the second section we give an outline of the proofs of the description of elliptic subgroups of Bir(P 2 C ), i.e. the subgroups of Bir(P 2 C ) whose all elements are elliptic: if G is such a group, either G is a bounded subgroup of Bir(P 2 C ), or G is a torsion subgroup ([Ure]). It is thus natural to describe torsion subgroups of Bir(P 2 C ). In the third section we give an outline of the proof of the fact that if G is a torsion subgroup of Bir(P 2 C ), then G is isomorphic to a bounded subgroup of Bir(P 2 C ); furthermore it is isomorphic to a subgroup of GL(48, C). Let us mention the surprising fact that the proof uses model theory as Malcev already did in [START_REF] Malcev | On isomorphic matrix representations of infinite groups[END_REF].

The fourth section deals with Tits alternative and Burnside problem. We recall the Ping Pong Lemma and give a sketch of the proof of the Tits alternative for the Cremona group, i.e. the proof of Theorem 8.1 ([Can11, Ure]). -Every subgroup of Bir(P 2 C ) either is virtually solvable, or contains a non-abelian free group.

One consequence is a positive answer to the Burnside problem for the Cremona group: every finitely generated torsion subgroup of Bir(P 2 C ) is finite. The study of solvable groups is a very old problem. For instance let us recall the Lie-Kolchin theorem: any linear solvable subgroup is up to finite index triangularizable ( [START_REF] Kargapolov | Fundamentals of the theory of groups[END_REF]). Note that the assumption "up to finite index" is essential: for instance the subgroup [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF]). The sixth section is devoted to normal subgroups of Bir(P 2 C ) and the non-simplicity of Bir(P 2 C ). Strategies of [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF] and [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF] are evoked. A consequence of one result of [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF] is the following property: the Cremona group contains infinitely many characteristic subgroups ( [START_REF] Cantat | The Cremona group in two variables[END_REF]).

1 0 1 -1 , -1 1 0 1 of PGL(2, C) is isomorphic to S 3 ,
Taking the results of the sixth section as a starting point Urech gives a classification of all simple groups that act non-trivially by birational maps on complex compact Kähler surfaces. In particular he gets the two following statements: Theorem 8.2 ([Ure20]). -A simple group G acts non-trivially by birational maps on a rational complex projective surface if and only if G is isomorphic to a subgroup of PGL(3, C).

Theorem 8.3 ([Ure20]

). -Let G be a simple subgroup of Bir(P 2 C ). Then G does not contain loxodromic elements; if G contains a parabolic element, then G is conjugate to a subgroup of J ; if G is an elliptic subgroup, then G is either a simple subgroup of an algebraic subgroup of Bir(P 2 C ), or conjugate to a subgroup of G.

In the last section we give a sketch of the proof of these results.

8.1. A group of rank 1.5 8.1.1. Rank 2 phenomenon. -Let k be a field. Consider a connected semi-simple algebraic group G defined over k. Let Ψ : G → Aut(G) be the mapping g → Ψ g where Ψ g denotes the inner automorphism given by

Ψ g : G → G, h → ghg -1 .
For each g in G one can define Ad g to be the derivative of Ψ g at the origin Ad g = (DΨ g ) id : g → g where D is the differential and g = T id G is the tangent space of G at the identity element of G. The map Ad : G → Aut(g), g → Ad g is a group representation called the adjoint representation of G. The k-rank of G is the maximal dimension of a connected algebraic subgroup of G which is diagonalizable over k in GL(g). Such a maximal diagonalizable subgroup is a maximal torus. Let us give an idea of the proof. Consider a finite group G of Bir(P 2 C ). It can be realized as a group of automorphisms of a rational surface S (see for instance [START_REF] De Fernex | Resolution of indeterminacy of pairs[END_REF]). Moreover one can assume that every birational G-equivariant morphism of S onto a smooth surface with a G-action is an isomorphism. Then according to [START_REF] Ju | Rational surfaces over perfect fields[END_REF] either G preserves a fibration π : S → P 1 with rational fibers, or Pic(S) G has rank 1.

In the first case G embeds in the group of automorphisms of the generic fibre P 1 C(t) of π and Beauville classified the p-elementary subgroups of Aut(P 1 C(t) ). In the last case S is a del Pezzo surface and the group Aut(S) is well known. Beauville also classified the p-elementary subgroups of such groups.

Combining this result of those recalled in Chapter 7, §7.5 Zhao get:

Theorem 8.8. -Let φ ∈ Bir(P 2 C ) be an element of infinite order. If the centralizer of φ is not virtually abelian, then either φ is an elliptic map, or a power of φ is conjugate to an automorphism of C 2 of the form (z 0 , z 1 ) → (z 0 , z 1 + 1) or (z 0 , z 1 ) → (z 0 , βz 1 ) with β ∈ C * . Remark 8.9. -This statement also holds for Bir(P 2 k ) where k is an algebraically closed field ([Zha19]).

8.1.2. Rank 1 phenomenon. -Generic elements of degree ≥ 2 of Bir(P 2 C ) are loxodromic and hence can not be conjugate to elements of the maximal torus D 2 . The description of their centralizer is given by: Theorem 8.10 ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). -Let φ be a loxodromic element of Bir(P 2 C ). The infinite cyclic subgroup of Bir(P 2 C ) generated by φ has finite index in the centralizer

Cent(φ) = ψ ∈ Bir(P 2 C ) | ψ • φ = φ • ψ of φ.
Remark 8.11. -Theorem 8.10 holds for any field k.

The centralizer of a generic element of SL(n + 1, C) is isomorphic to (C * ) n ; Theorem 8.10 suggests that Bir(P 2 C ) behaves as a group of rank 1. Sketch of the proof. -If ψ commutes to φ, then the isometry ψ * of H ∞ preserves the axis Ax(φ) of φ * and its two endpoints. Consider the morphism Θ which maps Cent(φ) to the group of isometries of Ax(φ). View it as a morphism into the group of translations R of the line. On the one hand the translation lengths are bounded from below by log(λ L ) where λ L is the Lehmer number, i.e. the unique root > 1 of the irreducible polynomial x 10 + x 9x 7x 6x 5x 4x 3 + x + 1 (see [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). On the other hand every discrete subgroup of R is trivial or cyclic. As a result the image of Θ is a cyclic group. Its kernel is made of elliptic elements of Cent(φ) fixing all points of Ax(φ). Denote by e φ the projection of e 0 on Ax(φ). Since ker Θ fixes e φ , the inequality dist(ψ * e 0 , e 0 ) ≤ 2dist(e 0 , e φ )

holds. As a consequence ker Θ is a group of birational maps of bounded degree. From [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF] the Zariski closure of ker Θ in Bir(P 2 C ) is an algebraic subgroup of Bir(P 2 C ). Let us denote by G the connected component of the identity in this group. Assume that ker Θ is infinite. Then dim G is positive and G is contained, after conjugacy, in the group of automorphisms of a minimal, rational surface ( [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF][START_REF] Enriques | Sui gruppi continui di transformazioni cremoniani nel piano[END_REF]). Therefore, G contains a Zariski closed abelian subgroup whose orbits have dimension 1. Those orbits are organised in a pencil of curves that is invariant under the action of φ: contradiction with the fact that φ * is loxodromic. As a result ker Θ is finite.

8.1.3. Rank 1 phenomenon. -To generalize Margulis work on linear representations of lattices of simple real Lie groups to non-linear representations Zimmer proposed to study the actions of lattices on compact varieties ([Zim86, Zim84, Zim87a, Zim87b]). One of the main conjectures of the program drawn by Zimmer is: let G be a connex, simple, real Lie group and let Γ be a lattice of G. If there exists a morphism from Γ into the diffeomorphisms group of a compact variety V with infinite image, then the real rank of G is less or equal to the dimension of V .

In the context of birational self maps one has the following statement that can be see as another rank one phenomenum: Theorem 8.12 ([Can11, D 06a]). -Let S be a complex projective surface. Let Γ be a countable group with Kazhdan property (T ).

If υ : Γ → Bir(P 2 C ) is a morphism with infinite image, then υ is conjugate to a morphism into PGL(3, C).

Remark 8.13. -Theorem 8.12 indeed holds for any algebraically closed field k.

Sketch of the proof. -The first step is based on a fixed point property: since Γ has Kazhdan property (T), then υ(Γ) acts by isometries on H ∞ and (υ(Γ)) * has a fixed point. Then according to [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger)[END_REF] all its orbits have bounded diameter. Hence ρ(Γ) has bounded degree. There thus exists a birational map π :

X P 2 C such that Γ S = π -1 • Γ • π is a subgroup of Aut(S); Aut(S) 0 ∩ Γ S has finite index in Γ S .
The classification of algebraic groups of maps of surfaces and the fact that every subgroup of SL(2, C) having Kazhdan property (T) is finite allow to prove that: since Aut(S) 0 contains an infinite group with Kazhdan property (T) the surface S is isomorphic to the projective plane P 2 C .

8.2. Subgroups of elliptic elements of Bir(P 2 C ) A subgroup G of the plane Cremona group is elliptic if any element of G is an elliptic birational map. Let us give an example: a bounded subgroup of Bir(P 2 C ) is elliptic. But not all elliptic subgroups are bounded; indeed for instance all elements of (z 0 , z 1 + a(z 0 )) | a ∈ C(z 0 ) are elliptic but (z 0 , z 1 + a(z 0 )) | a ∈ C(z 0 ) contains elements of arbitrarily high degrees;

Wright gives examples of subgroups of Bir(P 2 C ) isomorphic to a subgroup of roots of unity of C * that are not bounded [START_REF] Wright | Abelian subgroups of Aut k (k[X, Y ]) and applications to actions on the affine plane[END_REF]). Let us be more precise. Set ψ 0 : (z 0 , z 1 ) → (-z 0 , -z 1 ) and for any k ≥ 1

α k = exp iπ 2 k , φ k : (z 0 , z 1 ) → (z 1 , c k z 2 k +1 1 + z 0 ), ϕ k = φ 2 k • φ 2 k-1 • . . . • φ 2 1
where c k denotes an element of C * . Consider

ψ k = ϕ -1 k • (z 0 , z 1 ) → (α k z 0 , α p k z 1 ) • ϕ k where p is an odd integer. The group G = k≥0 ψ k
is an abelian group obtained as a growing union of finite cyclic groups that does not preserve any fibration ( [START_REF] Lamy | Dynamique des groupes paraboliques d'automorphismes polynomiaux de C 2[END_REF]). This gives all the possibilities for elliptic subgroups of Bir(P 2 C ): Theorem 8.14 ( [Ure]). -Let G be an elliptic subgroup of the plane Cremona group. Then one of the following holds:

G is a bounded subgroup; G preserves a rational fibration; G is a torsion group.

Furthermore he characterizes torsion subgroups of Bir(P 2

C ): Theorem 8.15 ( [Ure]). -Let G ⊂ Bir(P 2 C ) be a torsion group. Then G is isomorphic to a bounded subgroup of Bir(P 2 C ). Furthermore G is isomorphic to a subgroup of GL(48, C).

As a consequence he gets an analogue of the Theorem of Jordan and Schur:

Corollary 8.16 ([Ure]

). -There exists a constant γ such that every torsion subgroup of Bir(P 2 C ) contains a commutative normal subgroup of index ≤ γ.

Theorems 8.14 and 8.15 allow to refine the result of Cantat about Tits alternative ( §8.4); the description of solvable subgroups of Bir(P 2 C ) (see §8.5). The aim of the section is to prove Theorem 8.14. We need the following technical lemmas.

Lemma 8.17 [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]). -Let φ be a birational self map of the complex projective plane that fixes pointwise two different rational fibrations. Then φ is of finite order.

Proof. -The intersections of the generic fibres of these two fibrations are finite, uniformly bounded. But these intersections are invariant by φ so φ is of finite order. such that dim ker f ≥ 6. Assume by contradiction that there exists a second rational fibration π : F n → P 1 C preserved by G; this yields to a second homomorphism g : G → PGL(2, C).

One has dim ker π | ker π > 0; therefore, dim(ker f ∩ker g) > 0. In particular ker f ∩ker g contains an element of infinite order: contradiction with Lemma 8.17.

Lemma 8.19 ( [Ure]). -Let G ⊂ Bir(P 2 C ) be an algebraic subgroup isomorphic as an algebraic group to C * .

There exists a constant K(G) such that any elliptic element of

Cent(G) = ϕ ∈ Bir(P 2 C ) | ϕ • ψ = ψ • ϕ ∀ ψ ∈ G has degree ≤ K(G).
Proof. -Up to conjugacy by an element ψ ∈ Bir(P 2 C ) one can assume that G = (z 0 , z 1 ) → (αz 0 , z 1 ) | α ∈ C * .

An elliptic element of Cent(G) is of the following form ϕ : (z 0 , z 1 ) (z 0 ϕ 1 (z 1 ), ϕ 2 (z 1 ))

where ϕ 1 , ϕ 2 are rational functions. Since (deg ϕ n ) n is bounded, ϕ 1 is constant, and so -The group H is semi-simple; in particular its group of inner automorphisms has finite index in its group of algebraic automorphisms. As a result there exists N ∈ Z such that for any φ in G conjugation by φ N induces an inner automorphism of H. Hence, there exists an element ψ in H such that φ N • ψ centralizes H. By assumption H is semi-simple, so H contains a closed subgroup D isomorphic as an algebraic group to C * and this group is centralized by φ N • ψ. From Lemma 8.19 we get that deg(φ N • ψ) is bounded by a constant that depends neither on φ, nor on N. As H is an algebraic group both deg ψ and deg φ are also bounded independently of φ and N. Finally G is bounded.

ϕ 2 : z 1 → az 1 +b

Lemma 8.22 ([Ure]

). -Let G be a subgroup of Bir(P 2 C ) that fixes a point of H ∞ . Then the degree of all elements in G is uniformly bounded; there exist a smooth projective surface S and a birational map ϕ : If n = +∞, then take Γ a finitely generated subgroup of G such that dim Γ ≥ 9. By Lemma 8.18 the group Γ preserves a unique fibration and this fibration is, again by Lemma 8.18, preserved as well by Γ, φ for any φ in G. Assume now n ∈ N * . Let Γ be a finitely generated subgroup of G such that dim Γ = n. Let Γ 0 be the neutral component of Γ. For any ϕ ∈ G the group Γ

P 2 C S such that ϕ • G • ϕ -1 ⊂ Aut(S).
0 , ϕ • Γ 0 • ϕ -1 is con- nected and contained in Γ, ϕ • Γ • ϕ -
R ( +1) = id R ( ) ⊂ . . . ⊂ R (2) ⊂ R (1) ⊂ R (0) = R the derived series of R (i.e. R (k+1) = [R (k) , R (k) ]
). Note that dim R ( ) > 0 and R ( ) is abelian. This series is invariant under Aut(Γ 0 ), and so invariant under conjugation by elements of G. In particular G normalizes R ( ) . Since R ( ) is bounded, R ( ) is conjugate to one of the groups of Theorem 3.46; in particular R ( ) can be regularized. In other words, up to birational conjugacy, G is a subgroup of Bir(S) for some smooth projective surface S on which R ( ) acts regularly. If all the orbits of R The C-fibration of C * × C is given by the invertible functions; it is thus preserved by Aut(C * × C). In particular G preserves a rational fibration. If R ( ) C * × C * , then elements of G are monomial maps, and Lemma 8.20 allows to conclude.

Torsion subgroups of the Cremona group

As we have seen at the beginning of §8.2 some torsion groups can be embedded into Bir(P 2 C ) in such a way that they neither are bounded, nor preserve any fibration. However the group structure of torsion subgroups can be specified: Theorem 8.25 ( [Ure]). -A torsion subgroup G of Bir(P 2 C ) is isomorphic to a bounded subgroup of Bir(P 2 C ). Furthermore G is isomorphic to a subgroup of GL(48, C).

Malcev used model theory to prove that if for a given group G every finitely generated subgroup can be embedded into GL(n, k) for some field k, then there exists a field k such that G can be embedded into GL(n, k ). Let us briefly introduce the compactness theory from model theory; it states that a set of first order sentences has a model if and only if any of its finite subsets has a model. Definition. -Let {x i } i∈I be a set of variables. A condition is an expression of the form P(x i 1 , x i 2 , . . . , x i k ) = 0 or an expression of the form

P 1 (x i 1 , x i 2 , . . . , x i k ) = 0 ∨ P 2 (x i 1 , x i 2 , . . . , x i k ) = 0 ∨ . . . ∨ P (x i 1 , x i 2 , . . . , x i k ) = 0
where P and the P i 's are polynomials with integer coefficients.

Definition. -A mixed system is a set of conditions.

Definition. -A mixed system S is compatible if there exists a field k which contains values {y i } i∈I that satisfy S. Theorem 8.26 [START_REF] Malcev | On isomorphic matrix representations of infinite groups[END_REF]). -If every finite subset of a mixed system S is compatible, then S is compatible.

Let us now explain the proof of Theorem 8.25. Let G be a torsion subgroup of Bir(P 2 C ). If G is finite, then G is bounded; we can thus assume that G is infinite. Following Theorem 3.46 we will deal with different cases.

First assume that every finitely generated subgroup of G is isomorphic to a subgroup of PGL(3, C). Consider the closed embedding ρ of PGL(3, C) into GL(8, C) given by the adjoint representation. Let P 1 , P 2 , . . ., P n be polynomials in the set of variables {x i j } 1≤i, j≤8 such that ρ(PGL(3, C)) ⊂ GL(8, C) is the zero set of P 1 , P 2 , . . ., P n . To any element g ∈ G we associate a 8 × 8 matrix of variables (x g i j ). Consider the following mixed system S defined by

(1) the equations (x

f i j )(x g i j ) = (x h i j ) for all f , g, h ∈ G such that f • g = h; (2) the conditions i x g ii -1 = 0 ∨ i = j x g i j -1 = 0 ; (3) x id
ii -1 = 0 and x id i j = 0 for all 1 ≤ i = j ≤ N; (4) P k ({x i j }) = 0 for all 1 ≤ k ≤ n, for all g ∈ G;

(5) p = 0 for all p ∈ Z + primes. Lemma 8.27 ( [Ure]). -The system S is compatible.

Proof. -According to Theorem 8.26 it suffices to show that every finite subset of S is compatible. Let c 1 , c 2 , . . ., c n ∈ S be finitely many conditions. Only finitely many variables x g i j appear in c 1 , c 2 , . . ., c n . Let g 1 , g 2 , . . . , g ⊂ G be the finite set of all elements g ∈ G such that for some 1 ≤ i, j ≤ 8 the variable x g i j appears in one of the conditions c 1 , c 2 , . . ., c n .

Consider the finitely generated subgroup Γ = g 1 , g 2 , . . . , g of G. By Theorem 8.48 the group Γ is finite. Therefore, by assumption Γ has a faithful representation to PGL(3, C). This representation implies that C contains values that satisfy the conditions c 1 , c 2 , . . ., c n . In other words S is compatible.

As a result there exists a field k such that k contains values y g i j for all 1 ≤ i, j ≤ 8 and all g ∈ G satisfying conditions (1) to (5). Condition (5) asserts that the characteristic of k is 0. The group G has at most the cardinality of the continuum since G ⊂ Bir(P 2 C ); the values {y g i j } are thus contained in a subfield k of k that has the same cardinality as C. Hence k can be embedded into C as a subfield. Hence we may suppose that k = C. Consider the map ϕ : G → PGL(3, C), g → (y g i j ) i, j . Note that -conditions (1) imply that the image of any element of G is an invertible matrix and that ϕ is a group automorphism; -conditions (2) lead that this automorphism is injective; -conditions (3) imply ϕ(id) = id; -conditions (4) lead that ϕ(G) ⊂ PGL(3, C) ⊂ GL(8, C). Denote by S 6 the del Pezzo surface of degree 6. If any finitely generated subgroup of G can be embedded into Aut(S 6 ) D 2 Z 2Z × S 3 a similar reasoning leads to: G is isomorphic to a subgroup of Aut(S 6 ). If any finitely generated subgroup of G can be embedded into

Aut(P 1 C × P 1 C ) PGL(2, C) × PGL(2, C) Z 2Z ,
then G is isomorphic to a subgroup of Aut(P 1 C × P 1 C ). If any finitely generated subgroup of G can be embedded into

Aut(F 2n ) C[z 0 , z 1 ] 2n GL(2, C) µ 2n
for some n > 0 (and not necessarily the same for all finitely generated subgroups of G), then G is isomorphic to a subgroup of GL(2, C) and thus can be embedded in PGL(3, C).

It remains to consider the case where G contains -a finitely generated subgroup Γ 1 that can not be embedded into Aut(P 2 C ), -a finitely generated subgroup Γ 2 that can not be embedded into Aut(S 6 ), -a finitely generated subgroup Γ 3 that can not be embedded into Aut(P 1 C × P 1 C ), -a finitely generated subgroup Γ 4 that can not be embedded into Aut(F 2n ) for all n > 0. The finitely generated subgroup Γ = Γ 1 , Γ 2 , Γ 3 , Γ 4 is not isomorphic to any subgroup of infinite automorphisms group of a del Pezzo surface. Adding finitely many elements if needed we may assume that Γ has order > 648; as a consequence Γ is isomorphic neither to any subgroup of an automorphisms group of a del Pezzo surface (Theorem 3.39), nor to a subgroup of Aut(F 2n ) for all n > 0. Consider a finitely generated subgroup H of G. The finitely generated subgroup Γ, H , and in particular H, is isomorphic to a subgroup of (Theorem 3.46)

• either Aut(S, π) where π : S → P 1 C is an exceptional conic bundle, • or Aut(S, π) where (S, π) is a Z 2Z 2 -conic bundle and S is not a del Pezzo surface,

• or Aut(F 2n+1 ) for some n > 0. According to Lemmas 3.42, 3.43 and 3.44 the group H is isomorphic to a subgroup of PGL(2, C) × PGL(2, C). Therefore, every finitely generated subgroup of G is isomorphic to a subgroup of PGL(2, C) × PGL(2, C). The group G is thus isomorphic to a subgroup of PGL(2, C) × PGL(2, C) (Theorem 8.26) and hence to a subgroup of Aut(P 1 C × P 1 C ).

Lemma 8.28 ( [Ure]). -Every torsion subgroup of Bir(P 2 C ) is isomorphic to a subgroup of GL(48, C).

Proof. -Let G be a torsion group of Bir(P 2 C ). Assume that G is infinite. As we just see G is isomorphic to a subgroup of Aut(P 2 C ), Aut(P 1 C × P 1 C ), Aut(S 6 ) or Aut(F n ) for some n ≥ 2. According to the structure of Aut(F n ) and Lemma 3.42 all torsion subgroups of Aut(F n ) are isomorphic to a subgroup of GL(2, C) or PGL(2, C) × C * . But PGL(2, C) can be embedded into GL(3, C) and PGL(3, C) into GL(8, C), and Aut(S 6 ) into GL(6, C) (Lemma 3.41); the group G is thus isomorphic to a subgroup of GL(8, C). Suppose that G is finite and not contained in an infinite bounded subgroup. Then G is contained in the automorphism group (Theorem 3.46)

• either of a del Pezzo surface,

• or of an exceptional fibration,

• or of a ( Z 2Z ) 2 -fibration. In the first case we get from Lemma 3.47 that G is isomorphic to a subgroup of GL(8, C).

In the second case the group G can be embedded into PGL(2, C) × PGL(2, C) (Lemma 3.43).

In the last case G is isomorphic to a subgroup of GL(48, C) according to [Ure17, Lemma 6.2.12].

Tits alternative and Burnside problem

A group G is virtually solvable if G contains a finite index solvable subgroup. A group G satisfies Tits alternative if every subgroup of G either is virtually solvable or contains a non-abelian free subgroup.

A group G satisfies Tits alternative for finitely generated subgroups if every finitely generated subgroup of G either is virtually solvable or contains a non-abelian free subgroup.

Tits showed that linear groups over fields of characteristic zero satisfy the Tits alternative and that linear groups over fields of positive characteristic satisfy the Tits alternative for finitely generated subgroups [START_REF] Tits | Free subgroups in linear groups[END_REF]). Other well-known examples of groups that satisfy Tits alternative include mapping class groups of surfaces ( [START_REF] Ivanov | Algebraic properties of the Teichmüller modular group[END_REF]), the outer automorphisms group of the free group of finite rank n ([BFH00]), or hyperbolic groups in the sense of Gromov ([Gro87]). Lamy studied the group Aut(A 2 C ); in particular using its amalgamated product structure he showed that Tits alternative holds for Aut(A 2 C ) (see [START_REF] Lamy | L'alternative de Tits pour Aut[C 2[END_REF]). In [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF] Cantat established that Bir(P 2 C ) satisfies Tits alternative for finitely generated subgroups. Then Urech proved that Bir(P 2 C ) satisfies Tits alternative ([Ure20]).

On the contrary the group of C ∞ -diffeomorphisms of the circle does not satisfy Tits alternative ([BS85, GS87]). Note that since solvable subgroups have either polynomial or exponential growth, if G satisfies Tits alternative, G does not contain groups with intermediate growth.

The main technique to prove that a group contains a non-abelian free group is the ping-pong Lemma (for instance [dlH00]): Lemma 8.29. -Let S be a set. Let g 1 and g 2 be two bijections of S. Assume that S contains two non-empty disjoint subsets S 1 and S 2 such that

g m 1 (S 2 ) ⊂ S 1 g m 2 (S 1 ) ⊂ S 2 ∀ m ∈ Z {0}.
Then g 1 , g 2 is a free group on two generators.

Sketch of the Proof. -Let w = w(a, b) be a reduced word that represents a non-trivial element in the free group F 2 = a, b . Let us prove that w(g 1 , g 2 ) is a non-trivial map of S. Up to conjugacy by a power of g 1 assume that w(g 1 , g 2 ) starts and ends with a power of g 1 :

w(g 1 , g 2 ) = g n 1 g m n 2 . . . g m 1 2 g 0 1 .
One checks that g 0 1 maps S 2 into S 1 , then g m 1 2 g 0 1 maps S 2 into S 2 , . . . and w maps S 2 into S 1 . As S 2 is disjoint from S 1 one gets that w(g 1 , g 2 ) is non-trivial.

Consider a group Γ that acts on a hyperbolic space H ∞ and that contains two loxodromic isometries ψ 1 and ψ 2 whose fixed points in ∂H ∞ form two disjoint pairs. Let us take disjoint neighborhoods S i ⊂ H ∞ of the fixed point sets of ψ i , i = 1, 2. Then Lemma 8.29 applied to sufficiently high powers ψ n 1 and ψ n 2 of ψ 1 and ψ 2 respectively produces a free subgroup of Γ. This strategy can be used for the Cremona group acting by isometries on H ∞ (P 2 C ). More precisely Cantat obtained the following result: Theorem 8.30 [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let S be a projective surface S over a field k. The group Bir(S) satisfies Tits alternative for finitely generated subgroups.

Then Urech proves:

Theorem 8.31 ([Ure]). -Let S be a complex Kähler surface. Then Bir(S) satisfies Tits alternative.

Let us now give a sketch of the proof of this result in the case S = P 2 C .

8.4.1. Subgroups of Bir(P 2 C ) that contain a loxodromic element. -Recall that the subgroup of diagonal automorphisms

D 2 = (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C * ⊂ PGL(3, C) = Aut(P 2 C ) is a real torus of rank 2; a matrix A = (a i j ) ∈ GL(2, Z) determines a birational map of P 2 C (z 0 , z 1 ) z a 00 0 z a 01 1 , z a 10 0 z a 11 1 The normalizer of D 2 in Bir(P 2 C ) is the semidirect product Norm D 2 , Bir(P 2 C ) = φ ∈ Bir(P 2 C ) | φ • D 2 • φ -1 = D 2 = GL(2, Z) D 2 . If M ∈ GL(2,
Z) has spectral radius strictly larger than 1, the associated birational map is loxodromic. In particular there exist loxodromic elements that normalize an infinite elliptic subgroup. Up to conjugacy these are the only examples with this property: Theorem 8.32 ([DP12]). -Let G be a subgroup of Bir(P 2 C ) containing at least one loxodromic element. Assume that there exists a short exact sequence

1 -→ A -→ G -→ B -→ 1
where A is infinite and of bounded degree.

Then G is conjugate to a subgroup of GL(2, Z) D 2 .

Urech generalizes this result to the case where A is an infinite group of elliptic elements ([Ure]):

Theorem 8.33 ( [Ure]). -Let G be a subgroup of Bir(P 2 C ) containing at least one loxodromic element. Suppose that there exists a short sequence

1 -→ A -→ G -→ B -→ 1
where A is an infinite group of elliptic elements.

Then G is conjugate to a subgroup of GL(2, Z) D 2 .

In order to give the proof of Theorem 8.33 we need to establish some results.

Lemma 8.34 ([Ure]

). -Let φ be a loxodromic monomial map of the complex projective plane. Let ∆ 2 be an infinite subgroup of D 2 normalized by φ.

Then ∆ 2 is dense in D 2 with respect to the Zariski topology.

Proof. -Denote by ∆ 2 0 the neutral component of the Zariski closure of ∆ 2 .

If ∆ 2 0 has a dense orbit on P 2 C , then ∆ 2 is dense in D 2 . Otherwise the dimension of the generic orbits of ∆ 2 0 is 1. But φ normalizes ∆ 2 0 , so preserves its orbits. In particular φ thus preserves a rational fibration: contradiction with the fact that φ is loxodromic.

In [START_REF] Shepherd-Barron | Some effectivity questions for plane Cremona transformations[END_REF] the classification of tight elements of Bir(P 2 C ) is given: Theorem 8.35 ([SB13]). -Every loxodromic element of the plane Cremona group is rigid.

Let φ be a loxodromic birational self map of the complex projective plane; then if φ is conjugate to a monomial map, no power of φ is tight; otherwise φ n is tight for some integer n.

Consider a subgroup G of Bir(P 2 C ). Let φ ∈ G be a rigid element; then φ is also a rigid element in G. The same holds for tight elements but the converse does not: there exist loxodromic maps φ ∈ G such that φ is tight in G but not in Bir(P 2 C ). Proof of Theorem 8.35 and Lemma 8.34 imply the following: Theorem 8.36 ( [Ure]). -Let G be a subgroup of Bir(P 2 C ). Let φ be a loxodromic element. The following assertions are equivalent: no power of φ is tight in G; there is a subgroup ∆ 2 ⊂ G that is normalized by φ and a birational self map

ψ of P 2 C such that ψ • ∆ 2 • ψ -1 is a dense subgroup of D 2 and ψ • ϕ • ψ -1 belongs to GL(2, Z) D 2 .
Proof of Theorem 8.33. -The group A fixes a point p ∈ ∂H ∞ ∪ H ∞ (Theorem 8.44). Note that if p belongs to H ∞ , then A is bounded and Theorem 8.32 allows to conclude. Let us assume that p belongs to ∂H ∞ . Remark that if A fixes an other point q on ∂H ∞ , then A fixes the geodesic between p and q, and so A would be bounded again. Suppose thus that p is the only fixed point of A in ∂H ∞ . Consider a loxodromic map φ of N. It normalizes A and so φ fixes p. As φ is loxodromic, φ does not preserve any fibration; consequently p does not correspond to the class of a fibration. From Lemma 8.24 any finitely generated group of elliptic elements that fixes p is bounded. Let G be the subgroup of birational self maps of P 2 C that fix p. Denote by L the one-dimensional subspace of Z(P 2 C ) corresponding to p. The group G fixes p; hence its linear action on Z(P 2 C ) acts on L by automorphisms preserving the orientation. This implies the existence of a group homomorphism ρ : G → R * + . Note that G does not contain any parabolic element because p does not correspond to the class of a fibration and that loxodromic elements do not fix any vector in Z(P 2 C ). As a result ker ρ consists of elliptic elements. But 1 is the only eigenvalue of a map of Z(P 2 C ) induced by an elliptic birational self map ([Can11]); as a consequence any elliptic birational map of G is contained in ker ρ.

Take a loxodromic map φ in G. Let us show by contradiction that no power of φ is tight in G. So assume that there exists n ∈ Z such that φ n is tight in G. The subgroup N of G is infinite and φ has finite index in Cent(φ) (Theorem 8.10); there thus exists ψ ∈ G that do not commute with φ n . Since all non trivial elements of

φ n are loxodromic ([CL13]) the map ψ • φ n • ψ -1 • φ -n is loxodromic. But ρ ψ • φ n • ψ -1 • φ -n = 1, i.e. ψ • φ n • ψ -1 • φ -n
is elliptic: contradiction. Finally no power of φ is tight in G. According to Theorem 8.36 there exist ϕ ∈ Bir(P 2 C ) and ∆ 2 an algebraic subgroup of G such that

-ϕ • φ • ϕ -1 is monomial; -ϕ • ∆ 2 • ϕ -1 = D 2 . Consider a finitely generated subgroup Γ of ker ρ. The Zariski closure Γ of Γ is an algebraic subgroup of G because Γ is bounded. Set d = sup{dim Γ | Γ ⊂ ker ρ finitely generated}
We will distinguish the cases d is finite and d is infinite.

-First consider the case d < ∞. Note that ker ρ contains a subgroup conjugated to D 2 , so d ≥ 2. Take Γ a finitely generated subgroup of ker ρ such that dim Γ = d. Let Γ 0 be the neutral component of the algebraic group Γ. Let φ be an element of G. The group φ • Γ 0 • φ -1 is again an algebraic subgroup and Γ

0 , φ • Γ 0 • φ -1 is contained in Γ, φ • Γ • φ -1 .
According to [START_REF] Humphreys | Linear algebraic groups[END_REF] the group Γ

0 , φ • Γ 0 • φ -1 is closed and connected. On the one hand dim Γ 0 , φ • Γ 0 • φ -1 ≤ d and on the other hand Γ 0 ⊂ Γ 0 , φ • Γ 0 • φ -1 . As a consequence Γ 0 , φ • Γ 0 • φ -1 = Γ 0 .
In other words φ normalizes Γ 0 . But Γ ∩ Γ 0 is infinite, so there exists a birational self map ψ of Bir(P

2 C ) such that ψ • G • ψ -1 ⊂ GL(2, Z) D 2 (Theorem 8.32) and hence ψ • N • ψ -1 ⊂ GL(2, Z) D 2 .
-Now assume d = ∞. Let Γ be a finitely generated subgroup of ker ρ such that dim Γ ≥ 9.

The closure Γ of Γ preserves a unique rational fibration given by a rational map π : P 2 C P 1 C (Lemma 8.18). Consider an element φ of ker ρ. The algebraic group Γ, φ also preserves a unique rational fibration; since Γ ⊂ Γ, φ this fibration is given by π. As a result ker ρ preserves a rational fibration. Hence ker ρ is bounded and the group φ

• G • φ -1 is contained in GL(2, Z) D 2 (Theorem 8.32); in particular φ • N • φ -1 is a subgroup of GL(2, Z) D 2 .
Lemma 8.37 ([Ure]). -Let φ and ψ be two loxodromic elements of Bir(P 2 C ) such that Ax(φ) = Ax(ψ). Then either φ and ψ have not a common fixed point on ∂H ∞ , or φ, ψ contains a subgroup G and there exists a birational self map ϕ of the complex projective plane such that

-ϕ • φ, ψ • ϕ -1 ⊂ GL(2, Z) D 2 , -ϕ • G • ϕ -1 is a dense subgroup of D 2 .
Proof. -Suppose that φ and ψ have a common fixed point p ∈ ∂H ∞ . Denote by L the onedimensional subspace of Z(P 2 C ) corresponding to p. The group φ, ψ generated by φ and ψ fixes p, so its linear action on Z(P 2 C ) acts on L by automorphisms preserving the orientation. A reasoning analogous to that of the proof of Theorem 8.33 implies the existence of a group homomorphism ρ : φ, ψ → R * + whose kernel consists of elliptic birational maps (see Proof of Theorem 8.33). Assume that φ n is tight for some n. Since Ax(φ) = Ax(ψ) the maps φ n and ψ do not commute. According to [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF] any non trivial element of φ n is loxodromic. Therefore, on the one hand ψ • φ n • ψ -1 • φ -n is loxodromic, and on the other hand ρ(ψ

• ϕ n • ψ -1 ) = 1 hence ψ • ϕ n • ψ -1
is elliptic: contradiction. As a result for any k the map φ k is not tight in φ, ψ . Theorem 8.36 implies that there exist a birational self map ϕ of P 2 C and a bounded subgroup

∆ 2 ⊂ φ, ψ such that ϕ • φ • ϕ -1 is monomial; ϕ • ∆ 2 • ϕ -1 is a dense subgroup of D 2 . pairwise disjoint. Set U 1 = U + 1 ∪ U - 1 and U 2 = U + 2 ∪ U - 2 .
There exist n 1 , n 2 , n 3 , n 4 some positive integers such that [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let G be a finitely generated group. Assume that G is an extension of a virtually solvable group R of length r by an other virtually solvable group Q of length q

φ n 1 (U 2 ) ⊂ U + 1 , φ -n 2 (U 2 ) ⊂ U - 1 , ψ n 3 (U 1 ) ⊂ U + 2 , ψ -n 4 (U 1 ) ⊂ U - 2 . Set n = max(n 1 , n 2 , n 3 , n 4 ). Since φ(U + 1 ) ⊂ U + 1 φ -1 (U - 1 ) ⊂ U - 1 ψ(U + 2 ) ⊂ U + 2 ψ -1 (U - 2 ) ⊂ U - 2 one gets that for any k ≤ n φ k (U 2 ) ⊂ U 1 φ -k (U 2 ) ⊂ U 1 ψ k (U 1 ) ⊂ U 2 ψ -k (U 1 ) ⊂ U
1 -→ R -→ G -→ Q -→ 1.
Then G is virtually solvable of length ≤ q + r + 1.

Hence one has the following statement: Proposition 8.41 [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let G 1 and G 2 be two groups that satisfy Tits alternative.

If G is an extension of G 1 by G 2 , then G satisfies Tits alternative.

Proof. -Let Γ be a subgroup of G that does not contain a non abelian free subgroup. For i ∈ {1, 2} denote by pr i : G → G i the canonical projection. Since pr i (G) does not contain a non abelian free subgroup pr i (Γ) = Γ ∩ G i is virtually solvable (G i satisfies Tits alternative). Then according to Lemma 8.40 the group Γ is virtually solvable.

A first consequence of this result is the following one: [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let V be a Kähler compact manifold. Its automorphism group satisfies Tits alternative.

Theorem 8.42 ([
Proof. -The group Aut(V ) acts on the cohomology of V . This yields to a morphism ρ from Aut(V ) to GL(H * (V, Z)) where H * (V, Z) denotes the direct sum of the cohomology groups of V . According to [START_REF] Lieberman | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF] the kernel of ρ is a complex Lie group with a finite number of connected components; its connected component Aut 0 (V ) is an extension of a compact complex torus by a complex algebraic group. We get the result from Proposition 8.41 and classical Tits alternative.

A direct consequence of Proposition 8.41 and Tits alternative for linear groups is:

Proposition 8.43 ([ Can11
v, v = v 2 0 - ∞ ∑ i=1 v 2 i where the coordinates v i are the coordinates of v in B. The light cone of H is the set L(H ) = v ∈ H | v, v = 0 . Let H ∞ be the connected component of the hyperboloid v ∈ H | v, v = 1
that contains e 0 . Consider the metric defined on H ∞ by

d(u, v) := arccos( u, v ).
The space H ∞ is a complete CAT(-1) space, so is hyperbolic (Chapter 2). Its boundary ∂H ∞ can be identified to P(L(H )).

Theorem 8.44 [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let Γ be a subgroup of O(1, ∞).

1. If Γ contains a loxodromic isometry ψ, then one of the following properties holds: Γ contains a non-abelian free group, Γ permutes the two fixed points of ψ that lie on ∂H ∞ . 2. If Γ contains no loxodromic isometry, then Γ fixes a point of H ∞ ∪ ∂H ∞ .

Proof. -Assume first that Γ contains two loxodromic isometries φ and ψ such that the fixed points of φ and ψ on ∂H ∞ are pairwise distinct. According to the ping-pong Lemma (Lemma 8.29) there are two integers n and m such that φ n and ψ m generate a subgroup of Γ isomorphic to the free group F 2 .

Suppose that Γ contains at least one loxodromic isometry φ. Let α(φ) and ω(φ) be the fixed points of φ on ∂H ∞ . If Γ contains an element ψ such that α(φ), ω(φ) ∩ α(ψ), ω(ψ) = / 0 then φ and ψ • φ • ψ -1 are two loxodromic isometries to which we can apply the previous argument. Otherwise Γ fixes either α(φ), ω(φ) , or {α(φ)}, or {ω(φ)}. Then Γ contains a subgroup of index 2 that fixes α(φ) of ω(φ).

Assume that Γ contains two parabolic isometries φ and ψ whose fixed points α(φ) ∈ ∂H ∞ and α(ψ) ∈ ∂H ∞ are distinct. Take two elements of L(H ) still denoted α(φ) and α(ψ) that represent these two points of ∂H ∞ . Let be a point of H such that α(φ), < 0 α(ψ), > 0.

The hyperplane of H orthogonal to intersects H ∞ in a subspace L that "separates" α(φ)

and α(ψ). As a result there exist integers n and m such that φ m (L), φ -m (L), ψ n (L) and ψ -n (L) don't pairwise intersect. [START_REF] Schur | Über Gruppen linearer Substitutionen mit Koeffizienten aus einem algebraischen Zahlkörper[END_REF]). One of the tool of the proof is the Jordan-Schur Theorem.

In the 1930's Burnside asked another related question called the restricted Burnside problem: if it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on n and m ? In other words are there only finitely many finite groups with m generators of exponent n up to isomorphism ?

In 1958 Kostrikin was able to prove that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one: this provides a solution for the restricted Burnside problem for the case of prime exponent ( [START_REF] Kostrikin | On Burnside's problem[END_REF]).

Later Zelmanov solved the restricted Burnside problem for an arbitrary exponent ([Zel90, Zel91]).

Golod gave a negative answer to the Burnside problem for groups that have a complete system of linear representations ( [START_REF] Golod | On nil-algebras and finitely approximable p-groups[END_REF]).

Later many examples of infinite, finitely generated and torsion groups with even bounded ordres were exhibited ([NA68a, NA68b, NA68c, Ol '82,[START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF][START_REF] Lysënok | Infinite Burnside groups of even period[END_REF]).

The

LZ(P 2 C ) = d ∈ Z(P 2 C ) | d • d = 0 of Z(P 2
C ) still denoted by α(φ) that represents α(φ). Assume by contradiction that there exists ϕ in G such that ϕ(α(φ)) = α(φ). The map ψ = ϕ • φ • ϕ -1 is parabolic and fixes the unique element α(ψ) of LZ(P 2

C ) proportional to ϕ(α(φ)). If ε > 0 let us denote by U α, ε the set U α, ε = ∈ LZ(P 2 C ) | α • < ε . Take ε > 0 such that U α(φ), ε ∩U α(ψ), ε = / 0. Since ψ * is parabolic, ψ n * U α(φ), ε is contained in U α(ψ), ε for n large enough. For m sufficiently large the following inclusions hold φ m * • ψ n * U α(φ), ε ⊂ U α(φ), ε 2 U α(φ), ε . This implies that φ m * • ψ n * is loxodromic: contradiction. So α(φ * ) = α(ϕ * ) for any ϕ ∈ G.
Finally G is a subgroup either of J , or of the automorphism group of a Halphen surface.

3. If G is a group of elliptic elements, then according to Theorems 8.14 and 8.15 either G is a bounded subgroup, or G preserves a rational fibration.

Normal subgroups of the Cremona group

The strategy of Cantat and Lamy to produce strict, non-trivial, normal subgroups of Bir(P 2 k ) is to let Bir(P 2 k ) act on the hyperbolic space H ∞ (P 2 k ). In the first part of their paper they define the notion of tight element: an element φ of Bir(P 2 k ) is tight if it satisfies the following three properties:

φ * ∈ Isom(H ∞ ) is hyperbolic;
there exists a positive number ε such that: if ψ belongs to Bir(P 2 k ) and ψ * (Ax(φ)) contains two points at distance ε which are at distance at most 1 from Ax(φ), then ψ * (Ax(φ)) = Ax(φ); if ψ belongs to Bir(P 2 k ) and ψ

* (Ax(φ)) = Ax(φ), then ψ • φ • ψ -1 = φ or ψ • φ • ψ -1 = φ -1 .
The second property is a rigidity property of Ax(φ) with respect to isometries ψ * for ψ ∈ Bir(P 2 k ); we say that Ax(φ) is rigid under the action of Bir(P 2 k ). The third property means that the stabilizer of Ax(φ) coincides with the normalizer of the cyclic group φ .

Here since there is no confusion we write φ for φ Bir(P 2 k ) . Cantat and Lamy established the following statement: the base-points of φ and φ -1 belong to P 2 C ; Base(φ k ) ∩ Base(φ -i ) = / 0 for any k, i > 0. In [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF] Lonjou proved the following statement: Theorem 8.55 ( [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF]). -For any field k the plane Cremona group Bir(P 2 k ) is not simple. She did not use the notion of tight element but uses the WPD (weakly properly discontinuous) property. This property was proposed in the context of the mapping class group in [START_REF] Bestvina | Bounded cohomology of subgroups of mapping class groups[END_REF]. An element g of a group G satisfies the WPD property if for any ε ≥ 0 for any point p ∈ H ∞ there exists a positive integer N such that the set

S(ε, p; N) = h ∈ G | dist(h(p), p) ≤ ε, dist(h(g N (p)), g N (p)) ≤ ε
is finite. Since the elements studied by Lonjou have an axis she followed the terminology introduced in [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF] and said that the group G acts discretely along the axis of g.

In [START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF] the authors generalized the small cancellation theory for groups acting by isometries on δ-hyperbolic spaces.

Small cancellation theory and the WPD property are connected:

in the normal group generated by a family satisfying the small cancellation property elements have a large translation length ([Gui14]); if some element g satisfies WPD property then the conjugates of g n form a family satisfying the small cancellation property.

where the ϕ i 's are some elements of Bir(P If φ belongs to J , then φ = Bir(P 2 C ) (8.6.4) Indeed up to birational conjugacy φ : (z 0 , z 1 ) φ 1 (z 0 , z 1 ), γ(z 1 ) where γ is an homothety or a translation. Consider an element ψ :

(z 0 , z 1 ) ψ 1 (z 0 , z 1 ), z 1 of PGL(2, C(z 1 )). The map ϕ = [φ, ψ] belongs to φ ∩ PGL(2, C(z 1 )).
If ψ is well chosen, then ϕ is non trivial and from (8.6.3) one gets φ = Bir(P 2 C ).

As a result if φ is a birational self map of the complex projective plane such that there exists ψ ∈ Bir(P 2 C ) for which [φ, ψ] preserves a rational fibration, then from (8.6.4) φ = Bir(P 2 C ) (8.6.5) Let φ : (z 0 , z 1 ) → (z 1 , P(z 1 )δz 0 ), δ ∈ C * , P ∈ C[z 1 ], deg P ≥ 2, be a Hénon map. Then φ = Bir(P 2 C ). Indeed if ψ : (z 0 , z 1 ) → (z 0 , 2z 1 ), then [φ, ψ] preserves the rational fibration z 0 =cst; one concludes with (8.6.5).

More generally over any infinite field of characteristic which does not divide n the map h n does not satisfy the WPD property: this explains the assumptions of Proposition 8.57.

Let us mention that Lonjou got not only the non-simplicity of the plane Cremona group from [START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF] but also the following result: Theorem 8.59 [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF]). -Let k be a field. The plane Cremona group contains free normal subgroups; is SQ-universal, that is any countable subgroup embeds in a quotient of Bir(P 2 k ).

In [START_REF] Shepherd-Barron | Some effectivity questions for plane Cremona transformations[END_REF] the author proved that any loxodromic element in the Cremona group over any field k generates a proper normal subgroup; as a result the group Bir(P 2 k ) is not a simple group. He also gave a criterion in terms of the translation length of a loxodromic map φ to know if φ is tight and hence if φ n is a proper subgroup of Bir(P 2 k ) for some n.

Remark Then G is conjugate to a subgroup of J and is isomorphic to a subgroup of PGL(2, C).

Proof. -According to Lemma 8.46 one has the following alternative: G is conjugate

• either to a subgroup of the automorphisms group of a Halphen surface,

• or to a subgroup of J .

But automorphisms groups of Halphen surfaces are finite extensions of abelian subgroups (Theorem 8.47), so do not contain infinite simple subgroups. As a result G is conjugate to a subgroup of J . The short exact sequence from the semi-direct product of J is

1 -→ PGL(2, C(z 1 )) -→ J f -→ PGL(2, C) -→ 1
The group G is simple thus contained in the kernel of the image of f . In both cases G is isomorphic to a subgroup of PGL(2, C).

(ii) Suppose that G is an elliptic group.

Lemma 8.65 ([Ure20]

). -Let G be a simple subgroup of the plane Cremona group of elliptic elements. Then either G is a subgroup of an algebraic group of Bir(P 2 C ),

or G is conjugate to a subgroup of J .

Proof. -According to Theorems 8.14 and 8.15 one of the following holds: G is conjugate to a subgroup of an algebraic group; G preserves a rational fibration; G is a torsion group and G is isomorphic to a subgroup of an algebraic group.

In the first two cases we are done. Let us assume that we are in the third one. Then G is a linear group and according to the Theorem of Jordan and Schur G has a normal abelian subgroup of finite index. As a consequence G is finite, and so algebraic.

(iii) Finally we give a sketch of the proof of Theorem 8.66 ([Ure20]). -A simple subgroup of Bir(P 2 C ) does not contain any loxodromic element.

Let G be a simple subgroup of Bir(P 2 C ). Assume by contradiction that G contains a loxodromic map φ. Theorems 8.54 and 8.36 imply that φ is a monomial map up to birational conjugacy. Looking at the curves contracted by elements of G Urech gets that all loxodromic elements of G are contained in GL(2, Z) D 2 ([Ure, Lemmas 3.17. and 3.18.]). Consider ψ in G. As ψ

• φ • ψ -1 is loxodromic it is monomial. The axis of ψ • φ • ψ -1 is fixed pointwise by both ψ • D 2 • ψ -

CHAPTER 9 BIG SUBGROUPS OF AUTOMORPHISMS "OF POSITIVE ENTROPY"

In this chapter we will focus on automorphisms of surfaces with positive entropy. Recall that a K3 surface (1) is a complex, compact, simply connected surface S with a trivial canonical bundle. Equivalently there exists a holomorphic 2-form ω on S which is never zero; ω is unique modulo multiplication by a scalar. Let S be a K3 surface with a holomorphic involution ι. If ι has no fixed point, the quotient of S by ι is an Enriques surface, otherwise it is a rational surface. Recall that every non-minimal rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the complex projective plane, P 1 C × P 1 C and the Hirzebruch surfaces F n , n ≥ 2. If S is a complex, compact surface carrying a biholomorphism of positive topological entropy, then S is either a complex torus, or a K3 surface, or an Enriques surface, or a non-minimal rational surface ([Can99]). Although automorphisms of complex tori are easy to describe, it is rather difficult to construct automorphisms on K3 surfaces or rational surfaces. Constructions and dynamical properties of automorphisms of K3 surfaces can be found in [START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF] and [START_REF] Mcmullen | Dynamics on K3 surfaces: Salem numbers and Siegel disks[END_REF]. The first examples of rational surfaces endowed with biholomorphisms of positive entropy are due to Coble and Kummer ([Cob61]): the Coble surfaces are obtained by blowing up the ten nodes of a nodal sextic in P 2 C ; the Kummer surfaces are desingularizations of quotients of complex 2-tori by involutions with fixed points.

Obstructions to the existence of such biholomorphisms on rational surfaces are also known: if φ is a biholomorphism of a rational surface S such that h top (φ) > 0, then the representation Aut(S) → GL(Pic(S))

g → g *

(1) "so named in honor of Kummer, Kähler, Kodaira and of the beautiful mountain K2 in Kashmir" ( [START_REF] Weil | Scientific works[END_REF]).

has infinite image. Hence according to [START_REF] Harbourne | Rational surfaces with infinite automorphism group and no antipluricanonical curve[END_REF] its kernel is finite so that S has no non-zero holomorphic vector field. A second obstruction follows from [START_REF] Nagata | On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1[END_REF]: the surface S has to be obtained by successive blowups from the complex projective plane and the number of blowups must be at least ten. The first infinite families of examples have been constructed independently in [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF] and [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] by different methods. Since then many constructions have emerged (see for instance [BK10, BK12, Dil11, DG11, [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF][START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]).

In the first section we give three answers to the question "When is a birational self map of a complex projective surface birationally conjugate to an automorphism ?" In the second section we deal with constructions of automorphisms of rational surfaces with positive entropy. In the last section we explain how SL(2, Z) is realized as a subgroup of automorphisms of a rational surface with the property that every element of infinite order has positive entropy. 9.1. Birational maps and automorphisms 9.1.1. Definitions. -Given a birational map φ : S S of a projective complex surface its dynamical degree λ(φ) is a positive real number that measures the complexity of the dynamics of φ (see §2.3). The neperian logarithm log λ(φ) provides an upper bound for the topological entropy of φ : S S and is equal to it under natural assumptions ([BD05, DS05]). Let us give an alternative but equivalent definition to that of §2.3. A birational map φ : S S of a projective complex surface determines an endomorphism φ * : NS(S) → NS(S); the dynamical degree λ(φ) of φ is defined as the spectral radius of the sequence of endomorphisms (φ n ) * as n goes to infinity:

λ(φ) = lim n→+∞ ||(φ n ) * || 1/n
where || • || denotes a norm on the real vector space End(NS(S)). This limit exists and does not depend on the choice of the norm. For any ample divisor D ⊂ S λ(φ) = lim 2 . Note that all Pisot numbers between λ P and λ G have been listed.

A Salem number is an algebraic integer λ ∈]1, +∞[ whose other Galois conjugates are in the closed unit disk with at least one on the boundary. The minimal polynomial of λ has thus at least two complex conjugate roots on the unit circle, its roots are permuted by the involution z → 1 z and has degree at least 4. Let Sal be the set of Salem numbers. The unique root λ L > 1 of the irreducible polynomial x 10 + x 9x 7x 6x 5x 4x 3 + x + 1 is a Salem number. Conjecturally the infimum of Sal is larger than 1 and should be equal to λ L .

Remark that Pis is contained in the closure of Sal. 9.1.3. Dynamical degrees and Pisot and Salem numbers. -Let us recall that a birational map φ : S S of a compact complex surface is algebraically stable if (φ * ) n = (φ n ) * for all n ≥ 0 (see §2.3). If φ is algebraically stable, then so does φ -1 and λ(φ) is an algebraic integer. Any birational map of a compact complex surface is conjugate by a birational morphism to an algebraically stable map (Proposition 2.10). From this fact and the Hodge index theorem according to which the intersection form has signature (1, r S -1), where r S denotes the rank of S, Diller and Favre get the following statement: Theorem 9.1 ([DF01]). -Let φ be a birational self map of a complex projective surface. If λ(φ) is distinct from 1, i.e. if φ is loxodromic, then λ(φ) is a Pisot or a Salem number. 9.1.4. When is a birational map conjugate to an automorphism ? -A natural question is the following one; when is a birational self map of a complex projective surface birationally conjugate to an automorphism ? There are three answers to this question and we will detail it. C ) be a loxodromic map. Assume that φ is algebraically stable. The action of φ on H 1,1 (P 2 C ) admits the eigenvalue λ(φ) > 1 with eigenvector Θ(φ).

The map φ is birationally conjugate to an automorphism if and only if Θ(φ) • Θ(φ) = 0.

When φ is an automorphism, it is easy to check that Θ(φ) • Θ(φ) = 0. We will thus deal with the reciprocical property. Let φ be a birational self map of a complex projective surface S. Assume that φ is algebraically stable. Hence λ(φ) is equal to the spectral radius of φ * ∈ End(NS(R, S)) but also to the spectral radius of φ * = (φ -1 ) * ; indeed these endomorphisms are adjoint for the intersection form: As a result for all E j Θ(φ) • Θ(φ) = 0 ⇐⇒ Θ(φ) • E j = 0.

Assume now that Θ(φ) • Θ(φ) = 0; then Θ(φ) • E j = 0 for all E j . As the E j 's are effective and Θ(φ) is nef the Q-vector subspace of NS(Q, S) generated by the irreducible components of the divisors E j is contained in Θ(φ) ⊥ . On the orthogonal complement Θ(φ) ⊥ of the isotropic vector Θ(φ) the intersection form is negative and its kernel is the line generated by Θ(φ). Equation (9.1.1) implies

φ k * Θ(φ) = 1 λ(φ) k Θ(φ).
But λ(φ) > 1 and φ * preserves the lattice NS(Z, S), so Θ(φ) is irrational. Consequently the intersection form is negative definite on the Q-vector space generated by all classes of irreducible components of the divisors E j . According to the Grauert-Mumford contraction theorem ( [START_REF] Barth | Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]) there exists a birational morphism η : S → Y that contracts simultaneously all these components. Set ϕ = η • φ • η -1 . As Θ(φ) does not intersect the curves contracted by η the class η * Θ(φ) ∈ NS(R,Y ) is isotropic, and an eigenvector for ϕ * with eigenvalue λ(φ).

Let us iterate this process until ϕ -1 does not contract any curve, that is ϕ ∈ Aut(Y ). If Y is singular, then consider the minimal desingularization Y of Y ; the automorphism ϕ lifts to an automorphism ϕ of Y .

As a result one can state Theorem 9.3 ([DF01]). -Let S be a complex projective surface. Let φ be a loxodromic birational self map of S. Then all divisors E j are orthogonal to Θ(φ) if and only if Θ(φ) is an isotropic vector; if Θ(φ) is an isotropic vector, then there exists a birational morphism η : S → Y such that η • φ • η -1 is an automorphism of Y .

Then Diller and Favre prove the following statement:

Theorem 9.4 ([DF01]). -Let φ ∈ Bir(S) (resp. ψ ∈ Bir( S)) be an algebraically stable map of a complex projective surface S (resp. S). Assume that φ and ψ are conjugate via a proper modification. Suppose that λ(φ) > 1 (or equivalently that λ(ψ) > 1). Then Θ(φ) • Θ(φ) = 0 if and only if Θ(ψ) • Θ(ψ) = 0.

Theorem 9.2 follows from Theorems 9.3 and 9.4. 9.1.4.2. A second answer. -The following statement gives another characterization of birational maps conjugate to an automorphism of a smooth projective rational surface:

Theorem 9.5 ([DF01, BC16]). -Let φ be a birational map of a complex projective surface S.

If λ(φ) is a Salem number, then there exists a birational map ψ : S S that conjugates φ to an automorphism of S; if φ is conjugate to an automorphism, then λ(φ) is a quadratic integer or a Salem number.

Assume that λ(φ) is a Salem number. Denote by P(t) ∈ Z[t] the minimal polynomial of λ(φ). But λ(φ) is a Salem number, so there exists a root of P with modulus 1, denote it α. Hence fix an automorphism κ of the field C such that κ(λ(φ)) = α. According to Proposition 2.10 we can suppose that φ is algebraically stable up to birational conjugacy. The eigenvector Θ(φ) thus corresponds to the eigenvalue λ(φ), and so may be taken in NS(L, S) where L is the splitting field of P. The automorphism κ acts on NS(C, S) preserving NS(S) pointwise. Since φ * is defined over Z and φ * Θ(φ) = λ(φ)Θ(φ) one obtains ). -The dynamical number of base-points is an invariant of conjugation. In particular if φ is conjugate to an automorphism of a smooth projective surface, then µ(φ) = 0.

A base-point p of φ is a persistent base-point if there exists an integer N such that for any k ≥ N p ∈ Base(φ k ) p ∈ Base(φ -k ) Let p be a point of S or a point infinitely near S such that p ∈ Base(φ). Consider a minimal resolution of φ Z

π 1 0 0 π 2 Ð Ð S φ G G S
Because p is not a base-point of φ it corresponds via π 1 to a point of Z or infinitely near; using π 2 we view this point on S again maybe infinitely near and denote it φ • (p). For instance if S = P 2 C , p = (1 : 0 : 0) and φ is the birational self map of P 2 C given by (z 0 : z 1 : z 2 ) (z 1 z 2 + z 2 0 : z 0 z 2 : z 2 2 ) the point φ • (p) is not equal to p = φ(p) but is infinitely near to it. Note that if φ, ψ are two birational self maps of S and p is a point of S such that p ∈ Base(φ), φ(p) ∈ Base(ψ), then (ψ • φ) • (p) = ψ • (φ • (p)). One can put an equivalence relation on the set of points of S or infinitely near S: the point p is equivalent to the point q if there exists an integer k such that (φ k ) • (p) = q; in particular p ∈ Base(φ k ) and q ∈ Base(φ -k ). Note that the equivalence class is the generalization of set of orbits for birational maps.

A base-point is periodic if either (φ k ) • (q) = q for some k ≥ 0, or q ∈ Base(φ k ) for any k ∈ Z {0} (in particular (φ k ) • (p) is never defined for k = 0).

Let P be the set of periodic base-points of φ. Denote by P the finite set of points equivalent to a point of P . Both b(φ) and b(φ -1 ) are finite, so there exists n ∈ N such that for any p ∈ Base(φ) ). -Let S be a smooth projective surface. Let φ be a birational self map of S. Then µ(φ) coincides with the number of equivalence classes of persistent base-points of φ. In particular µ(φ) is an integer.

The following statement gives another characterization of birational maps which are conjugate to an automorphism of a projective surface; contrary to the two previous one it works for all maps of Bir(S). Let us give an example of [START_REF] Déserti | Automorphisms of rational surfaces with positive entropy[END_REF]. Consider the birational self map of P 2 C given by ψ : (z 0 : z 1 : z 2 ) (z 0 z 2 2 + z 3 1 : z 1 z 2 2 : z 3 2 ); it has five base-points: p = (1 : 0 : 0) and four points infinitely near. Denote by P 1 the collection of these points. Similarly ψ -1 has five base-points: (1 : 0 : 0) and four points infinitely near; let P 2 be the collection of these points. Consider the automorphism A given by A : (z 0 : z 1 : z 2 ) → αz 0 + 2(1α)z 1 + (2 + αα 2 )z 2 : -z 0 + (α + 1)z 2 : z 0 -2z 1 + (1α)z 2 with α ∈ C {0, 1}. Then P 1 , A( P 2 ), and (A • ψ • A)( P 2 ) have distinct supports;

P 1 = (A • ψ) 2 • A( P 2 ).
As a result the base-points of φ = A•ψ are non-persistent, so φ is conjugate to an automorphism of a rational surface; this rational surface is P 2 C blown up in P 1 , A( P 2 ), and (A • ψ • A)( P 2 ). Furthermore λ(A • ψ) > 1.

Proof of Theorem 9.10. -Lemma 9.7 shows that if φ is conjugate to an automorphism of a smooth projective surface, then µ(φ) = 0.

Let us prove the converse. Assume that µ(φ) = 0. One can suppose that by blowing-up points φ is algebraically stable (Proposition 2.10). Therefore, φ has no periodic base point and B ++ = / 0. Furthermore µ(φ) = 0 corresponds to B +-= B -+ = / 0. All base-points thus belong to B --. Assume that φ is not an automorphism of S. Let τ : Z → S be the blow-up of the base-points of φ. The morphism χ = φ • τ : Z → S is the blow-up of the base-points of φ -1 . Consider a (-1)-curve E ⊂ Z contracted by χ. The image χ(E) of E is a proper point of S that belongs to Base(φ -1 ). Since φ is algebraically stable, then for all k ≥ 0 χ(E) ∈ Base(φ k ).

As a result φ k • χ : Z S is well-defined at any point of E. The curve C = τ(E) is thus an irreducible curve of S contracted by φ k+1 ; any base-point of φ k+1 that belongs to C as proper of infinitely near point is also a base-point of φ. This finite set of points is contained in B --; so there is n > 0 such that no base-point of φ n belongs to C. Since C is blown down by φ n , C is a (-1)-curve of S. Contracting C conjugates φ to an algebraically stable birational map whose all base-points are in B --. The rank of the Picard group of this new surface is strictly less than the rank of Pic(S). Consequently if we repeat this process, it has to stop. In other words φ is conjugate to an automorphism of a smooth projective surface. 9.2. Constructions of automorphisms with positive entropy 9.2.1. McMullen's idea. -In [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF] McMullen establishes a result similar to Torelli's theorem for K3 surfaces: he constructs automorphisms on some rational surfaces prescribing the action of the automorphisms on cohomological groups of the surface.

The relationship between the Coxeter group and the birational geometry of the plane, used by McMullen, is discussed since 1895 (see [START_REF] Kantor | Theorie der endlichen Gruppen von eindeutigen Transformationen in der ebene[END_REF]) and has been much developed since then (see for instance [Cob61, DO88, DZ01, Har88, Giz80]).

A rational surface S is a marked blow-up of P 2 C if it is presented as a blow-up π : S → P 2 C of P 2 C at n distinct points p 1 , p 2 , . . ., p n . The marking determines the basis for Pic(S) given by the hyperplane bundle and the classes of the exceptional curves over the p j . The first step toward finding an automorphism φ of S is to construct a plausible candidate for its linear action φ * on the Picard group. Note that candidate actions must preserve the intersection form, the class of the canonical divisor, and the set of effective classes. Let us mention two sorts of involutions on Pic(S) that satisfy these restrictions: an abstraction of the involution σ 2 , the involution that swaps the basis elements corresponding to two different exceptional curves.

If we compose such involutions one gets a Coxeter group W n that is infinite as soon as n ≥ 9, has elements with positive spectral radius when n ≥ 10. Furthermore except in some degenerate situations an element w ∈ W n transforms the basis of Pic(S) corresponding to the given marking into a basis corresponding to some other marking ϕ : S → P 2 C . If the base-points of the new marking coincide, up to an element of Aut(P 2 C ), with those of the original, then one obtains an automorphism φ = ϕ -1 • ϕ of S with φ * = w. The main problem with this approach is that it is not easy, given w ∈ W n , to see how the basepoints of the two markings are related. The problem is easier if the base-points of the original marking lie along an elliptic curve; indeed in that case the new base-points also lie on this elliptic curve. Computations are thus computations on a curve so simpler. The best case is the case of a cuspidal cubic as there is a one-parameter subgroup of Aut(P 2 C ) fixing such a curve. McMullen proved Theorem 9.12 ( [START_REF] Mcmullen | Dynamics on blowups of the projective plane[END_REF]). -For any n ≥ 10 the standard element w of W n may be realized by an automorphism φ of a marked blow-up S with an invariant cuspidal anticanonical curve. The entropy of φ is the spectral radius of w which is positive.

The following question "What are the elements of w ∈ W n which may be realized by rational surface automorphisms ?" was also considered in [START_REF] Diller | Cremona transformations, surface automorphisms, and plane cubics[END_REF] and [START_REF] Uehara | Rational surface automorphisms with positive entropy[END_REF]. Diller gave a rather thorough enumeration of the possibilities for quadratic birational maps which have an invariant curve. Such maps are determined by the data consisting of three orbit lengths (n 1 , n 2 , n 3 ) and a permutation of {1, 2, 3}. Diller also showed that not all orbit data, and not all w ∈ W n , are realizable by maps with invariant curve. Uehara established the following statement: Theorem 9.13 ([Ueh16]). -For every w ∈ W n with spectral radius > 1 there is a rational surface automorphism φ such that the spectral radius of φ * is the same as the spectral radius of w.

Uehara's method combines elements of McMullen's and Diller's approaches. Given w ∈ W n he prescribed a set of orbit data and proved that these orbit data can be realized by an automorphism φ. The induced φ * has the same spectral radius as w, although the two may not be conjugate.

Remark 9.14. -While McMullen's and Diller's constructions involve automorphisms with invariant curves note that in [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] the authors showed that rational surface automorphisms of positive entropy do not necessarily possess invariant curves. 9.2.2. Bedford and Kim construction. -In [START_REF] Bedford | Periodicities in linear fractional recurrences: degree growth of birational surface maps[END_REF] and [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] the authors found automorphisms within a specific two-parameters family of plane birational maps. The initial observation in the two papers is the same: for certain parameter pairs all points of indeterminacy for all iterates of the map in question can be eliminated by performing finitely many point blow-ups. The map then lifts to an automorphism of the resulting rational surface. This idea was "systematized" in [START_REF] Déserti | Automorphisms of rational surfaces with positive entropy[END_REF].

In [START_REF] Bedford | Dynamics of rational surface automorphisms: linear fractional recurrences[END_REF] the authors prove that essentially all examples of rational surfaces automorphisms associated to Coxeter elements can be found within the two-parameter birational family ( f a,b ) (a,b) given by f a,b (z 0 , z 1 ) = z 1 , z 1 +a z 0 +b . 9.3. Automorphisms are pervasive 9.3.1. Automorphisms of del Pezzo surfaces. -Any del Pezzo surface S contains a finite number of (-1)-curves (i.e. smooth curves isomorphic to P 1 C and of self-intersection -1). Each of them can be contracted to get another del Pezzo surface of degree (K S ) 2 + 1. There are, moreover, the only reducible curves of S of negative self-intersection. If S = P 2 C , then there is a finite number of conic bundles S → P 1 C up to automorphism of P 1 C and each of them has exactly 8 -(K S ) 2 singular fibers.

This latter fact can be found by contracting one component in each singular fiber which is the union of two (-1)-curves, obtaining a line bundle on a del Pezzo surface, isomorphic to P 1 C × P 1 C or to the Hirzebruch surface F 1 and having degree 8. For more details see [START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF][START_REF] Yu | of North-Holland Mathematical Library[END_REF].

Automorphisms of del Pezzo surfaces of order 4. -Set S = (z 0 : z 1 : z 2 : z 3 ) ∈ P(2, 1, 1, 1) | z 2 0z 4 1 = z 2 z 3 (z 2 + z 3 )(z 2 + µz 3 )

where µ belongs to C {0, 1}. The surface S is a del Pezzo one of degree 2. The automorphism β given by β : (z 0 : z 1 : z 2 : z 3 ) → (z 0 : iz 1 : z 2 : z 3 ) fixes pointwise the elliptic curve given by z 0 = 0. When µ varies all possible elliptic curves are obtained. Moreover rk Pic(S) β = 1. There are other automorphisms β of order 4 of rational surfaces S such that β 2 fixes an elliptic curve but none for which rk Pic(S) β = 1 (see [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]).

Automorphisms of del Pezzo surfaces of order 6. -Let us give explicit possibilities for automorphisms of order 6. i) Set S = (z 0 : z 1 : z 2 : z 3 ) ∈ P(3, 1, 1, 2) | z 2 0 = z 3 3 + µz 4 1 z 3 + z 6 1 + z 6 2 for some general µ ∈ C such that S is smooth. The surface S is a del Pezzo surface of degree 1. Consider on S α : (z 0 : z 1 : z 2 : z 3 ) → (z 0 : z 1 : -jz 2 : z 3 )

where j = e 2iπ/3 . The automorphism α fixes pointwise the elliptic curve given by z 2 = 0. When µ varies all possible elliptic curves are obtained. The equality rk Pic(S) α = 1 holds (see [DI09, Corollary 6.11]). ii) Set S = (z 0 : z 1 : z 2 : z 3 ) ∈ P 3 C | z 0 z 2 1 + z 3 0 + z 3 2 + z 3 3 + µz 0 z 2 z 3 = 0 where µ is such that the cubic surface is smooth. The surface is a del Pezzo surface of degree 3. Consider on S the automorphism α given by α : (z 0 : z 1 : z 2 : z 3 ) → (z 0 : -z 1 : jz 2 : j 2 z 3 ).

Remark that α 3 fixes pointwise the elliptic curve z 1 = 0 and α acts on it via a translation of order 3. When µ varies all possible elliptic curves are obtained. The equality rk Pic(S) α = 1 holds ([DI09]). iii) Set S = (z 0 : z 1 : z 2 : z 3 ) ∈ P 3 C | z 3 0 + z 3 1 + z 3 2 + (z 1 + µz 2 )z 2 3 = 0 where µ ∈ C is such that the cubic surface is smooth. It is a del Pezzo surface of degree 3. Consider α defined by α : (z 0 : z 1 : z 2 : z 3 ) → (jz 0 : z 1 : z 2 : z 3 ). The automorphism α 3 fixes pointwise the elliptic curve z 3 = 0 and α acts on it via an automorphism of order 3 with three fixed points. When µ varies the birational class of α changes but not the isomorphism class of the elliptic curve fixed by α 3 . M is elliptic if and only if M has finite order; M is parabolic if and only if M has infinite order and its trace is ±2; M is loxodromic if and only if M has infinite order and its trace is = ±2.

Up to conjugacy the elliptic elements of SL(2, Z) are

-1 0 0 -1 , 0 1 -1 -1 , 0 1 -1 0 , 0 -1 1 0 , 0 -1 1 1 .
In particular an element of finite order is of order 2, 3, 4 or 6.

A parabolic element of SL(2, Z) is up to conjugacy one of the following one 1 a 0 1

-1 a 0 -1 with a ∈ Z. Since B 2 ∈ SL(2, Z) is an involution its image by any embedding θ : SL(2, Z) → Bir(P 2 C ) is a birational involution. As we have seen in §6.1 an element of order 2 of the Cremona group is up to conjugacy one of the following an automorphism of P 2 C , a Jonquières involution of degree ≥ 2, a Bertini involution, a Geiser involution.

Since B 2 commutes with SL(2, Z) the group θ SL(2, Z) is contained in the centralizer of θ(B 2 ). But if θ(B 2 ) is a Bertini involution or a Geiser involution, then the centralizer of θ(B 2 ) is finite ( [START_REF] Blanc | On birational transformations of pairs in the complex plane[END_REF]). As a result θ(B 2 ) is conjugate either to an automorphism of P 2 C , or to a Jonquières involution. Assume that θ(B 2 ) is not linearisable; θ(B 2 ) fixes thus pointwise a unique irreducible curve Γ of genus ≥ 1. Denote by G the image of θ. The group G preserves Γ and the action of G on Γ gives the exact sequence

1 -→ G -→ G -→ H -→ 1
where H is a subgroup of Aut(Γ) and G contains θ(B 2 ) and fixes Γ. The genus of Γ is positive; hence H cannot coincide with G θ(B 2 ) , a free product of Z 2Z and Z 3Z . As a consequence G G strictly contains θ(B 2 ) ; thus G is infinite and not abelian. In particular the group of birational maps fixing pointwise Γ is infinite and not abelian. So according to [START_REF] Blanc | Sur un théorème de Castelnuovo[END_REF] the curve Γ has genus 1. One can now state: Pick µ ∈ C * such that the point p = (µ : 1 : 0) ∈ L z 2 has a trivial isotropy group under the action of PSL(2, Z). Fix an even integer k > 0; consider ψ the conjugation of ψ : (z 0 : z 1 : z 2 ) (z k 0 : z k-1 0 z 1 + z k 2 : z k-1 0 z 2 ) by (z 0 : z 1 : z 2 ) → (z 0 + µz 1 : z 1 : z 2 ). Then define the morphism θ k : SL(2, Z) → Bir(P 2 C ) as follows θ k (B) = θ e (B) : (z 0 : z 1 : z 2 ) → (z 1 : -z 0 : z 2 ) θ k (AB) = ψ • θ e (AB) • ψ -1 .

The map ψ restricts to an automorphism of the affine plane where z 0 = 0, commutes with θ k (B 2 ) = θ e (B 2 ) = (z 0 : z 1 : -z 2 ) ∈ Aut(P 2 C ) and acts trivially on L z 2 . Since ψ commutes with θ k (B 2 ) the map θ k (AB) commutes with θ k (B 2 ). As a result θ k is a well-defined morphism. As ψ |L z 2 {p} = id the actions of θ e and θ k on L z 2 are the same; θ k is thus an embedding. has degree k 2n and has exactly 2n proper base-points, all lying on L z 2 .

More precisely the base-points are p, (AB) a 1 -1 (p), B b 1 (AB) a 1 -1 (p), (AB) a 2 B b 1 (AB) a 1 -1 (p), . . . , (AB) a n B b n-1 (AB) a n-1 . . . B b 1 (AB) a 1 -1 (p), B b n (AB) a n B b n-1 (AB) a n-1 . . . B b 1 (AB) a 1 -1 (p).

This result implies the existence of infinitely many loxodromic embeddings of SL(2, Z) into Bir(P 2 C ): Corollary 9.17 Y 4 conjugating α 3 to β 2 (which exists if and only if the elliptic curves are isomorphic), which is general enough, we obtain a loxodromic embedding SL(2, Z) → Bir(P 2 C ). To prove that there is no other relation in α, β and that all elements of infinite order are loxodromic the morphisms X → X 4 and Y → Y 4 and the actions of α and β on Pic(X)

  has a pole of order k on D; ν f (D) = 0 otherwise. To any rational function f ∈ C(V ) * we associate a divisor div( f ) defined by div( f ) = ∑ D prime divisor ν f (D)D.

  Proposition 1.3 ([Har77]). -Let S be a smooth projective surface. There exists a unique bilinear symmetric form Div(S) × Div(S) → Z (C, D) → C • D such that if C and D are smooth curves with transverse intersections, then C • D = #(C ∩ D); if C and C are linearly equivalent, then C • D = C • D for any D.In particular this yields an intersection formPic(S) × Pic(S) → Z (C, D) → C • D.Let π : Bl p S → S be the blow-up of the point p ∈ S. The morphism π induces the map π * : Pic(S) → Pic(Bl p S), C → π -1 (C).

  m p (C) ≥ 0, m p (C) = 0 if and only if p does not belong to C, m p (C) = 1 if and only if p is a smooth point of C. Assume that C and D are distinct curves with no common component ; we can define an integer (C • D) p which counts the intersection of C and D at p: if either C or D does not pass through p, it is equal to 0; otherwise let f , resp. g be some local equation of C, resp. D in a neighborhood of p, and define (C • D) p to be the dimension of O p,S ( f , g) . This number is related to C • D by the following statement: Proposition 1.4 ([Har77]). -If C and D are distinct curves without any common irreducible component on a smooth surface, then C • D = ∑ p∈C∩D (C • D) p .

π

  * C • π * D = C • D for any C, D in Pic(S); π * C • E = 0 for any C in Pic(S); E 2 = E • E = -1; C 2 = C 2 -1 for any smooth curve C passing through p and where C is the strict transform of C.

v, y e ⊥ 0 ;

 0 = v e 0 e 0 + v e ⊥ 0 where v e 0 belongs to R and v e ⊥ 0 belongs to e ⊥ 0 . Consider the symetric bilinear form B of H defined by B(x,y) = x e 0 y e 0x e ⊥ 0

π

  : Y →S NS(Y ) thus exists. It is called the Picard Manin space of S. The intersection forms on the groups NS(Y ) induce a quadratic form on Z(S) of signature (1, ∞).

  Remark 2.4. -An invariant structure is given by the canonical form. The canonical class of P 2 C blown up in n points p 1 , p 2 , . . ., p n is equal to -3e 0 -n ∑ j=1 e p j . By taking intersection products one obtains a linear form ω ∞ defined by ω ∞ : Z(P 2 C ) → Z, m 0 e 0 -Since the isometric action of Bir(P 2 C ) on Z(P 2 C ) preserves the linear form ω ∞ we get the following equalities already obtained in §1.3: if φ * e 0 = de 0 -n ∑ j=1 m j e p j , then

  S is a del Pezzo surface of degree 4 obtained by blowing up 5 points in P 2C and H S is the group of automorphisms of P 2 C that preserve this set of points; (8) Aut(S) where S is a del Pezzo surface of degree 3 of the following form the triple cover of P 2 C ramified along a smooth cubic Γ. If S is the Fermat cubic, then Aut(S) = Z 3Z 3 S 4 , otherwise we have an exact sequence

  Lemmas 3.40, 3.41 and Theorem 3.46 allow to prove the following statement: Lemma 3.47 ([Ure]). -Let G be a subgroup of the plane Cremona group. Assume that G is conjugate to an automorphism group of a del Pezzo surface S. Then G can be embedded into GL(8, C). Proof. -If deg S ≤ 5, then Aut(S) is finite and Lemma 3.40 allows to conclude. If deg S = 6, then Aut(S) can be embedded into GL(8, C) (Lemma 3.41). If deg S = 7, then Aut(S) is conjugate to a subgroup of Aut(P 1 C

  Theorem 3.48 ([START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). -Let n ≥ 1 be an integer. The group Bir(P n C ) is topologically simple when endowed with the Zariski topology.Remark 3.49. -This statement was proved in dimension 2 byBlanc ([Bla10]) using the classical Noether and Castelnuovo Theorem.

  Theorem 3.48. -Let us first prove the statement for n = 1. Lemma 3.51. -Let n ≥ 2 be an integer. The group PSL(n, C) is dense in PGL(n, C) with respect to the Zariski topology. Moreover every non-trivial normal subgroup of PGL(n, C) contains PSL(n, C). In particular PGL(n, C) does not contain any non-trivial normal strict subgroups closed for the Zariski topology. Proof. -The group morphism det : GL(n, C) → C * yields a group morphism det : PGL

  and the induced morphism φ : X → Y is an open immersion. We can thus state: Lemma 3.57. -Let X and Y be two varieties. Let φ : X Y be a birational map. Then the set Breg(φ) = p ∈ X | φ is biregular in p is open and dense in X.

  3.5.3. Proof of Theorem 3.58. -Let us start with the following statement: Lemma 3.69 ([Kra18]

  Theorem 4.1 ([Cas01]). -The group Bir(P 2 C ) is generated by Aut(P 2 C ) = PGL(3, C) and the standard quadratic involution

  Remark 4.2. -One consequence of Noether and Castelnuovo theorem is: the Jonquières group and Aut(P 2 C ) = PGL(3, C) generate Bir(P 2 C

  Theorem 4.6 ([Wri92]). -The plane Cremona group is the free product of PGL(3, C), Aut(P 1 C × P 1 C ) and J amalgamated along their pairwise intersections in Bir(P 2 C ). Twenty years later Blanc proved: Theorem 4.7 ([Bla12]). -The group Bir(P 2

  2.1) The maps τ and ρ belong to PGL(3, C) and (τ • ρ) 3 = id in PGL(3, C); as a consequence (τ • ρ) 3 = id in G. One has σ 3 2 = σ 2 in Aut(P 1 C × P 1 C ) so in G. From (4.2.1) one gets (τ • ε) 3 = σ 2 in G. Consequently β induces a map β : Bir(P 2 C ) → G with the following property: β restricts to the identity on J and id, τ . According to Theorem 4.5, α • β = id. The image of β contains J ⊂ G and τ ∈ (PGL(3, C) ∩ Aut(P 1 C × P 1 C )) ⊂ G. But both PGL(3, C) and Aut(P 1 C × P 1 C ) are generated by their intersection with J (in Bir(P 2 C )) together with τ; hence PGL(3, C) and Aut(P 1 C

  According to Theorem 4.7 the plane Cremona group is isomorphic to PGL(3, C) * PGL(3,C)∩J J divided by the relation τ • σ 2 • τ • σ 2 where τ : (z 0 : z 1 : z 2 ) → (z 1 : z 0 : z 2 ). Note that this relation holds as well in G. As a consequence ϕ factors through the quotient PGL(3, C)

First

  case: deg(σ 2 • φ n • σ 2 ) = 1. According to Lemma 4.28 the word σ 2 • φ n • σ 2 can be replaced by the linear map φ n = σ 2 • φ n • σ 2 using relations (R 1 ), (R 3 ) and (R 4 ). We thus get a new pair (D , n ) with D ≤ D; moreover if D = D , then n < n.

  [START_REF] Gizatullin | On some tensor representations of the Cremona group of the projective plane[END_REF] Gizatullin has considered the following question: can a given group homomorphism ϕ : PGL(3, C) → PGL(n + 1, C) be extended to a group homomorphism Φ : Bir(P 2 C ) → Bir(P n C ) ? He answers yes when ϕ is the projective representation induced by the regular action of PGL(3, C) on the space of plane conics, plane cubics, or plane quartics. To construct these homomorphisms Gizatullin uses the following construction. Denote by Sym(n, C) the C-algebra of symmetric n × n matrices. Define S(2, n) as the quotient Sym(n, C)3 GL(n, C) where the regular action of GL(n, C) is given by

  Theorem 4.35 ([Hud27, Pan99]). -Any set of group generators of Bir(P n C ), n ≥ 3, contains uncountably many elements of Bir(P n C ) PGL(n + 1, C). Let us first recall the following construction of Pan which given a birational self map of P n C allows one to construct a birational self map of P n+1

  Corollary 5.3 ([D 06a]). -If a morphism from a subgroup of finite index of SL(n, Z) into Bir(P 2 C ) has infinite image, then n ≤ 3. 5.1. The group Bir(P 2 C ) is not linear Cantat and Lamy proved that Bir(P 2 C ) is not simple but the non-existence of a faithful representation does not imply the non-existence of a non-trivial representation. So let us deal with the following statement: Proposition 5.4 ([CD13]). -The plane Cremona group has no faithful linear representation in characteristic zero. Before giving the proof let us mention that making an easy refinement of it provides the following stronger result: Proposition 5.5 ([Cor]). -If k is an algebraically closed field, then there is no non-trivial finite dimensional linear representation for Bir(P 2 k ) over any field. Let us recall the following statement due to Birkhoff: Lemma 5.6 ([Bir36]). -Let k be a field of characteristic zero. Let A, B, and C be three elements of GL(n, k) such that [A, B] = C, [A,C] = [B,C] = id, C has prime order p. Then p ≤ n. Proof. -Assume that k is algebraically closed. Since C is of order p its eigenvalues are p-rooth of unity. If the eigenvalues of C are all equal to 1, then C is unipotent and p ≤ n. Otherwise C admits an eigenvalue α = 1. Consider the eigenspace E α = v |Cv = αv of C associated to the eigenvalue α. By assumption A and B commute to C, so E α is invariant by A and B

  As a consequence the inclusion σ 2 Bir(P 2 C ) ⊂ A Bir(P 2 C ) holds. But σ 2 Bir(P 2 C ) = Bir(P 2 C ) (Proposition 5.10) hence Proposition 5.11 ([CD13]). -Let A be a non-trivial automorphism of the complex projective plane. Then A Bir(P 2 C ) = Bir(P 2 C ).

  Lemma 5.23 ([D 06a]). -Let υ be an embedding of H k into PGL(3, C). Up to linear conjugacy υ( f ) :

  are, for some integer n, some automorphisms of a minimal rational surface, that is ofP 2 C or of F n , n ≥ 2. Let us mention the case F n , n ≥ 2 (see [D 06a]for more details) and detail the case P 2 C . Lemma 5.27 ([D 06a]). -Let ρ be a morphism from a congruence subgroup Γ(3, q) of SL(3, Z) in the plane Cremona group. Assume that ρ(e q 12 ), ρ(e q 13 ) and ρ(e q 23 ) belong to Aut(F n ), n ≥ 2, for some integer . Then the image of ρ is either finite, or contained in PGL(3, C) = Aut(P 2 C ) up to conjugacy. Lemma 5.28 ([D 06a]). -Let ρ be an embedding of a congruence subgroup Γ(3, q) of SL(3, Z) into Bir(P 2 C ). If ρ(e qn 12 ), ρ(e qn 13 ) and ρ(e qn 23

  Lemma 5.30 ([D 06a]). -Let φ be an element of the plane Cremona group. Assume that Exc(φ) and Exc(φ 2 ) are non-empty and contained in the line at infinity. If Ind(φ) is also contained in the line at infinity, then φ is a polynomial automorphism of A 2 C .Proof of Lemma 5.28. -Lemma 5.23 allows us to assume that ρ(e qn 13 ) :(z 0 , z 1 ) → (z 0 + qn, z 1 ), ρ(e qn 12 ) : (z 0 , z 1 ) → (z 0 + ζz 1 , z 1 + β), ρ(e qn 23 ) : (z 0 , z 1 ) → (z 0 + γz 1 , z 1 + δ) where ζδβγ = q 2 n 2 .Let us first suppose that βδ = 0. Since [ρ(e qn 13 ), ρ(e qn 21 )] = ρ(e -q 2 n 2 23 ) the curves blown down by ρ(e qn 21 ), if they exist, are of the type z 1 = constant. As ρ(e qn 21 ) and ρ(e qn 23 ) commute, the sets Exc(ρ(e qn 21 )) and Ind(ρ(e qn 21 )) are invariant by ρ(e qn 23 ). As a result Exc(ρ(e qn 21 )), Ind(ρ(e qn 21 )) and Exc((ρ(e qn 21 )) 2 ) are contained in the line at infinity. Hence ρ(e qn 21 ) belongs to either PGL(3, C) or Aut(A 2 C ) (Lemma 5.30). Note that if ρ(e qn 21 ) belongs to PGL(3, C), then ρ(e qn 21 ) preserves the line at infinity because [ρ(e qn 21 ), ρ(e qn 23 )] = id. In other words ρ(e qn 21 ) also belongs to Aut(A 2 C ). Using the relations [ρ= id we get that ρ(e qn 23 ) belongs to Aut(A 2 C

  z 1 does change neither ρ(e qn 13 ), nor ρ(e qn 12 ), and sends ρ(e qn 23 ) onto (z 0 , z 1 ) → (z 0 , z 1 + δ). One can thus assume that ρ(e qn 13 ) : (z 0 , z 1 ) → (z 0 + qn, z 1 ), ρ(e qn 12 ) : (z 0 , z 1 ) → (z 0 + ζz 1 , z 1 ) ρ(e qn 23 ) : (z 0 , z 1 ) → (z 0 , z 1 + δ). The map ρ(e qn 21 ) satisfies the relations [ρ(e qn 13 ), ρ(e qn 21 )] = ρ(e -q 2 n 2 23 ), and [ρ(e qn 21 ), ρ(e qn 23 )] = id so does the element ψ : (z 0 , z 1 ) → (z 0 , δnz 0 + z 1 ) of PGL(3, C). Remark that the map φ = ρ(e qn 21 ) • ψ -1 commute to both ρ(e qn 13 ) and ρ(e qn 23 ). As a consequence φ : (z 0 , z 1 ) → (z 0 + a, z 1 + b) for some a, b in C. Finally up to conjugacy by (z 0 , z 1 ) → z 0 + b δ , z 1 one has ρ(e qn 21 ) : (z 0 , z 1 ) → (z 0 + a, δz 0 + z 1 ); 5.3. REPRESENTATIONS OF SL(n, Z) INTO Bir(P 2 C ) FOR n ≥ 3 109 in particular ρ(e qn 21 ) belongs to PGL(3, C). Similarly if ϕ is the map given by (z 0 , z 1 ) → z 0 1 + ζz 1 , z 1 1 + ζz 1 then the map ρ(e qn 32 ) • ϕ -1 commute to both ρ(e qn 13 ) and ρ(e qn 12 ). Therefore ρ(e qn 32 ) • ϕ -1 : (z 0 , z 1 ) → (z 0 + b(z 1 ), z 1 ) and ρ(e qn 32 ) :

  = id and [ρ(e qn 12 ), ρ(e qn 31 )] = ρ(e -q 2 n 2 32 ) we get ρ(e qn 21 ) : (z 0 , z 1 ) → (z 0 , δz 0 + z 1 ). Finally since ρ(e qn 31 ) and ρ(e qn 32 ) commute, b ≡ 0 and im ρ ⊂ PGL(3, C).

  Theorem 5.32. -Since PGL(3, C) is simple the restriction ϕ |PGL(3,C) is either trivial or injective. Let us first suppose that ϕ |PGL(3,C) is trivial. Consider the element of PGL(3, C) given by

  Since PGL(3, C) = U, SL(3, Z) the inclusion ϕ(PGL(3, C)) ⊂ PGL(3, C) holds. According to [BT73] the action of ϕ on PGL(3, C) comes, up to inner conjugacy, from an embedding of the field C into itself. Assume now that ϕ |SL(3,Z) is A → A ∨ . Similar computations and [BT73] imply that ϕ |PGL(3,C) comes, up to inner conjugacy, from the composition of A → A ∨ and an embedding of the field C into itself.

Lemma 6 . 2 C

 62 15 ([DI09]). -Let (S, ψ) and (S , ψ ) be two rational G-surfaces. Let φ : S P and φ : S P 2 C be two birational maps. The subgroups ι φ (G) and ι φ (G) are conjugate if and only if there exists a birational map of G-surfaces S S.

  Proof. -If φ belongs to G, we denote by dom(φ) an open subset on which φ is defined. Set D = φ∈G dom(φ). Then U = φ∈G g(D) is an open invariant subset of P 2 C on which φ acts biregularly. Consider U = U G the orbit space; it is a normal algebraic surface. Let us choose any normal projective completion X of U . Consider S the normalization of X in the field of rational functions of U. It is a normal projective surface on which G acts by biregular transformations. A G-invariant resolution of singularities S of S suits ([dFE02]).

  P 3 C given by φ : (z 0 : z 1 : z 2 : z 3 ) → z 1 : z 0 : z 2 : ζz 3 where ζ 3 = 1, ζ = 1 has only four fixed points while φ 2 fixes the elliptic curve z 3 = 0. Definition. -Let φ ∈ Bir(P 2 C ) be a non-trivial element of finite order n. Then NFCA(φ) is the sequence of isomorphism classes of pairs NFC(φ k ), φ |NFC(φ k ) n-1 k=1 where φ |NFC(φ k ) is the automorphism induced by φ on the curve NFC(φ k ) (if NFC(φ k ) = / 0, then φ acts trivially on it).

P 2 C

 2 invariant by G. Brunella, McQuillan and Mendes have classified, up to birational equivalence, singular holomorphic foliations on projective, compact, complex surfaces ([Bru15,[START_REF] Mcquillan | Diophantine approximations and foliations[END_REF][START_REF] Mendes | Kodaira dimension of holomorphic singular foliations[END_REF]

Corollary 7 .

 7 15 ([D 06b]). -The automorphism group of C(z 0 , z 1 ) is isomorphic to the automorphisms group of Bir(P 2 C ). Remark 7.16. -According to [Bea07] the groups Bir(P n C ) and Bir(P 2 C ) are isomorphic if and only if n = 2.

  if and only if γ = 1. Then b(αz 0 ) = b(z 0 ) + 1 -1 can be rewritten b(z 0 ) = b(αz 0 ). 7.5.2. Centralizers of Jonquières twists. -Recall that the subgroup J of Jonquières maps is isomorphic to PGL(2, C(z 1 )) PGL(2, C). Let us denote by pr 2 the morphism pr 2 : J → PGL(2, C).

  Theorem 8.4 ([START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF][START_REF] Enriques | Sui gruppi continui di transformazioni cremoniani nel piano[END_REF]). -Let G m be the multiplicative group over C. Let r be an integer.If G r m embeds as an algebraic subgroup in Bir(P n C ), then r ≤ n. If r = n, then the embedding is conjugate to an embedding into the group of diagonal matrices in PGL(n + 1, C). Remark 8.5. -Theorem 8.4 not only holds for C but also for any algebraically closed field k. In other words the group of diagonal automorphisms D n plays the role of a maximal torus in Bir(P n C ) and the Cremona group "looks like" a group of rank n. Furthermore Beauville has shown a finite version of Theorem 8.4 in dimension 2: Theorem 8.6 ([Bea07]). -Let p ≥ 5 be a prime. If the abelian group Z pZ r embeds into Bir(P 2 C ), then r ≤ 2. Moreover if r = 2, then the image of Z pZ r is conjugate to a subgroup of the group D 2 of diagonal automorphisms of P 2 C . Remark 8.7. -This statement not only holds for C but also for any algebraically closed field k.

  Lemma 8.18 ([Ure]). -An algebraic subgroup G of Bir(P 2 C ) of dimension ≤ 9 preserves a unique rational fibration.Proof. -According to Theorem 3.46 the group G is conjugate to a subgroup of Aut(F n ) for some Hirzebruch surface F n , n ≥ 2. As a consequence G preserves a rational fibration π :F n → P 1C . The fibres of π are permuted by G, this yields to a homomorphism f : G → PGL(2, C)

  Proof. -Denote by p ∈ H ∞ the fixed point of G, and by e 0 ∈ H ∞ the class of a line in P 2 C . Take an element ψ of G. The action of G on H ∞ is isometric hence d(ψ(e 0 ), p) = d(e 0 , p), and so d(ψ(e 0 ), p) ≤ 2d(e 0 , p). This implies ψ(e 0 ), e 0 ≤ cosh(2d(e 0 , p)) ∀ ψ ∈ G. Since ψ(e 0 ), e 0 = deg ψ the previous inequality can be rewritten as follows deg ψ ≤ cosh(2d(e 0 , p)) ∀ ψ ∈ G, i.e. the degrees of all elements in G are uniformly bounded.According to Weil G can be regularized ( §3.5).

  ( ) have dimension ≤ 1, then G preserves a rational fibration. Assume that R ( ) has an open orbit O. The group G normalizes R ( ) , so G acts regularly on O. The action of R ( ) is faithful; as a result dim R ( ) = 1 and R ( ) C 2 , or R ( ) C * × C, or R ( ) C * × C * . If R ( ) C 2 , then O is isomorphic to the affine plane, and the action of R ( ) on O is given by translations. But the normalizer of C 2 in Aut(A 2 C ) is the group of affine maps GL(2, C) C 2 hence G is bounded. If R ( ) C * × C, then we similarly get the inclusion G ⊂ Aut(C * × C).

  Theorem 8.54 ([START_REF] Cantat | Normal subgroups in the Cremona group[END_REF]). -Let k be an algebraically closed field. If φ ∈ Bir(P 2 k ) is tight, then there exists a non-zero integer n such that for any non-trivial elementψ of φ n deg ψ ≥ deg(φ n ).In particular φ n is a proper subgroup of Bir(P 2 k ). In the second part of their article Cantat and Lamy showed that Bir(P 2 k ) contains tight elements. They distinguished two cases: k = C and k = C. Let us focus on the case k = C. They proved that an element φ of Bir(P 2 C ) of the form a • j, where a is a general element of PGL(3, C) and j is a Jonquières twist, is tight. Let us explain what general means in this context: any element of PGL(3, C) suits after removing a countable number of Zariski closed subsets of PGL(3, C). More precisely they needed the two following conditions:

  n→+∞ (D • (φ n ) * D) 1/n . The Néron-Severi group of P 2 C coincides with the Picard group of P 2 C , has rank 1, and is generated by the class e 0 of a line NS(P 2 C ) = Pic(P 2 C ) = Ze 0 . A map φ ∈ Bir(P 2 C ) acts on Pic(P 2 C ) by multiplication by deg φ. 9.1.2. Pisot and Salem numbers. -We will give the definitions of Pisot and Salem numbers, for more details see [BDGGH + 92]. A Pisot number is an algebraic integer λ ∈]1, +∞[ whose other Galois conjugates lie in the unit disk. Let us denote by Pis the set of Pisot numbers. It includes all integers ≥ 2 as well as all reciprocical quadratic integers λ > 1. The set Pis is a closed subset of the real line; its infimum is equal to the unique root λ P > 1 of the cubic equation x 3 = x + 1. The smallest accumultation point of Pis is the golden mean λ G = 1+ √ 5

  9.1.4.1. A first answer. -Diller and Favre give the first characterization of loxodromic birational maps which are conjugate to an automorphism of a projective surface: Theorem 9.2 ([DF01]). -Let φ ∈ Bir(P 2

φ

  * C • D = C • φ * Dfor all C, D divisor classes. One can factorize φ as φ = η • π -1 where η : Z → S and π = π 1 • . . . • π m : Z → S are two sequences of point blowups. Denote by F j ⊂ Z the total transform of the indeterminacy point of π -1 j under the map π j • . . . • π m . For 1 ≤ j ≤ m let E j be the direct image of F j by η. Each E j , if not zero, is an effective divisor. According to[START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] we get the following formula called push-pull formulaφ * φ * C = C + m ∑ j=1 (C • E j )E j (9.1.1)for all curves (resp. divisor classes) C in S. Since φ * and φ * are adjoint endomorphisms of NS(R, S) for the intersection form we getφ * C • φ * C = C •C m ∑ j=1 (E j •C) 2 (9.1.2)This formula and the Hodge index theorem imply that λ(φ) is a Pisot number or Salem number. The endomorphisms φ * and φ * preserve both the pseudo effective and nef cones of NS(R, S). Suppose that λ(φ) > 1. According to the Perron-Frobenius theorem there exists an eigenvector Θ(φ) for φ * in the nef cone of NS(S) such that φ * Θ(φ) = λ(φ)Θ(φ) (9.1.3) Note that furthermore this vector is unique up to scalar form ([DF01]). Both (9.1.2) and (9.1.3) imply that(λ(φ) 2 -1)Θ(φ) • Θ(φ) = m ∑ j=1 (E j • Θ(φ)) 2 .

φ

  * (κ(Θ(φ)) = κ(λ(φ))κ(Θ(φ)) = ακ(Θ(φ))that is φ * Θ = α Θ where Θ = κ(Θ(φ)). The divisor classes of the E j 's belong to NS(S), so they are κ-invariant. As a consequence (9.1.1) impliesφ * φ * Θ = Θ + m ∑ j=1 ( Θ • E j )E j (9.1.4)Denote by Θ the conjugate of Θ and by α the conjugate of α; from (9.1.4) one gets(αα) θ • θ = φ * Θ • φ * Θ As |α| = αα = 1 one gets that E j • Θ = 0 for any 1 ≤ j ≤ m and E j • Θ(φ) = 0 for any 1 ≤ j ≤ m.Theorem 9.5 follows from Theorem 9.3.Remark 9.6. -Theorem 9.5 does not extend to quadratic integers (see[START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). 9.1.4.3. A third answer. -As we have seen in §1.1 if S is a projective smooth surface, then every φ ∈ Bir(S) admits a minimal resolution, i.e. there exist π 1 : Z → S, π 2 : Z → S two sequences of blow ups such that no (-1)-curves of Z is contracted by both π 1 and π 2 ; φ = π 2 • π -1 1 . Denote by b(φ) the number of base points of φ; note that b(φ) is equal to the difference of the ranks of Pic(Z) and Pic(S); thus b(φ) is equal to b(φ -1 ). Let us introduce the dynamical number of the base-points of φ: it isµ(φ) = lim k→+∞ b(φ k ) k Since b(φ • ψ) ≤ b(φ) + b(ψ) for any φ, ψ in Bir(S), µ(φ) is a non-negative real number. As b(φ) = b(φ -1 ) one gets µ(φ k ) = |kµ(φ)| for any k ∈ Z. Furthermore if ψ : S Z is a birational map between smooth projective surfaces and if φ ∈ Bir(S), then for all n ∈ Z -2b(ψ) + b(φ n ) ≤ b(ψ • ϕ n • ψ -1 ) ≤ 2b(ψ) + b(φ n );hence µ(φ) = µ(ψ • φ • ψ -1 ). One can thus state the following result: Lemma 9.7 ([START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]

  periodic and for any j, ≥ N p ∈ Base(φ j ) ⇐⇒ p ∈ Base(φ ) p ∈ Base(φ -j ) ⇐⇒ p ∈ Base(φ -) Let us decompose Base(φ) into five disjoint sets:B ++ = p | p ∈ P , p ∈ Base(φ j ), p ∈ Base(φ -j ) ∀ j ≥ N B +-= p | p ∈ P , p ∈ Base(φ j ), p ∈ Base(φ -j ) ∀ j ≥ N B -+ = p | p ∈ P , p ∈ Base(φ j ), p ∈ Base(φ -j ) ∀ j ≥ N B --= p | p ∈ P , p ∈ Base(φ j ), p ∈ Base(φ -j ) ∀ j ≥ Nand P . Remarks 9.8. -Note that: B +-is the set of persistent base-points of φ; B -+ is the set of persistent base-points of φ -1 ; two equivalent base-points of φ belong to the same subsets of Base(φ). Take k ≥ 2N an integer. Let us compute b(φ k ). Any base-point of φ k is equivalent to a base-point of φ. Let us thus consider a base-point p of φ and determine the number m p,k of base-points of φ k which are equivalent to p. a) If p belongs to P , then the number of points equivalent to p is less than #P and m p,k ≤ #P . b) If p does not belong to P , then any point equivalent to p is equal to (φ i ) • (p) for some i; furthermore these points all are distinct. Hence m p,k = #I p,k whereI p,k = i ∈ Z | p ∈ Base(φ i ), p ∈ Base(φ i+k ) .b)i) Suppose that p belongs to B ++ . Since p does not belong to Base(φ i ), the following inequalities hold: -N < i < N, and som p,k < 2N. b)ii) If p belongs to B --, then p belongs to Base(φ i+k ) hence -N < i + k < N and m p,k < 2N. b)iii) Assume that p belongs to B -+ . As p ∈ Base(φ i ) (resp. p ∈ Base(φ i+k )), one has -N < i (resp. i + k ≤ N). These two conditions imply -N < i ≤ Nk. But k > 2N, so m p,k = 0.b)iv) Finally consider a point p in B +-. The fact that p ∈ Base(φ i ) (resp. p ∈ Base(φ i+k ))yields i < N (resp. -N < i + k).As a result -Nk < i < N and m p,k ≤ 2N + k. Conversely if i ≤ -N and i + k ≥ N, then p ∈ Base(φ i ) and p ∈ Base(φ i+k ), i.e. i ∈ I p,k . As a consequence m p,k ≥ #[Nk, -N] = k -2N + 1. Finally -2N ≤ m p,kk ≤ 2N. Consequently there exist two constants α, β (independent on k) such that for all k ≥ 2N νk + α ≤ b(φ k ) ≤ νk + β where ν is the number of equivalence classes of persistent base-points of φ (recall that B +- is the set of persistent base-points of φ). But µ(φ) = lim k→+∞ b(φ k ) k , so µ(φ) = ν. One can thus state: Proposition 9.9 ([BD15]

  Theorem 9.10 ([START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]). -Let φ be a birational self map of a smooth projective surface. Then φ is conjugate to an automorphism of a smooth projective surface if and only if µ(φ) = 0. Remark 9.11. -This characterization was implicitely used in [BK09, BK10, BK12, DG11].

  9.3.2. Outline of the construction. -9.3.2.1. The central involution of SL(2, Z) and its image into Bir(P 2 C ).-Set A = of SL(2, Z) is given by (see[START_REF] Newman | Integral matrices[END_REF])A, B | B 4 = (AB) 3 = 1, B 2 (AB) = (AB)B 2 .As as result the quotient of SL(2, Z) by its center is a free product of Z 2Z and Z 3Z generated by the classes[B] of B and [AB] of AB PSL(2, Z) = [B], [AB] | [B] 2 = [AB] 3 = id .Recall that SL(2, R) acts on the upper half planeH = x + iy ∈ C | x, y ∈ R, y > 0 by Möbius transformations SL(2, R) × H → H, a b c d , z → az + b cz + dthe hyperbolic structure of H being preserved. This yields to a natural notion of elliptic, parabolic and loxodromic elements of SL(2, R). If M belongs to SL(2, Z) one can be more precise and check the following observations:

  Lemma 9.15 ([START_REF] Blanc | Embeddings of SL(2, Z) into the Cremona group[END_REF]). -Let θ be an embedding of SL(2, Z) into the plane Cremona group. Then up to birational conjugacy either θ(B 2 ) is an automorphism of P 2 C of order 2, or θ(B 2 ) is a Jonquières involution of degree 3 fixing (pointwise) an elliptic curve. 9.3.2.2. Existence of infinitely many loxodromic embeddings of SL(2, Z) into Bir(P 2 C ). -Let us consider the standard embedding θ e : SL(2, Z) → Bir(P 2 C ) a b c d → (z 0 : z 1 : z 2 ) → (az 0 + bz 1 : cz 0 + dz 1 : z 2 ) . Note that θ e SL(2, Z) is a subgroup of PGL(3, C) that preserves the line L z 2 of equation z 2 = 0 and acts on it via the maps SL(2, Z) → PSL(2, Z) ⊂ PSL(2, C) = Aut(L z 2 ).

  Lemma 9.16 ([BD12]). -Let n be a positive integer. Let a 1 , . . ., a n , b 1 , . . ., b n be 2n elements in {-1, 1}. The birational self map of P 2 C θ k B b n (AB) a n B b n-1 (AB) a n-1 . . . B b 1 (AB) a 1

  [START_REF] Blanc | Embeddings of SL(2, Z) into the Cremona group[END_REF]). -Let n be a positive integer. Let a 1 , a 2 , . . .,a n , b 1 , b 2 , . . ., b n be 2n elements in {-1, 1}. The birational self map of P 2 C θ k B b n (AB) a n B b n-1 (AB) a n-1 . . . B b 1 (AB) a 1rank 2 and are generated by the fibers of the two conic bundles on X 4 and Y 4 . Choosing a birational map X 4

  Bl p 1 P 2 C to P 2 C . If p 2 is a point of indeterminacy of ϕ 1 , we blow up p 2 via π 2 : Bl p 1 ,p 2 P 2

	6 we get
	
	     
	     

C → P 2 C , and we set

E 2 = π -1 2 (p 2 ). Again the map ϕ 2 = ϕ 1 • π 1 : Bl p 1 ,p 2 P 2 C P 2

C is a birational map. We iterate this processus until ϕ r becomes a morphism. Set

E i = (π i+1 • . . . • π r ) * E i and = (π 1 • . . . • π r ) * L

where L is the divisor of a line. Applying r times Proposition 1.Pic(Bl p 1 ,p 2 ,...,p r P 2 C

  The image pr 2 (Y ) of Y by the second projection pr 2 is closed in W d since W d n-1 is a complete variety and pr 2 a Zariski closed morphism. One can write H d as U ∩ pr 2 (Y ) where U ⊆ W d is the open set of elements having a nonzero Jacobian. As a result H d is locally closed in W d and closed in U.

  set is thus closed in W d , and so in H d . If is a positive integer and F a closed subset of H , then we denote by Y F the subset ofY × F (where Y ⊂ W d n-1 ×W d is as above and F is the closure of F in W ) given by Hence Y F is closed in Y × F and also in W d n-1 ×W d ×W . The subset pr 2 (Y F ) of W d is closed in W d ,and so in pr 2 (Y ); as a result pr 2 (Y F ) ∩ U is closed in pr 2 (Y ) ∩ U. We conclude using the fact that pr 2 (Y F ) ∩ U = (π d ) -1 (π (F)) and pr 2 (Y ) ∩ U = H d .

	Lemma 3.7 ([BF13]
	Y F = ((ζ, φ), ϕ) | φ and ϕ yield the same map P n C	P n C .
	In other words
	Y F = ((ζ, φ), ϕ) | φ i ϕ j = φ j ϕ i ∀ i, j .

  we repeat this process for any point of V we get an affine covering.

	Corollary 3.8 ([BF13]). -	A set S ⊆ Bir(P n C ) is closed if and only if π -1 d (S) is closed in
	H d for any d ≥ 1.
	For any d,the set Bir ≤d (P n C ) is closed in Bir(P n C ). For any d, the map π d : H d → Bir ≤d (P n C ) is surjective, continuous and closed. In partic-
	ular it is a topological quotient map.
	Proof. -Let us prove the first assertion. Assume that S is closed in Bir(P n C ). Recall that a subset of Bir(P n C ) is closed in Bir(P n C ) if and only if its preimage by any morphism is closed. Since any π d : H d → Bir(P n C ) is a morphism π -1 d (S) is thus closed in H d . Conversely suppose that π -1 d (S) is closed in H d for any d. Let V be an irreducible algebraic variety, and let υ : V → Bir(P n C ) be a morphism. According to Lemma 3.7 there exists an open
	affine covering (V i ) i∈I of V such that for any i there exist
	an integer d i ,
	a morphism υ
	Lemma 3.7 implies the following one:

i :

  . -Let us first prove that G has a finite number of irreducible components. The group G is closed in Bir ≤d (P n C ) hence its preimage (π d ) -1 (G) is also closed in H d . It thus has a finite number of irreducible components C 1 , C 2 , . . ., C r . The sets π d (C 1 ), π d (C 2 ), . . ., π d (C r ) are closed and irreducible and cover G (third assertion of Corollary 3.8). If we keep the maximal ones with respect to inclusion we get the irreducible components of G.

	As for algebraic groups ([Hum75, §7.3]) one can show that:
	exactly one irreducible component of G passes through id;
	this irreducible component is a closed normal subgroup of finite index in G whose cosets
	are the connected as well as irreducible components of G.

  The Euclidean topology on Bir ≤d (P n C ) is the quotient topology induced by the surjective map π d : H d → Bir ≤d (P n C ) where we put the Euclidean topology on H d

d be a closed subset. To prove that π d (F) is closed in Bir ≤d (P n C ) amounts to prove that the saturated set F = (π d ) -1 (π d (F)) is closed in H d . Consider a sequence (ϕ i ) i∈N of elements in F which converges to ϕ ∈ H d . Let us show that ϕ belongs to F. Since π d is by construction continuous, the sequence π d (ϕ i ) i∈N converges to π d (ϕ) in Bir ≤d (P n C ). Taking a subsequence of π d (ϕ i ) i∈N if needed, we may suppose that the degree of all π

  ≤d-1 (P n C )) is closed in H d , H d,d is open in H d for the Zariski topology and hence also for the Euclidean topology; π d restricts to a homeomorphism H d,d → Bir d (P n The construction of the topology implies that W d and H d are metric spaces. As W d is compact and H d is locally closed in W d (Lemma 3.5) the set H d is locally compact. But metric spaces are sequential spaces and quotients of sequential spaces are sequential ([Fra65]). We now would like to prove that the topological map π d : H d → Bir ≤d (P n C ) is proper and the topological space Bir ≤d (P n C ) is locally compact. Recall that a map f : X → Y between two topological spaces is proper if it is continuous and universally closed: for each topological spaceZ the map f × id Z : X × Z → Y × Z is closed ([Bou98]).A topological space is locally compact if it is Hausdorff and if each of its points has a compact neighborhood. If f : X → Y is a quotient map between topological spaces such that X is locally compact, then f is proper if and only if it is closed and the preimages of points are compact. This implies furthermore that Y is locally compact. According to Lemma 3.5 for any φ in Bir ≤d (P n C ) the set (π d ) -1 (φ) is closed in the compact space W d , so (π d ) -1 (Y ) is compact. The topological space H d being locally compact (Lemma 3.14), to prove that π d is proper it suffices to prove that π d is closed.

	C ) for any
	d ≥ 1.
	Lemma 3.14 ([BF13]). -Let d ≥ 1 be an integer. The spaces W d and H d are locally compact
	metric spaces endowed with the Euclidean topology.
	In particular the sets W d , H d and Bir ≤d (P n C ) are sequential spaces: a subset F is closed if
	the limit of every convergent sequence with values in F belongs to F.
	Proof. -Claim 3.15. -The map π d : H d → Bir ≤d (P n C ) is proper.
	Proof. -Let F ⊂ H

  be an integer. Then the topological map π d : H d → Bir ≤d (P n

	C ) is proper (and closed); C ) is locally compact (and Hausdorff). the topological space Bir ≤d (P n
	Lemma 3.17 ([BF13]). -Let d ≥ 0 be an integer. The natural injection
	ι d : Bir ≤d (P n C ) → Bir ≤d+1 (P n C )
	is a closed embedding, that is a homeomorphism onto its image which is closed in
	Bir ≤d+1 (P n C ).
	Proof. -Consider the map

  is open in Bir ≤d (P n C ).

	3.1.5. The Euclidean topology on Bir(P n C ). -Thanks to Lemma 3.17 one can put on Bir(P n C ) the inductive limit topology given by the Bir ≤d (P n C ): a subset of Bir(P n C ) is closed (resp. open) if and only if its intersection with any Bir ≤d (P n C ) is closed (resp. open). In particular the injections Bir ≤d (P n C ) → Bir(P n C ) are closed embeddings. This topology is called the Euclidean topology of Bir(P n C ). Let us now prove that Bir(P n C ) is a topological
	group endowed with the Euclidean topology.
	Lemma 3.18 ([BF13]). -Let d ≥ 1 be an integer. The map

  and χ d,k is continuous.

	3.1.6. Properties of the Euclidean topology of Bir(P n C ). -
	Lemma 3.24. -Any convergent sequence of Bir(P n C ) has bounded degree.
	Proof. -If the sequence (ϕ i ) i∈N of Bir(P n C ) converges to ϕ, then ϕ i | i ∈ N ∪ {ϕ} is com-pact, so contained in Bir ≤d (P n C ) for some d.
	Lemma 3.25. -The topological group Bir(P n C ) is Hausdorff.
	Proof. -According to [Bou98, III, §2.5, Prop. 13] a topological group is Hausdorff if and
	only if the trivial one-element subgroup is closed. Any point of Bir(P n C ) is closed in some Bir ≤d (P n C ) (Lemma 3.5), so is closed in Bir(P n C ). As a result Bir(P n C ) is Hausdorff.
	Lemma 3.26. -Any compact subset of Bir(P n C ) is contained in Bir ≤d (P n C ) for some d.
	Proof. -Assume by contradiction that Bir(P n C ) contains a compact subset K such that
	According to Lemma 3.20 the map
	χ d,k : Bir ≤d (P n C ) × Bir ≤k (P n C ) → Bir ≤dk (P n C )
	is continuous for each d, k ≥ 1. As a consequence by definition of the topology of Bir(P n C ) we
	get:
	Corollary 3.21 ([BF13]). -The map
	Bir(P n C ) × Bir(P n C ) → Bir(P n C ),	(φ, ψ) → φ • ψ
	is continuous.
	Corollaries 3.19 and 3.21 complete the proof of:
	Theorem 3.22 ([BF13]). -The n-dimensional Cremona group endowed with the Euclidean
	topology is a topological group.
	Let us give a statement about the restriction of the topology on algebraic subgroups:
	Proposition 3.23 ([BF13]). -Let G be a Zariski closed subgroup of Bir(P n C ) of bounded
	degree, let K be its associated algebraic group (Corollary 3.11). We put on G the restriction
	of the Euclidean topology of Bir(P n C ), we get the Euclidean topology on K via the bijection
	π : K → G which becomes a homeomorphism.

  Proof. -Let us denote by Ω the closure of φ k | k ∈ Z in Bir(P n C ). For any j ∈ Z the set φ j (Ω) is a closed subset of Bir(P n C

	there is a closed subset S ⊂ Bir(P n C ) such that algebraic elements of S are semi-simple
	and correspond to elements of
	(a, ξ) ∈ A 1 C × (A 1 C {0}) | a = ξ k
	for some k ∈ Z.
	In particular Bir(P n C ) alg and the set of unipotent elements of Bir(P n C ) are not closed in Bir(P n C ).
	Furthermore we will see that Bir(P n C ) alg is a countable union of closed sets of Bir(P n C ).
	Lemma 3.32 ([Bla16]). -Let φ be an element of Bir(P n C ). The closure of φ k | k ∈ Z in Bir(P n C ) is a closed abelian subgroup of Bir(P n C ).
	An element φ ∈ Bir(P n C ) is algebraic if it is contained in an algebraic subgroup G of Bir(P n C ). Let us denote by Bir(P n C ) alg the set of algebraic elements of Bir(P n C ).
	Theorem 3.31 ([Bla16]). -Let n ≥ 2.
	There are a closed subset U ⊂ Bir(P n C ) canonically homeomorphic to A 1 C and a family of birational maps U → Bir(P n C ) such that algebraic elements of U are unipotent and
	correspond to elements of the subgroup of (C, +) generated by 1;

  not the point of highest multiplicity, the complexity of the system decreases (Lemma 4.14); otherwise if p 0 is the point of highest multiplicity, then the complexity remains constant (Lemma 4.14). Furthermore p 0 belongs to C (Lemma 4.13). Since m p i > c, m p j > c and νm p 0 = 2c, then m p i < c and m p

j < c, i.e. p i ∈ C and p j ∈ C . As a consequence n = n -2.

Lemma 4.16. -Assume there exists a base-point p k in C that is not infinitely near p 0 . Then after composition by a general quadratic birational map, one can disperse the points above p 0 and p k .

  Bir(P 2 C ) 4.2.1. It is not an amalgamated product of two groups. -Let us recall that the group Aut(A 2

C ) ⊂ Bir(P 2 C ) of polynomial automorphisms of the plane is the amalgamated product of the affine group Aff 2

  Bir(P 2 C ). Let us consider an action of Bir(P 2 C ) on T . By Steps 2 and 4 it is sufficient to consider the case when PGL(3, C) C ) has no fixed point. Denote by T i the set of fixed points of H i , i = 1, 2. These two trees are exchanged by σ 2 , and as H 1 and H 2 generate Bir(P 2

	has a fixed point. Let us prove that Bir(P 2 C ) has a fixed point; suppose by contradiction that Bir(P 2

C ) they are disjoint. Denote by S = [p 1 , p 2 ], p i ∈ T i , the minimal segment joining the two trees, and by s > 0 its length. The segment S is thus fixed by C ⊂ H 1 ∩ H 2 , and reversed by σ 2 .

  4.3. Two presentations of the Cremona group 4.3.1. A simple set of generators and relations for Bir(P 2 C ). -In [Bla12] Blanc gives a simple set of generators and relations for the plane Cremona group Bir(P 2 C ). Namely he shows: Theorem 4.23 ([Bla12]). -The group Bir(P 2 C ) is the amalgamated product of the Jonquières group with the group Aut(P 2 C ) of automorphisms of the plane, divided by the relation σ 2

  n C ) are pairwise non-isomorphic, thereby solving an open problem for n ≥ 4. Corollary 7.19 ([Can14]). -Let n and k be natural integers. The group Bir(P n C ) embeds into Bir(P k C ) if and only if n ≤ k. In particular Bir(P n C ) is isomorphic to Bir(P k C ) if and only if n = k.

  so τ does not belong to Cent(φ). As a result ker pr 2|Cent(φ) = J F . Any ϕ ∈ Cent(φ) has to preserve C and the fibration z 1 =cst; the restriction ϕ |C of ϕ to C is an automorphism of C that commutes with the involution τ |C . The group Aut τ (C ) of such automorphisms is a finite group (more precisely if F is generic, then Aut τ

  so is solvable but is not triangularizable. The fifth section dedicated to a sketch of the proof of the characterization of the solvable subgroups of the plane Cremona group ([D 15a, Ure]).

	Let us recall a very old question, already asked in 1895 in [Enr95]:
	"Tuttavia altre questioni d'indole gruppale relative al
	gruppo Cremona nel piano (ed a più forte ragione in S n ,
	n > 2) rimangono ancora insolute; ad esempio l'importante
	questione se il gruppo Cremona contenga alcun sottogruppo
	invariante (questione alla quale sembra probabile si debba
	rispondere negativamente)".
	In 2013 Cantat and Lamy established that Bir(P 2 k ) is not simple as soon as k is algebraically closed ([CL13]). Then in 2016 Lonjou proved that Bir(P 2 k ) is not simple over any field
	([

  The group G is contained in GL(2, Z) (C * ) 2 . Consider the projection π : G → GL(2, Z). On the one hand ker π is bounded, on the other hand all elements of π(G) are bounded. All elliptic elements in GL(2, Z) ⊂ Bir(P 2 C ) are of finite order, so π(G) is a torsion subgroup of GL(2, Z). Since there are only finitely many conjugacy classes of finite subgroups in GL(2, Z) the group π(G) is finite. Therefore, G is a finite extension of a bounded subgroup hence G is bounded.Lemma 8.21 ([Ure]). -Let H be a semi-simple algebraic subgroup of Bir(P 2 C ). Let G ⊂ Bir(P 2 C ) be a group of elliptic elements that normalizes H. Then G is bounded. Proof.

	cz 1 +d for some matrix	a b c d	of PGL(2, C). In particular deg ϕ ≤ 2. The constant
	K(G) thus only depends on the degree of ψ.
	Lemma 8.20 ([Ure]). -A group G ⊂ Bir(P 2 C ) of monomial elliptic elements is bounded.
	Proof. -

  1 which is finitely generated and thus of dimension less or equal to n. As a consequence Γ

	0 , ϕ • Γ 0 • ϕ -1 = Γ 0 for any ϕ ∈ G and Γ 0 is nor-
	malized by G. If Γ 0 is semi-simple, Lemma 8.21 allows to conclude. Assume that Γ 0 is
	not semi-simple. Denote by R the radical of Γ 0 , i.e. R is the maximal connected normal
	solvable subgroup of Γ 0 . Since Γ 0 is semi-simple the inequality dim R > 0 holds. The
	radical is unique hence preserved by Aut(Γ	0 ) and in particular normalized by G. Denote
	by

  2According to Ping Pong Lemma applied to φ n , ψ n together with U 1 and U 2 we get that ψ n , φ n generates a non-abelian free subgroup of G.

	8.4.2. Tits alternative for finitely generated subgroups for automorphisms groups and
	Jonquières group. -
	Lemma 8.40 ([

  Weak alternative" for isometries of H ∞ . -Let us recall some notations and definitions introduced in Chapter 2. Let H be a seperable Hilbert space. Let us fix a Hilbert basis B = (e i ) i on H . Consider the scalar product defined on H by

	]). -The Jonquières group
	J PGL(2, C) PGL(2, C(z 0 ))
	satisfies Tits alternative.
	8.4.3. "

  The isometry φ m • ψ n has thus two distinct fixed points on ∂H Assume that G is a group of elliptic elements. -According to Theorems 8.14 and 8.15 one of the following holds: Suppose that G is isomorphic to a bounded subgroup; in particular G is isomorphic to a subgroup of linear groups, and so satisfies Tits alternative.If G preserves a rational fibration, then G satisfies Tits alternative (Proposition 8.43).8.4.5. A consequence ofTits alternative: the Burnside problem. -The Burnside problem posed by Burnside in 1902 asks whether a finitely generated torsion group is finite. Schur showed in 1911 that any finitely generated torsion group that is a subgroup of invertible n × n complex matrices is finite ([

	8.4.4.3. G is isomorphic to a bounded subgroup;
	G preserves a rational fibration.
	Let us finish by assuming that all elements of Γ are elliptic ones. According to [GdlH90,
	Chapter 8, Lemma 35 and Corollary 36]
	-either the orbit of any point of H ∞ is bounded;
	-or the limit set of Γ is a point.
	From [dlHV89, Chapter 2, §b.8] one gets the following alternative: Γ fixes
	-either a point of H ∞ ;
	-or a point of ∂H ∞ .
	8.4.4. Proof of Theorem 8.31. -
	8.4.4.1. Assume that G contains a loxodromic element. -Let G be a subgroup of Bir(P 2 C )
	that contains a loxodromic element. According to Lemma 8.39 we have to consider the three
	following cases:
	G is conjugate to a subgroup of GL(2, Z) D 2 and Tits alternative holds by Proposi-
	tion 8.41;
	G contains a subgroup of index at most 2 that is isomorphic to Z H where H is a finite
	group, in other words G is cyclic up to finite index, so Tits alternative holds;

∞ ; hence it is a loxodromic one. Applying the above argument we get that φ, ψ contains a free group. Therefore, if Γ contains at least one parabolic isometry, then -either Γ contains a non-abelian free group; -or Γ fixes a point of ∂H ∞ that is the unique fixed point of the parabolic isometries of Γ.

  problem raised by Burnside is still open for homeomorphism (resp. diffeomorphism) groups on closed manifolds. Very few examples are known. Cantat gave a positive answer to the Burnside problem for the Cremona group: G is a subgroup of elliptic elements, in particular G is isomorphic either to a solvable subgroup of J , or to a solvable subgroup of a bounded group; G is conjugate to a subgroup of J ; G is conjugate to a subgroup of the automorphism group of a Halphen surface; G is conjugate to a subgroup of GL(2, Z) D 2 where D 2 = (z 0 , z 1 ) → (αz 0 , βz 1 ) | α, β ∈ C * ; G contains a loxodromic element and there exists a finite subgroup H of Bir(P 2 C ) such that G = Z H. Remark 8.50. -A solvable subgroup of a bounded group is a solvable subgroup from one of the groups that appear in Theorem 3.46. Remark 8.51. -The centralizer of a birational self map of P 2C that preserves a unique fibration that is rational is virtually solvable ( §7.5.2.4); this example illustrates the second case. The map φ preserves a unique fibration F that is elliptic or rational. Let us prove that any element of G preserves F . Denote by α(φ) ∈ ∂H ∞ the fixed point of φ * .

	8.5. Solvable subgroups of Bir(P 2 C ) either G is conjugate to a subgroup of GL(2, Z) D 2 ,
	or G contains a subgroup of index at most two that is isomorphic to Z H where The study of the solvable subgroups of the plane Cremona group starts in [D 15a] and goes H is a finite group. on in [Ure]. 2. Suppose now that G does not contain a loxodromic element but G contains a parabolic
	Theorem 8.49 ([Ure]). -Let G be a solvable subgroup of Bir(P 2 C ). Then one of the following holds: element φ. Take one element in the light cone
	Before giving the proof let us state some consequences.
	The soluble length of a nilpotent subgroup of Bir(P 2 C ) can be bounded by the dimension of P 2 C as Epstein and Thurston did in the context of Lie algebras and rational vector fields on a
	connected complex manifold ([ET79]):
	Corollary 8.52 ([D 15a]). -Let G be a nilpotent subgroup of Bir(P 2 C ) that is not a torsion
	group. The soluble length of G is bounded by 2.
	Theorem 3.46 allows to prove:
	Corollary 8.53 ([Ure]). -The derived length of a bounded solvable subgroup of Bir(P 2 C ) is
	≤ 5.
	The derived length of a solvable subgroup of Bir(P 2 C ) is at most 8.
	Theorem 8.48 ([Can11]). -Every finitely generated torsion subgroup of Bir(P 2 C ) is finite. Proof of Theorem 8.49. -It decomposes into three parts: G contains a loxodromic element;
	G does not contain a loxodromic element but G contains a parabolic element; G is a group of
	Proof. -Let G be a finitely generated torsion subgroup of Bir(P 2 C ). From Tits alternative elliptic elements.
	(Theorem 8.30) G is solvable up to finite index. Since any torsion, solvable, finitely generated 1. Assume first that G contains a loxodromic element. Then Tits alternative and Lemma
	group is finite, G is finite. 8.39 imply the following alternative

  8.60. -Let us give the relationship between tight element and element that satisfies WPD property. When we study the action of the Cremona group on H ∞ (P 2 k ) the axis of any loxodromic element φ is rigid and the stabiliser Stab(Ax(φ)) = ψ ∈ Bir(P 2 k ) | ψ(Ax(φ)) = Ax(φ) of the axis Ax(φ) is virtually cyclic if and only if some positive iterate of φ is tight ([CL13, Lon16, SB13]). As a result for N large the set S(ε, p; N) is contained in Stab(Ax(φ)). The map φ thus satisfies the WPD property if and only if some positive iterate of φ is tight.

	8.7.1. Simple subgroups of Bir(P 2 C ). -Let us first prove Theorem 8.2. Consider a simple
	group acting non-trivially on a rational complex surface. Then according to Theorems 8.3 and
	3.46 the group G is isomorphic to a subgroup of PGL(3, C).
	Conversely the group PGL(3, C) = Aut(P 2 C ) acts by birational maps on S.
	Let us now deal with the proof of Theorem 8.3. Let G be a simple subgroup of the plane
	Cremona group. We distinguish three cases:
	(i) G contains no loxodromic element but a parabolic one;
	(ii) G is an elliptic group;
	(iii) G contains a loxodromic element.
	(i) Assume that G contains no loxodromic element but a parabolic one.
	Lemma 8.64 ([Ure20]). -Consider a simple subgroup G of Bir(P 2 C ) that contains no
	loxodromic element but a parabolic element.
	Hénon maps
	h n : (z 0 : z 1 : z 2 )	z 1 z n-1 2	: z n 1 -z 0 z n-1 2	: z n 2
	of degree n which is not divisible by the characteristic of k. The group of automorphisms of
	Bir(P 2 C ) is generated by inner automorphisms and the action of Aut(C, +, •) (see §7.2). As h n is defined over Z the subgroup h m is a characteristic subgroup of Bir(P 2 C ). One has the
	following result:
	Proposition 8.62 ([Can13]). -The plane Cremona group contains infinitely many charac-
	teristic subgroups.
	8.7. Simple groups of Bir(P 2 C )
	This section is devoted to the classification of simple subgroups of Bir(P 2 C ) (Theorems 8.2
	and 8.3) but also to the proof of the following statement:
	Theorem 8.63 ([Ure20]). -Let S be a complex surface.
	If G is a finitely generated simple subgroup of Bir(S), then G is finite.

Remark 8.61. -Let us recall that a subgroup H of a group G is called a characteristic subgroup of G if for every automorphism ϕ of G the inclusion ϕ(H) ⊂ H holds.

Recall that the examples of elements having the WPD property given by Lonjou are the

  1 and D 2 . The group H generated by ψ • φ • ψ -1 and D 2 is thus bounded and according to Theorem 8.33 conjugate to a subgroup of D 2 . Hence ψ • D 2 • ψ -1 is contained in D 2 and ψ belongs to GL(2, Z) D 2 . Consequently we have the inclusion G ⊂ GL(2, Z) D 2 and get a non trivial morphism υ : G → GL(2, Z). The kernel of υ contains an infinite subgroup of D 2 normalized by φ (Lemma 8.34): contradiction with the fact that G is simple.Note that the conjugacy classes of these finite groups are also described in[START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF].Lemma 8.72 ([Ure20]). -Let C be a curve and let G ⊂ Bir(P 1 C × C ) be a finitely Proposition 8.23 asserts that either G is conjugate to a subgroup of J , or G is contained in an algebraic subgroup of Bir(P 2 C ). In the first case Lemma 8.72 allows to conclude. Let us focus on the last case: algebraic subgroups of Bir(P 2 C ) are linear hence G is linear and therefore finite since linear groups satisfy Malcev property.

	generated simple subgroup that preserves the P 1 C -fibration given by the projection to C .
	Then G is finite.
	Proof. -The group G being simple, G is isomorphic either to a subgroup of
	PGL(2, C), or to a subgroup of Aut(C ). But both PGL(2, C) and Aut(C ) satisfy
	Malcev property, so G is finite.
	(ii) It remains to look at G when G is a finitely generated, simple, elliptic subgroup
	of Bir(P 2
	8.7.2. Finitely generated simple subgroups of Bir(P 2 C ). -We finish the chapter by giving
	a sketch of the proof of the following statement:
	Theorem 8.67 ([Ure20]). -Any finitely generated simple subgroup of the plane Cremona
	group is finite.
	This result and the classification of finite subgroups of Bir(P 2 C ) (see [DI09]) imply:
	Corollary 8.68 ([Ure20]). -A finitely generated simple subgroup of Bir(P 2 C ) is isomorphic to
	either Z or A 5 ;	pZ for some prime p;
	or A 6 ;
	or PSL(2, C).
	Remark 8.69. -Theorem 8.67 also holds for the group of birational self maps of a surface
	over a field k.
	Let G be a finitely generated subgroup of Bir(P 2 C ). Let first see that G does not contain
	loxodromic elements:

C

).

  α 3 and Pic(Y ) β 2 are described ; furthermore the composition of the elements does what is expected.

	INDEX
	Index Index notations
	INDEX bounded degree (group of), 37 Breg(φ), 64 bubble space, 2 Div(V ), 6 D n , 3 Cartan subgroup, 82 G(χ), 143 characteristic (group), 186 closed subset (of Bir(P n Ind(φ), 1 C )), 33 NS(S), 18 comparison point, 16 Nef(V ), 9 comparison triangle, 16 Pic(V ), 6 compatible (mixed system), 167 Pis, 192 complete (linear system), 9 Sal, 193 complexity (linear system), 74 div( f ), 6 condition, 167 e i j , 102 congruence subgroups of SL(n, Z), 102 µ(φ), 196 conic bundle, 53 connected (graph), 80 ν f (D), 6 φ t Z , 143 Cremona group, 2 ∂H ∞ , 17 Cremona map, viii φ • , 197 cycle (of a graph), 81 φ • , 3 algebraically stable (birational map of P 2 k-rank, 159 Z 2Z (C • D) p , 7 2 H d , 34 -conic bundle, 56 δ-hyperbolic space, 15 S(ε, p; N), 183 V (K ), 84 δ-slim, 15 CAT(-1) inequality, 16 CAT(-1) space, 16 CAT(0) inequality, 17 W d , 34 X reg , 65 [H 1 , H 2 ], 99 [g, h], 99 CAT(0) space, 17 Γ(n, q), 102 G-equivariant (birational map), 53 G-equivariant (rational map), 64 G-regular (point), 65 G-surface, 126 k-connected (complex), 85 k-connected (simplices), 85 above (morphism), 18 above (surface), 3 Λ L , 160 Λ φ , 9 λ(φ), 23 λ G , 193 λ L , 193 λ P , 192 g G , 101 H ∞ , 19 B(S), 2 adjoint representation, 159 admissible (model), 85 algebraic birational map, 47 algebraic subgroup of Bir(P n C ), 37 C ), 25 algebraically stable (birational map of a surface), 25 axis (of a birational map), 23 F χ , 143 π d , 34 degree (of a del Pezzo surface), 53 degree (of a rational map), 2 Γ(n, q), 102 H k , 103 I B , 117 ∨ , 102 del Pezzo surface, 53 I G , 116 m p (C), 7 derived series, 99 dimension (linear system), 9 dimension (of a simplicial complex), 84 b(φ), 196 dimension (of a simplex), 84 Z(S), 18 C J A 2 , 80 descending central series, 99 J , 4 m p (φ), 89
	Aff(C), 138	discrete action along an axis, 183
	base-point (birational map), 3 base-point (fibration by lines), 142 base-point (linear system), 9 Aff 2 , 80 Aut(S, F ), 135 Aut(A n C ), 2	distorted (element), 103 dynamical degree, 23 dynamical number of the base-points, 196
	Bertini involution, 117 Ax(φ), 23
	Bertini type, 118 Base(φ), 3	elementary transformation, 55
	birational map, 1 birationally equivalent (maps), 119 biregular (rational map), 63 blow-up of a point, 1 boundary of H ∞ , 17 Bir(S, F ), 135 Bir(P n C ), 2 Bir(P n C ) alg , 47 Bir d (P n C ), 34 Bir ≤d (P n C ), 34	elliptic (birational map), 23 elliptic (isometry), 17 end (of a tree), 81 Enriques surface, 191 equivalent (point), 197

t , 102

3.1. TOPOLOGIES AND ALGEBRAIC SUBGROUPS OF Bir(P nC )

3.3. CLASSIFICATION OF MAXIMAL ALGEBRAIC SUBGROUPS OF Bir(P 2C )

Acknowledgments

I would like to thank all people working on the Cremona groups and around. I have a very special thought for Serge Cantat and Dominique Cerveau who made me discover this wonderful group.

I am grateful to the referees for very constructive recommendations. Enfin merci à mon petit univers...

Since φ belongs to J it fixes the point (1 : 0 : 0), and so permutes (0 : 1 : 0) and (0 : 0 : 1). As a result there exist ϕ in S 3 ∩ J and d in D 2 such that φ = d • ϕ. Hence

Lemma 4.29 ([UZ19]

). -Let φ be an element of PGL(3, C) ∩ J . Suppose that no three of the base-points of σ 2 and σ 2 • φ are collinear.

Then there exist ϕ, ψ in PGL(3, C) ∩ J such that σ 2 • φ • σ 2 = ϕ • σ 2 • ψ. Furthermore this expression is generated by relations (R 1 ), (R 3 ), (R 4 ) and (R 5 ).

Proof. -The assumption deg(σ 2 • φ • σ 2 ) = 2 implies that σ 2 and σ 2 • φ have exactly two common base-points (Lemma 4.26), among them (1 : 0 : 0) because σ 2 • φ and σ 2 belong to J . One can assume up to coordinate permutation that the second point is (0 : 1 : 0). More precisely there exist t 1 , t 2 in S 3 ∩ J such that t 1 • t 2 fixes (1 : 0 : 0) and (0 : 1 : 0). As a result

for some complex numbers a 1 , a 2 , b 1 , b 2 , c. Since no three of the base-points of σ 2 and σ 2 • φ are collinear, a 2 b 2 is non-zero. There thus exist d 1 , d 2 in D 2 such that

We get

and the above relation is generated by relations

Let us recall the following statement due to Cantat: [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). -Let Γ be a finitely generated subgroup of Bir(P 2 C ) of elliptic elements. Then either Γ is bounded, or Γ preserves a rational fibration, i.e. Γ ⊂ J up to birational conjugacy.

C ) be a group of elliptic elements. Then one of the following holds:

G preserves a fibration, and so up to birational conjugacy either

where S is a Halphen surface. every finitely generated subgroup of G is bounded.

Furthermore if G fixes a point p ∈ ∂H ∞ that does not correspond to the class of a rational fibration, then the second assertion holds.

Proof. -The group G fixes a point p ∈ H ∞ ∪ ∂H ∞ (Theorem 8.44).

If p belongs to H ∞ , then G is bounded.

Let us now assume that p belongs to ∂H ∞ . Then either p corresponds to the class of a general fibre of some fibration, or not.

If p corresponds to the class of a general fibre of some fibration π : Y → P 1 C where Y is a rational surface, then G preserves this fibration and is thus conjugate to a subgroup of J (if the fibration is rational), or to a subgroup of Aut(S) where S is a Halphen surface (if the fibration consists of curves of genus 1). Suppose now that p does not correspond to the class of a fibration. Let Γ be a finitely generated subgroup of G. Then either Γ is bounded, or Γ preserves a rational fibration (Proposition 8.23). If Γ preserves a rational fibration F , then Γ fixes a point q ∈ ∂H ∞ that corresponds to the class of F . Hence p and q are two distinct points preserved by G and G fixes the geodesic line through p and q. In particular G fixes a point in H ∞ and according to Lemma 8.22 the degrees of all elements in G are uniformly bounded.

Proof of Theorem 8.14. -Consider a subgroup G of Bir(P 2 C ) of elliptic elements. According to Lemma 8.22 either G preserves a rational fibration, or any finitely generated subgroup of G is bounded.

Assume that any finitely generated subgroup of G is bounded. Set

If n = 0, then G is a torsion group.

In particular ker ρ ⊃ ∆ 2 is thus infinite. Theorem 8.33 allows to conclude.

Lemma 8.38 ( [Ure]). -Let φ and ψ be two loxodromic elements of Bir(P 2 C ) such that Ax(φ) = Ax(ψ). Then φ and ψ have not a common fixed point on ∂H ∞ .

Proof. -Assume by contradiction that φ and ψ have no common fixed point on ∂H ∞ . Lemma 8.37 thus implies that up to birational conjugacy

The group D 2 fixes the axes of all the monomial loxodromic elements; in particular m 1 and m 2 have the same fixed points on ∂H ∞ as φ and ψ. But the group m 1 , m 2 does not contain any infinite abelian group, so according to Lemma 8.37 the birational maps m 1 and m 2 have not a common fixed point on ∂H ∞ : contradiction.

Lemma 8.39 ([Ure]

). -Let G be a subgroup of the plane Cremona group that contains a loxodromic element. Then one of the following holds:

G is conjugate to a subgroup of GL(2, Z) D 2 ; G contains a subgroup of index at most 2 that is isomorphic to Z H where H is a finite group; G contains a non-abelian free subgroup.

Proof. -Let φ be a loxodromic map of G.

Assume first that all elements in G preserve the axis Ax(φ) of φ. The group G contains a subgroup H of index at most 2 with the following property: H preserves the orientation of the axis. As a result any element ψ ∈ H translates the points on Ax(φ) by a constant c ψ . This yields a group morphism

Suppose that there is an element ψ ∈ G that does not preserve Ax(φ). Denote by α(φ) and ω(φ) (resp. α(ψ) and ω(ψ)) the attractive and repulsive fixed points of φ • (resp.

2 ) be a small neighborhood of α(φ) (resp.

G contains a non-abelian free subgroup, and Tits alternative holds.

We where S is a Halphen surface.

Proof.

-By Theorem 8.44 the group G fixes a point p ∈ H ∞ ∪ ∂H ∞ . Consider a parabolic element ϕ of G; then ϕ has no fixed point in H ∞ and a unique fixed point q in ∂H ∞ . As a consequence p = q. According to Theorem 2.9 there exist a surface S, a birational map ψ : P 2 C S, a curve C, and a fibration π :

C ) corresponding to F, so satisfies m • m = 0. Therefore, q ∈ ∂H ∞ corresponds to the line passing through the origin and m. It follows that any element in G fixes m, and so preserves the divisor class of F. In other words any element in ϕ • G • ϕ -1 permutes the fibres of the fibration π : S → C. If the fibration is rational, then up to birational conjugacy G ⊂ J . If the fibration is a fibration of genus 1 curves, there exists a Halphen surface S such that up to birational conjugacy G is contained in Bir(S ) and preserves the Halphen fibration. By Lemma 2.6 the group G is contained in Aut(S ).

Assume first that up to birational conjugacy G ⊂ J PGL(2, C(z 1 )) PGL(2, C). Tits alternative for linear groups in characteristic 0 and Proposition 8.41 imply Tits alternative for G.

Finally suppose that G ⊂ Aut(S) where S is a Halphen surface. The automorphisms groups of Halphen surfaces have been studied ([Giz80, CD12a, Gri16]). In particular Cantat and Dolgachev prove Theorem 8.47 ( [START_REF] Cantat | Rational surfaces with a large group of automorphisms[END_REF]). -Let S be a Halphen surface. There exists a homomorphism ρ : Aut(S) → PGL(2, C) with finite image such that ker ρ is an extension of an abelian group of rank ≤ 8 by a cyclic group of order dividing 24.

In other words the automorphism group of a Halphen surface is virtually abelian hence G is solvable up to finite index.

Combining these two statements the following holds: Theorem 8.56 ( [START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF]). -Let ε be a positive real number. Let G be a group acting by isometries on a δ-hyperbolic space X. Let g be a loxodromic element of G. If G acts discretely along the axis of g, then there exists n ∈ N such that for any h ∈ g n {id} the translation length L(h) of h satisfies L(h) > ε.

In particular, for n big enough g n is a proper subgroup of G. Furthermore this subgroup is free.

As a result to prove Theorem 8.55 Lonjou needed to exhibit elements satisfying the WPD property:

Proposition 8.57 [START_REF] Lonjou | Non simplicité du groupe de Cremona sur tout corps[END_REF]). -Let n ≥ 2 and let k be a field of characteristic which does not divide n. Consider the action of

where k is the algebraic closure of k. The group Bir(P 2 k ) acts discretely along the axis of the loxodromic map h n : (z 0 :

Remark 8.58. -If k is an algebraically closed field of characteristic p > 0, then for any ≥ 1 one has ([CD13]) h p = Bir(P 2 k ). Let us explain why when k = C.

Let us first establish that σ 2 = Bir(P 2 C ).

(8.6.1)

Let φ be a birational self map of the complex projective plane. According to the Noether and Castelnuovo Theorem

where the A i 's belong to PGL(3, C). The group PGL(3, C) is simple; as a result any A i can be written as

The involutions (z 0 , z 1 ) → (-z 0 , -z 1 ) and σ 2 are conjugate; therefore, φ can be written as a composition of conjugates of σ 2 . Since PGL(3, C) is simple, for any non-trivial element A of PGL(3, C) the involution ι : (z 0 , z 1 ) → (-z 0 , z 1 ) can be written as a composition of conjugates of A. The involutions ι and σ 2 being conjugate one has

). -Let G be a finitely generated subgroup of Bir(P 2 C ). If G contains a loxodromic element, then G is not simple.

To prove it we need the following statement. If φ n is tight in G for some integer n, then Theorem 8.54 allows to conclude. If no power of φ is tight, then G contains an infinite subgroup ∆ 2 that is normalized by φ and that is conjugate either to a subgroup of D 2 , or to a subgroup of C 2 (Theorem 8.36). In particular the degrees of the elements of ∆ 2 are uniformly bounded by an integer N. According to Proposition 8.71 there exist a finite field k and a non trivial morphism υ : G → Bir(P 2 k ) such that for all φ in G deg υ(φ) ≤ deg φ.

In Bir(P 2

k ) there exist only finitely many elements of degree ≤ N. As a result υ(∆ 2 ) is finite. The morphism υ has thus a proper kernel and G is not simple: contradiction.

We now have the following alternative (i) G contains a parabolic element, (ii) G is an elliptic subgroup.

Let us look at these two possibilities.

(i) If G contains a parabolic element, then G is conjugate either to a subgroup of the automorphism group Aut(S) of a Halphen surface, or to a subgroup of the Jonquières group J .

Assume first that, up to conjugacy, G ⊂ Aut(S) where S is a Halphen surface.

Recall that a group G satisfies Malcev property if every finitely generated subgroup Γ of G is residually finite, i.e. for any g ∈ Γ there exist a finite group H and a morphism υ : Γ → H such that g does not belong to ker υ. Malcev showed that linear groups satisfy this property ([Mal40]). In [START_REF] Bass | Automorphisms of groups and of schemes of finite type[END_REF] the authors proved that automorphism groups of scheme over any commutative ring also satisfy this property. Consequently if G contains a parabolic element, then G is, up to conjugacy, a subgroup of J . Suppose that G ⊂ J up to birational conjugacy. Then G is finite. Indeed:

has dynamical degree k 2n . In particular, θ k is a loxodromic embedding and

-Let us consider an element of infinite order of SL(2, Z); it is conjugate to

Idea of the proof of Lemma 9.16. -We proceed by induction on n. Let us detail the case n = 1. The birational map ψ has degree k and has a unique proper base-point p = (µ : 1 : 0) ∈ L z 2 . The same holds for ψ -1 . Moreover ψ |L z 2 {p} = ψ -1 |L z 2 {p} = id. Since θ e (AB) a 1 ∈ Aut(P 2 C ) moves the point p onto another point of L z 2 , the map θ k (AB) a 1 has degree k 2 and exactly two proper base-points which are p and (AB)

) has also degree k 2 and two proper base-points which are p and (AB) a 1 -1 (p). 9.3.2.3. Description of loxodromic embeddings for which the central element fixes (pointwise) an elliptic curve. -Let us note that

(take the presentation we gave before and set α 2 = AB, β = B) and that

In this section we will use this last presentation.

We say that a curve is fixed by a birational map if it is pointwise fixed, and say that a curve is invariant or preserved if the map induces a birational action on the curve.

All conjugacy classes of elements of order 4 and 6 in Bir(P 2 C ) have been classified in [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]. Many of them can act on del Pezzo surfaces of degree 1, 2, 3 or 4.

del Pezzo surfaces X, Y of degree ≤ 4 and automorphisms α ∈ Aut(X), resp. β ∈ Aut(Y ) of order 6, resp. 4 so that α 3 and β 2 fix pointwise an elliptic curve, and that Pic(X) α , Pic(Y ) β both have rank 1 are defined to create the embedding. Contracting (-1)-curves invariant by the involutions α 3 and β 2 (but not by α, β which act minimally on X and Y ) we get rational morphisms X → X 4 and Y → Y 4 where X 4 , Y 4 are del Pezzo surfaces on which α 3 and β 2 act minimally. Furthermore X 4 and Y 4 are del Pezzo surfaces of degree 4, both Pic(X 4 ) α 3 and Pic(Y 4 )