Serge Cantat

Jérémy Blanc

Xavier Caruso

Thomas Delzant

Julie Déserti

Igor Dolgachev

The Cremona group

Keywords: 2010 Mathematics Subject Classification. Primary 14E07 ; Secondary 20E32, 32M05, 37F99 Birational geometry, Cremona group, geometric group theory, holomorphic dynamical systems, Tits alternative

We survey a few results concerning groups of birational transformations. The emphasis is on the Cremona group in two variables and methods coming from geometric group theory.

An introduction based on examples

1.1. Cremona groups and groups of birational transformations. Let k be a field and n be a positive integer. The Cremona group Cr n (k) is the group of k-automorphisms of k(X 1 , . . . , X n ), the k-algebra of rational functions in n independent variables. Given n rational functions F i ∈ k(X 1 , . . . , X n ) there is a unique endomorphism of this algebra that maps X i onto F i . This endomorphism is an automorphism of k(X 1 , . . . , X n ) if, and only if the rational transformation f (X 1 , . . . , X n ) = (F 1 , . . . , F n ) is a birational transformation of the affine space A n k , i.e. an element of the group of birational transformations Bir(A n k ). This correspondence identifies Cr n (k) with the group Bir(A n k ). Compactify A n k into the projective space P n k , and denote by [x 1 : . . . : x n+1 ] a system of homogeneous coordinates with X i = x i /x n+1 . Every birational transformation of the affine space corresponds to a unique birational transformation of the projective space, and vice versa. Geometrically, one restricts elements of Bir(P n k ) to the Zariski open subset A n k (resp. one extends elements of Bir(A n k ) to the compactification P n k ). In terms of formulas, a rational transformation f of A n k which is defined by rational fractions F i , as above, gives rise to a rational transformation of the projective space which is defined by homogeneous polynomials f i in the x i : To obtain the f i one just needs to homogenize the F i and to multiply them by the lowest common multiple of their denominators. For instance, the birational transformation h(X 1 , X 2 ) = (X 1 /X 2 , X 2 + 17) of A 2 k corresponds to the birational transformation h[x 1 : x 2 :

x 3 ] = [x 1 x 3 : (x 2 + 17x 3 )x 2 :

x 3 x 2 ].
To sum up, one gets three incarnations of the same group,

Cr n (k) = Bir(A n k ) = Bir(P n k ). (1)
Moreover, every birational transformation f of P n k can be written as f [x 1 : . . . : x n+1 ] = [ f 1 : . . . :

f n+1 ] (2)
where the f i are homogeneous polynomials in the variables x i , of the same degree d, and without common factor of positive degree. This degree d is the degree of f . Birational transformations of degree 1 are linear projective transformations: They form the subgroup

PGL n+1 (k) = Aut(P n k ) ⊂ Bir(P n k ) (3)
of automorphisms of the projective space. More generally, two groups of transformations are naturally associated to any given variety Y : The group Aut(Y ) of its (regular) automorphisms, and the group Bir(Y ) of its birational transformations. If M is a complex manifold, one can consider its group of holomorphic diffeomorphisms and its group of bi-meromorphic transformations. They coincide with the aforementionned groups Aut(M) and Bir(M) when M is the complex manifold determined by a (smooth) complex projective variety.

Examples, indeterminacy points, and dynamics. The group of automorphisms of P n

k is the group PGL n+1 (k) of linear projective transformations. In dimension 1, Cr 1 (k) is equal to PGL 2 (k), because a rational transformation f (X 1 ) ∈ k(X 1 ) is invertible if and only if its degree is equal to 1.

Monomial transformations.

The multiplicative group G n m of dimension n can be identified to the Zariski open subset (A 1 k \ {0}) n of P n k . Thus, Cr n (k) contains the group of all algebraic automorphisms of the group G n m i.e. the group of monomial transformations GL n (Z).

A first example is given by the monomial transformation of the plane (X 1 , X 2 ) → (1/X 1 , 1/X 2 ). It is denoted by σ 2 in what follows; it can be written as

σ 2 [x 1 : x 2 : x 3 ] = [x 2 x 3 : x 3 x 1 : x 1 x 2 ] (4)
in homogeneous coordinates, and is therefore an involution of degree 2. By definition, σ 2 is the standard quadratic involution.

A second example is given by a(X 1 , X 2 ) = (X 2 1 X 2 , X 1 X 2 ). If k is the field of complex numbers C, this transformation a preserves the 2-dimensional real torus

T := {(X 1 , X 2 ) ∈ C * ; |X 1 | = |X 2 | = 1}
and induces a diffeomorphism of T . This torus is uniformized by the plane R 2 , with covering map (t 1 ,t 2 ) → (exp(2π √ -1t 1 ), exp(2π √ -1t 2 )), and the birational transformation a is covered by the linear transformation A(t 1 ,t 2 ) = (2t 1 + t 2 ,t 1 + t 2 ) of R 2 . The dynamics of a is quite rich, as explained in [START_REF] Brin | Introduction to dynamical systems[END_REF]. The linear transformation A has two eigenvalues,

λ A = 3 + √ 5 2
, 1

λ A = 3 - √ 5 2 ,
with λ A > 1, and the affine lines which are parallel to the eigenline for λ A (resp. for λ -1 A ) give rise to a linear foliation of the torus T whose leaves are uniformly expanded under the dynamics of a (resp. uniformly contracted). Periodic points of a |T : T → T correspond to rational points (t 1 ,t 2 ) ∈ Q × Q and form a dense subset of T ; on the other hand, there are points whose orbit is dense in T , and points whose orbit is dense in a Cantor subset of T . The action of a on T preserves the Lebesgue measure and acts ergodically with respect to it.

1.2.2. Indeterminacy points. Birational transformations may have indeterminacy points. The set of indeterminacy points of a birational transformation of a smooth projective variety Y is a Zariski closed subset of co-dimension ≥ 2, and is therefore a finite set when dim(Y ) = 2. For example, σ 2 is not defined at the three points [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1].

Consider the involution of the projective space which is defined by

σ 3 [x 1 : x 2 : x 3 : x 4 ] = 1 x 1 : 1 x 2 : 1 x 3 : 1 x 4 = [ x 2 x 3 x 4 : x 1 x 3 x 4 : x 1 x 2 x 4 : x 1 x 2 x 3 ].
Let ∆ denote the tetrahedron with faces {x i = 0}, 1 ≤ i ≤ 4, and vertices [1 : 0 : 0 : 0], . . ., [0 : 0 : 0 : 1]. The transformation σ 3 blows down each face of ∆ on the opposite vertex. Blow up these four vertices, to get a new projective variety Y together with a birational morphism π : Y → P 3 k . Then, σ 3 lifts to a birational transformation σ3 = π -1 • σ 3 • π of Y ; this birational transformation does not contract any hypersurface but it has indeterminacies along the strict transforms of the edges L i j = {x i = x j = 0}, i = j, of the tetrahedron ∆. Now, fix a field k of characteristic 0, and consider the birational transformation of the plane which is defined by g(X 1 , X 2 ) = (X 1 + 1, X 1 X 2 + 1). The line {X 1 = 0} is contracted to the point [START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF][START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF]. The forward orbit of this point is the sequence g n (1, 1) = (n, y n ) with y n+1 = ny n + 1; since y n grows faster than (n -1)!, one easily checks that this orbit (g n (1, 1)) n≥0 is Zariski dense. ( 1 ) Thus, the indeterminacy points of the iterates of g form a Zariski dense set. Similarly, each vertical line {X 1 = -m}, m ∈ Z + , is contracted by some iterate g m of g. With these remarks in mind, one can show that there is no birational mapping π : X P 2 k such π • g • π -1 becomes a regular automorphisms of (a non-empty Zariski open subset of) X. See also Remark 5.5 for other examples of this type.

Hénon mappings. The group Aut(A n

k ) of polynomial automorphisms of the affine space A n k is contained in the Cremona group Cr n (k). In particular, all transformations (X 1 , . . . , X n ) → (X 1 + P(X 2 , . . . , X n ), X 2 , . . . , X n ), 1 Another argument works as follows. Assume that this orbit is contained in a curve C, and fix an irreducible component D of C. The strict transform of D under the action of g intersects D infinitely many times, and must therefore coïncide with D. One checks that this is impossible by writing down an equation for D.

with P in k[X 2 , . . . , X n ], are contained in Cr n (k). This shows that Cr n (k) is "infinite dimensional" when n ≥ 2.

A striking example of automorphism is furnished by the Hénon mapping h a,c (X 1 , X 2 ) = (X 2 + X 2 1 + c, aX 1 ),

for a ∈ k * and c ∈ k. When a = 0, h a,c is not invertible: The plane is mapped into the line {X 2 = 0} and, on this line, h 0,c maps X 1 to X 2 1 + c. The dynamics of h 0,c on this line coincides with the dynamics of the upmost studied transformation z → z 2 + c , which, for k = C, provides interesting examples of Julia sets (see [START_REF] Milnor | Dynamics in one complex variable[END_REF]). For a ∈ C * , the main features of the dynamics of h 0,c survive in the dynamical properties of the automorphism h a,c : A 2 C → A 2 C , such as positive topological entropy and the existence of infinitely many periodic points [START_REF] Bedford | Polynomial diffeomorphisms of C 2 . IV. The measure of maximal entropy and laminar currents[END_REF].

1.3. Subgroups of Cremona groups. Birational transformations are simple objects, since they are determined by a finite set of data, namely the coefficients of the homogeneous polynomials defining them. On the other hand, they may exhibit very rich dynamical behaviors, as shown by the previous examples. Another illustration of the beauty of Cr n (k) comes from the study of its subgroups.

1.3.1. Mapping class groups. Let Γ be a group which is generated by a finite number of elements γ i , 1 ≤ i ≤ k. Consider the space R Γ of all homomorphisms from Γ to SL 2 (k): It is an algebraic variety over k of dimension at most 3k. The group SL 2 (k) acts on R Γ by conjugacy; the quotient space R Γ // SL 2 (k) , in the sense of geometric invariant theory, is an algebraic variety. The group of all automorphisms of Γ acts on R Γ by pre-composition. This determines an action of the outer automorphism group Out(Γ) by regular tranformations on R Γ // SL 2 (k) . (Out(Γ) is the quotient of Aut(Γ) by the subgroup of all inner automorphisms.)

There are examples for which this construction provides an embedding of Out(Γ) in the group of automorphisms of R Γ // SL 2 (k) . Fundamental groups of closed orientable surfaces of genus g ≥ 3 or free groups F g with g ≥ 2 provide such examples. Thus, the mapping class groups Mod(g) and the outer automorphism groups Out(F g ) embed into groups of birational transformations [START_REF] Magnus | Rings of Fricke characters and automorphism groups of free groups[END_REF][START_REF] Bass | Automorphisms of groups and of schemes of finite type[END_REF].

1.3.2.
Analytic diffeomorphisms of the plane. Consider the group Bir ∞ (P 2 R ) of all elements f of Bir(P 2 R ) such that f and f -1 have no real indeterminacy point: Over C, indeterminacy points come in complex conjugate pairs. Based on the work of Lukackiı, Kollár and Mangolte observed that Bir ∞ (P 2 R ) determines a dense subgroup in the group of diffeomorphisms of P 2 (R) of class C ∞ (see [START_REF] Kollár | Cremona transformations and diffeomorphisms of surfaces[END_REF] for stronger results). A similar result holds if we replace the projective plane by other rational surfaces, for instance by the sphere S 2 R . This implies that all dynamical features that can be observed for diffeomorphisms of P 2 (R) (resp. of S 2 (R)) and are stable under small perturbations are realized in the dynamics of birational transformations. For instance, there are elements f ∈ Bir ∞ (P 2 R ) with a horse-shoe in P 2 (R) (see [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF], Chapter 2.5.c for the definition of horse-shoes, and Chapter 18.2 for their stability). And there are elements of Bir ∞ (S 2 R ) which are not conjugate to a linear projective transformation in Bir(S 2 R ) but exhibit a simple, north-south dynamics: There is one repulsive fixed point, one attracting fixed point, and all orbits in the complement of the two fixed points go from the first to the second as time flows from -∞ to +∞ (see [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF], Chapter 1.6).

1.3.3. Groups of birational transformations. One says that a group Γ is linear if there is a field k, a positive integer n, and an embedding of Γ into GL n (k). Similarly, we shall say that Γ is a group of birational transformations over the field k if there is a projective variety Y k , and an embedding of Γ into Bir(Y k ). The following properties are obvious.

(1) Linear groups are groups of birational transformations.

(2) The product of two groups of birational transformations over k is a group of birational transformations over k.

(3) Any subgroup of a group of birational transformations is also a group of birational transformations.

In certain cases, one may want to specify further properties: If Γ acts faithfully by birational transformations on a variety of dimension d over a field of characteristic p, we shall say that Γ is a group of birational transformations in dimension at most d in characteristic p. For instance, (4) Every finite group is a group of birational transformations in dimension 1 and characteristic 0. (see [START_REF] Greenberg | Maximal groups and signatures[END_REF], Theorem 6') [START_REF] Beauville | De combien de paramètres dépend l'equation générale de degré n?[END_REF] The mapping class group Mod(g) of a closed, orientable surface of genus g ≥ 3 and the group Out(F g ) are groups of birational transformations in dimension ≤ 6g, but Out(F g ) is not linear if g ≥ 4 (see [START_REF] Bass | Automorphisms of groups and of schemes of finite type[END_REF][START_REF] Formanek | The automorphism group of a free group is not linear[END_REF][START_REF] Macbeath | Spaces of subgroups and Teichmüller space[END_REF]).

1.4. Aims and scope. This survey is organized in three main chapters. The leitmotiv is to compare groups of birational transformations, for instance Cremona groups, to classical Lie groups and to groups of diffeomorphisms of smooth compact manifolds.

We first look at the groups Bir(X) as (infinite dimensional) analogues of algebraic groups (see Sections 2 to 3). Then, we focus on recent results on groups of birational transformations of surfaces, with an emphasis on the most interesting example Cr 2 (k) (see Sections 4 to 7). The last chapters review several open problems concerning groups of birational transformations in dimension > 2.

There are several geometrical aspects of the theory which are not described at all, including classical features such as the geometry of homaloidal nets and the Noether-Fano inequality, as well as more recent developments like the Sarkisov program and the geometry of birationally rigid varieties. The lectures notes [START_REF] Dolgachev | Lectures on Cremona transformations[END_REF] and the books [START_REF] Dolgachev | Classical ALgebraic Geometry: A Modern View[END_REF][START_REF] Kollár | Rational and Nearly Rational Varieties[END_REF][START_REF]Explicit birational geometry of 3-folds[END_REF] are good introductions to these topics. Dynamical properties of birational transformations are also not discussed; this would require a much longer report [START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Guedj | Propriétés ergodiques des applications rationnelles[END_REF]. In this first part, the main emphasis is on the Zariski topology of the Cremona group and the structure of its algebraic subgroups. We compare Cr n (k) to linear algebraic groups: If Cr n (k) were such a group, what kind of linear group would it be ? 2.1. Zariski topology (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF][START_REF] Serre | Le groupe de Cremona et ses sous-groupes finis[END_REF]). Let B be an irreducible algebraic variety. A family of birational transformations of P n k parametrized by B is, by definition, a birational transformation f of B × P n k such that (i) f determines an isomorphism between two open subsets U and V of B × P n k such that the first projection maps both U and V surjectively onto B, and

(ii) f (b, x) = (b, p 2 ( f (b, x)))
where p 2 is the second projection; thus, each

f b := p 2 ( f (b, •)) is a birational transforma- tion of P n k . The map b → f b is called a morphism from the parameter space B to the Cremona group Cr n (k).
Then, one says that a subset S of Cr n (k) is closed if its preimage is closed for the Zariski topology under every morphism B → Cr n (k). This defines a topology on Cr n (k) wich is called the Zariski topology. Right and left translations whose elements are given by (n+1)-tuples of homogeneous polynomial functions h i (x 0 , . . • the Zariski topology on the Cremona group is the inductive limit topology of the topologies of Cr n (k; ≤ d).

g → g • h, g → h • g
These properties are described in [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]. The following example shows that morphisms into Cr n (k; ≤ d) do not always lift to morphisms into Form n (k; d) when the degree of the formulas varies with the parameter.

Example 2.1. A formula like f = [x 0 R : x 1 R : x 2 R],
for R a homogeneous polynomial of degree d -1 is a non-reduced expression for the identity map; thus, the map from formulas to actual birational transformations contracts sets of positive dimension (here, a projective space onto the point {id P 2 k }). A family of birational transformations of P2 k of degree d which depends on a parameter c may degenerate (for certain values c i of c) onto a non-reduced expression of this type. Assume that the parameter c varies on a smooth curve D, that the general member of the family has degree d, and that for two distinct points c 1 and c 2 the formulas become of type [x 0 R c i : x 1 R c i : x 2 R c i ] with R c 1 and R c 2 two homogeneous polynomials which are not proportional. Glue the two points c 1 and c 2 to obtain a nodal curve C, the normalization of which is D. Then, for every point of C, one gets a well defined birational transformation of the plane parametrized by C; but there is no globally defined morphism from C to the space of homogeneous formulas of degree d that determines globally these birational transformations: The two branches of C through its singularity would lead to two distinct expressions at the singular point, one for R c 1 , one for R c 2 .

An explicit example is described in [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF], with C ⊂ A 2 k the nodal plane cubic a 3 + b 3 = abc and f a,b,c [x 0 :

x 1 : x 2 ] = [x 0 P : x 1 Q : x 2 P],
where

P = ax 2 2 + cx 0 x 2 + bx 2 0 and Q = ax 2 2 + (b + c)x 0 x 2 + (a + b)x 2 0 .
The family of transformations f a,b,c is globally defined by formulas of degree 3; but each element f a,b,c has degree ≤ 2 and there is no global parametrization by homogeneous formulas of degree 2. More precisely, abP = (a

2 x 2 + b 2 x 0 )(bx 2 + ax 0 ) abQ = (a 2 x 2 + b(a + b)x 0 )(bx 2 + ax 0 )
so that we can factor out the linear term (bx 2 + ax 0 ). Thus, f a,b,c is a morphism from C to Cr 2 (k) which lifts to a morphism into Form 2 (k; 3), but each f a,b,c is in fact a birational map of degree ≤ 2 (the degree is indeed equal to 2 if [a : b : c] = [0 : 0 : 1]). On the other hand, there is no regular lift to the space of formulas Form 2 (k; 2). ( 2)

In some sense, the following example is even worse; it shows that there is no structure of algebraic variety on Cr n (k; ≤ d) (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). The sets Cr n (k; d) behave well, but the sets Cr n (k; ≤ d) don't.

Example 2.2. Consider the variety V that one obtains by removing p = [0 : 1 : 0] and q = [0 : 0 : 1] from the plane P 2 k . Use homogeneous coordinates [a : b : c] for this parameter space V ⊂ P 2 k . Note that V contains the line L = {b = c} (the two points p and q are not on this line). Now, consider the family g = g a,b,c of birational transformations defined by g[x 0 :

x 1 : x 2 ] = [x 0 (ax 2 + cx 0 ) : x 1 (ax 2 + bx 0 ) : x 2 (ax 2 + cx 0 )],
i.e. g(x, y) = ay + b ay + c x, y in affine coordinates. One gets a family of birational transformations of degree 2, except that all points of L are mapped to the identity (one factors out the linear term (ax 2 + bx 0 )). Thus, as a map from V to Cr 2 (k; ≤ 2), it contracts L to a point, but it is not constant on the {b = c + εa}, ε = 0. This prevents Cr 2 (k; ≤ 2) to be a bona fide algebraic variety! 2.2. Algebraic subgroups (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF][START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF]). An algebraic subgroup of the Cremona group is a subgroup G < Cr n (k) which is the image of an algebraic group K by a homomorphism ρ such that ρ : K → Cr n (k) is a morphism with respect to the Zariski topology. In particular, any algebraic group has bounded degree: It is contained in Cr n (k; ≤ d) for some d.

Let G be a subgroup of Cr n (k), closed for the Zariski topology, and of bounded degree. One can then prove that there is an algebraic group K and a morphism ρ : K → Cr n (k) such that ρ is a group homomorphism and ρ is a homeomorphism from K onto its image G for the Zariski topology; moreover, morphisms B → Cr n (k) with values in G correspond to algebraic morphisms into the algebraic variety K via ρ. Thus, algebraic subgroups correspond exactly to closed subgroups of bounded degree.

By a theorem of Weil, every subgroup G of bounded degree in Cr n (k) can be regularized: There is a projective variety X and a birational mapping π : X P n k such that G X := π -1 Gπ is contained in the group of regular automorphisms Aut(X) (see [START_REF] Cantat | Morphisms between Cremona groups, and characterization of rational varieties[END_REF] for a description of Weil theorem and references). Moreover, the identity component Aut(X) 0 is a linear algebraic group (because X is rational), and the intersection G X ∩ Aut(X) 0 has finite index in G X (see [START_REF] David | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF]).

Thus, algebraic subgroups of Cr n (k) correspond to algebraic groups of automorphisms of rational varieties X P n k .

2.3. Algebraic tori, rank, and an infinite Weyl group. (k). By stereographic projection, the quadric is birationally equivalent to the plane, so that Bir(P 2 k ) contains also a copy of PGL 2 (k) × PGL 2 (k). More generally, if V = G/P is a homogeneous variety of dimension n, where G is a semi-simple algebraic group and P is a parabolic subgroup of G, then V is rational; once a birational map π : V P n k is given, πGπ -1 determines an algebraic subgroup of Cr n (k).

Example 2.3. An important subgroup of Cr 2 (k) which is not algebraic is the Jonquières group3 Jonq 2 (k), of all transformations of P 1 k ×P 1 k that permute the fibers of the projection onto the first factor. It is isomorphic to the semi-direct product PGL 2 (k) PGL 2 (k(x)); for example, it contains all transformations (X 1 , X 2 ) → (aX 1 , Q(X 1 )X 2 ) with a in k * and Q in k(X 1 ) \ {0}, so that its "dimension" is infinite.

Rank and Weyl group.

Let k be a field. Let S be a connected semi-simple algebraic group defined over k. The group S acts on its Lie algebra s by the adjoint representation; the k-rank of S is the maximal dimension dim k (A) of a connected algebraic subgroup A of S which is diagonalizable over k in GL(s). Such a maximal diagonalizable subgroup is called a maximal torus. For example, the R-rank of SL n (R) is n -1, and diagonal matrices form a maximal torus. If k = C and the rank of S is equal to r, the centralizer of a typical element g ∈ S has dimension r. Thus, the value of the rank reflects the commutation properties inside S.

Theorem 2.4 (Enriques, Demazure, [START_REF] Enriques | Sui gruppi continui di trasformazioni cremoniane nel piano[END_REF][START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF]). Let k be an algebraically closed field, and G m be the multiplicative group over k. Let r be an integer. If G r m embeds as an algebraic subgroup in Cr n (k), then r ≤ n and, if r = n, the embedding is conjugate to an embedding into the group of diagonal matrices in PGL n+1 (k).

In other words, in Cr n (k) the group of diagonal matrices ∆ n plays the role of a maximal torus (more precisely, a torus of maximal dimension, see Remark 2.5 below). The normalizer of ∆ n in Cr n (k) is the semi-direct product of ∆ n with the group of monomial transformations GL n (Z), thus Cr n (k) looks like a group of rank n with maximal torus equal to the diagonal group ∆ n and an infinite Weyl group isomorphic to GL n (Z).

This property is reflected by the structure of its finite subgroups, as we shall see below. Nevertheless, for n = 2, we shall explain in Section 4 that Cr 2 (k) is better understood as a group of rank 1, and I expect similar rank n -1 phenomena for all dimensions n ≥ 2.

Remark 2.5. Theorem 2.4 is a bit misleading. If maximal tori are defined in terms of dimension, then maximal tori in Cr n (C) have dimension n and are all conjugate to the diagonal group. On the other hand, for n ≥ 5, Cr n (C) contains tori of dimension n -3 which are not contained in higher dimensional algebraic tori, and are therefore "maximal" in terms of inclusion; since they are maximal, they are not conjugate to a subgroup of PGL n+1 (C). This phenomenon has been discovered by Popov; we refer to [START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF][START_REF] Vladimir | Some subgroups of the Cremona groups[END_REF][START_REF] Popov | Tori in the Cremona groups[END_REF] for a study of maximal algebraic groups in Cr n (C) or Aut(A n C ).

2.4. Finite subgroups. The Cremona group Cr 1 (k) is isomorphic to PGL 2 (k). Thus, if G is a finite subgroup of Cr 1 (k) whose order is prime to the characteristic of k, then G is cyclic, dihedral, or isomorphic to A 4 , S 4 , or A 5 ; if k is algebraically closed, each of these groups occurs in Cr 1 (k) in a unique way modulo conjugacy. (here, A m and S m stand for the alternating group and the symmetric group on m symbols). One of the rich and well understood chapters on Cr 2 (k) concerns the study of its finite subgroups. While there is still a lot to do regarding fields of positive characteristic and conjugacy classes of finite groups, there is now a list of all possible finite groups and maximal algebraic subgroups that can be realized in Cr 2 (C). We refer to [START_REF] Serre | Le groupe de Cremona et ses sous-groupes finis[END_REF][START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF][START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF][START_REF] Blanc | Sous-groupes algébriques du groupe de Cremona[END_REF] for details and references, to [START_REF] Prokhorov | Simple finite subgroups of the Cremona group of rank 3[END_REF] for finite simple subgroups of Cr 3 (C), and to [START_REF] Beauville | De combien de paramètres dépend l'equation générale de degré n?[END_REF] for applications to the notion of essential dimension. In what follows, we only emphasize a few results.

2.4.1. Rank, and p-elementary subgroups. A finitary version of Theorem 2.4 has been observed by Beauville in [START_REF] Beauville | p-elementary subgroups of the Cremona group[END_REF] for n = 2.

Theorem 2.6. Let k be an algebraically closed field. Let p ≥ 5 be a prime number with p = char(k). Assume that the abelian group (Z/pZ) r embeds into Cr 2 (k). Then r ≤ 2 and, if r = 2, the image of (Z/pZ) r is conjugate to a subgroup of the group of diagonal matrices of PGL 3 (k).

Similarly, Prokhorov proved that the rank r of any p-elementary abelian group (Z/pZ) r of Cr 3 (C) is bounded from above by 3 if p ≥ 17 (see [START_REF] Prokhorov | p-elementary subgroups of the Cremona group of rank 3[END_REF][START_REF] Prokhorov | 2-elementary subgroups of the space Cremona group[END_REF]). One may ask whether there exists a function n → p(n) ∈ Z + such that p ≤ p(n) if p is prime and (Z/pZ) n+1 embeds in Cr n (C). In [START_REF] Serre | A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field[END_REF], Serre asks much more precise questions concerning the structure of finite subgroups of Cr n (k). One of them concerns the Jordan property: Does every finite subgroup G of Cr n (C) contain an abelian subgroup of rank ≤ n whose index in G is bounded by a constant j(n) depending only on the dimension n ? These questions were answered positively by Prokhorov and Shramov, assuming the so-called Borisov-Alexeev-Borisov conjecture on the boundedness of families of Fano varieties with terminal singularities (see [START_REF] Prokhorov | Jordan property for groups of birational selfmaps[END_REF][START_REF] Prokhorov | Jordan property for cremona groups[END_REF]). Amazingly, a recent preprint of Birkar delivers a proof of this conjecture (see [START_REF] Birkar | Singularities of linear systems and boundedness of fano varieties[END_REF]).

2.4.2. Finite simple subgroups (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF][START_REF] Igorevich | The conjugacy classes of finite nonsolvable subgroups in the plane Cremona group[END_REF]). There is one, and only one simple subgroup in Cr 1 (C), namely A 5 , the symmetry group of the icosaehdron.

Theorem 2.7. If G is a finite, simple, non-abelian subgroup of Cr 2 (C), then G is isomorphic to one of the groups PSL 2 (F 7 ), A 5 , and A 6 .

• There are two conjugacy classes of subgroups isomorphic to PSL 2 (F 7 ). First, PSL 2 (F 7 ) embeds in PGL 3 (C), preserving the smooth quartic curve x 3 0 x 1 + x 3 1 x 2 + x 3 2 x 0 = 0; then, it also embeds as a group of automorphisms of the double cover of the plane, ramified along the same quartic curve.

• There are three embeddings of A 5 in Cr • There is a unique copy of A 6 up to conjugacy, given by a linear projective action on P 2 C that preserves the curve

10x 3 0 x 3 1 + 9x 2 x 5 0 + 9x 2 x 5 1 + 27x 6 2 = 45x 2 0 x 2 1 x 2 2 + 135x 0 x 1 x 4 2 .
Note that, given an embedding ι : G → Cr 2 (C), one can twist it by an automorphism ϕ of G. When G is isomorphic to PSL 2 (F 7 ) or A 6 , ι is conjugate to ι • ϕ in Cr 2 (C) if and only if ϕ is an inner automorphism of G; thus, there are 4 distinct embeddings of A 6 (resp. PSL 2 (F 7 )) in Cr 2 (C) up to conjugacy. On the other hand, ι is always conjugate to ι • ϕ when G = A 5 ; thus, A 5 has exactly three embeddings in Cr 2 (C) up to conjugacy.

Remark 2.8. If G is a finite subgroup of Cr 2 (k) and the characteristic p of the field k does not divide the order of G, then G "lifts" in characteristic zero; but there are new examples of simple subgroups of Cr 2 (k) if we allow p to divide |G| (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]

for a classification).

There is also a classification, due to Prokhorov [START_REF] Prokhorov | Simple finite subgroups of the Cremona group of rank 3[END_REF], of finite simple subgroups of Cr 3 (C) up to isomorphism, but a complete list of their conjugacy classes is not available yet. Besides A 5 , A 6 , and PSL 2 (F 7 ), there are three new players: A 7 , PSL 2 (F 8 ), and PSP 4 (F 3 ), with respective orders 2520, 504, 25920. See [START_REF] Cheltsov | Three embeddings of the Klein simple group into the Cremona group of rank three[END_REF][START_REF] Cheltsov | Five embeddings of one simple group[END_REF][START_REF] Cheltsov | Cremona groups and the icosahedron[END_REF] for the study of their conjugacy classes in Cr 3 (C).

Closed normal subgroups.

Let us assume, for simplicity, that k is algebraically closed. In dimension n = 1, the Cremona group PGL 2 (k) is a simple group. As we shall see in Section 7, Cr 2 (k) is not simple, and contains many normal subgroups. But J. Blanc and S. Zimmerman observed that Cr n (k) behaves as a simple group if one restricts our study to closed, normal subgroups. Theorem 2.9 ( [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF][START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). Let k be an algebraically closed field. Every non-trivial normal subgroup of Cr n (k) which is closed for the Zariski topology coincides with Cr n (k).

This result explains why there is no construction from algebraic geometry that produces interesting normal subgroups in Cr n (k).

Assume now that k is a local field; this means that k is a locally compact topological field with respect to a non-discrete topology. The examples are R, C, and finite extensions of Q p and F q ((t)). (Here, Q p is the field of p-adic numbers and F q is a finite field with q elements) Then, there exists a group-topology on Cr n (k) that extends the "transcendental, euclidean" topology of PGL n+1 (k) (see [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]). Blanc and Zimmermann also prove that every normal subgroup that is closed for this topology is either trivial or equal to Cr n (k) (see [START_REF] Blanc | Topological simplicity of the Cremona groups[END_REF]). The first main result on Cr 2 (k) is due to Noether and Castelnuovo [START_REF] Noether | Über flächen, welche scharen rationaler curven besitzen[END_REF][START_REF] Castelnuovo | Le trasformazioni generatrici del gruppo cremoniano nel piano[END_REF]. It exhibits two sets of generators for Cr 2 (k). Theorem 3.1 (Noether, Castelnuovo). Let k be an algebraically closed field. The group Cr 2 (k) is generated by PGL 3 (k) and the standard quadratic involution σ 2 . It is also generated by Jonq 2 (k) and the involution η(X 1 , X 2 ) = (X 2 , X 1 ).

Generating sets and relations

Identify Jonq 2 (k) to the group of birational transformations of P 2 k that preserve the pencil of lines through the point [1 : 0 : 0], and η to the involution [x 1 : [START_REF] Blanc | Simple relations in the Cremona group[END_REF][START_REF] Iskovskikh | Proof of a theorem on relations in the two-dimensional Cremona group[END_REF] and [START_REF] Marat | Defining relations for the Cremona group of the plane[END_REF][START_REF] Marat | On some tensor representations of the Cremona group of the projective plane[END_REF] for former presentations of Cr 2 (k)). Thus, one knows a presentation of Cr 2 (k) by generators and relations.

x 2 : x 3 ] → [x 2 : x 1 : x 3 ]. With such a choice, η is in PGL 3 (k) and σ 2 is in Jonq 2 (k). Then, Cr 2 (k) is the amalgamated product of Jonq 2 (k) and PGL 3 (k) along their intersection, divided by one more relation, namely σ • η = η • σ (see
Example 3.2. Let k be an algebraically closed field. Consider the set of generators of Cr 2 (k) given by σ 2 and the group of automorphisms PGL 3 (k) of P 2 k . The following relations are satisfied

• σ 2 • τ = τ • σ 2 for every permutation τ of the three coordinates x i ; • σ 2 • a = a -1 • σ 2 for every diagonal automorphism a[x 0 : x 1 : x 2 ] = [ux 0 : vx 1 : wx 2 ]. • If h is the linear projective transformation h[x 1 : x 2 : x 3 ] = [x 1 , x 1 -x 2 , x 1 -x 3 ], then (h • σ 2 )
3 is the identity (see [START_REF] Marat | Defining relations for the Cremona group of the plane[END_REF]).

The first and second list of relations occur in the semi-direct product of the group GL 2 (Z) of monomial transformations and the diagonal group G m (k) × G m (k) (i.e. in the normalizer of the maximal torus).

Remark 3.3. Similarly, Jung's theorem asserts that the group of polynomial automorphisms of the affine plane is the free product of two of its subgroups, amalgamated along their intersection (see [START_REF] Lamy | Une preuve géométrique du théorème de Jung[END_REF] for example); the two subgroups are the group of affine transformations, and the group of elementary shears (x, y) → (ax, by

+ p(x)), with p ∈ k[x].
Note that this result holds for every field k, algebraically closed or not. This is related to the following geometric fact: If h is a polynomial automorphism of the affine plane, then h -1 has at most one indeterminacy point in P 2 (k), this point is the image of a general point of the line at infinity under the action of h and, as such, is contained in P 2 (k); thus, the first blow-up that is required to resolve the indeterminacy point is defined over k.

Elementary shears are examples of Jonquières transformations, preserving the pencil of vertical lines x = c st ; one feature of these shears is that there degrees remain bounded under iteration: If g(x, y) = (ax, by + p(x)) and p(x) has degree d, then all iterates g n are shears of degree at most d. This is not typical among Jonquières transformations (see Section 4.2).

3.2. Dimension ≥ 3. In dimension 2, the indeterminacy locus of a birational transformation is a finite set, and the curves that appear by blow-up are smooth rational curves. This simple picture changes dramatically in higher dimension: As we shall see below, for every smooth irreducible curve C, there is a birational transformation g of P 3

k and a surface X ⊂ P 3 k such that (i) X is birationally equivalent to C × P 1 k and (ii) g contracts X onto a subset of codimension ≥ 2. This new feature leads to the following result (see [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF]). Theorem 3.4 (Hudson, Pan). Let n ≥ 3 be a natural integer. Let k be an algebraically closed field. To generate Cr n (k), one needs as many algebraic families of generators, as families of smooth hypersurfaces of P n-1 k of degree ≥ n + 2; one cannot generate the Cremona group by generators of bounded degree.

Obviously, this is loosely stated, and we only present a sketch of the proof (see [START_REF] Pan | Une remarque sur la génération du groupe de Cremona[END_REF][START_REF] Cantat | Generators of the Cremona group in n > 2 variables (after Hudson and Pan)[END_REF] for details). Let [x] = [x 0 : . . . : x n-1 ] be homogeneous coordinates for P n-1 k and [y 0 : y 1 ] be homogeneous coordinates for P 1 k . Let Y be an irreducible hypersurface of degree d in P n-1 k , which is not the plane x 0 = 0, and let h be a reduced homogeneous equation for Y . Define a birational transformation

f Y of P n-1 k × P 1 k by f Y ([x], [y 0 : y 1 ]) = ([x], [y 0 x d 0 : h(x 0 , . . . , x n-1 )y 1 ]).
The transformation f Y preserves the projection onto the first factor P n-1 k , and acts by linear projective transformations on the general fibers P 1 k ; more precisely, on the fiber over [x], f Y is the projective linear transformation which is determined by the 2 by 2 matrix

x d 0 0 0 h(x 0 , . . . , x n-1 )
.

This matrix is invertible if and only if x 0 = 0 and h(x) = 0, and f Y contracts the hypersurface Y × P 1 k to the codimension 2 subset Y × {[1 : 0]}. Thus, given any irreducible hypersurface Y in P n-1 k , one can construct a birational transformation of P n k that contracts a hypersurface which is birationally equivalent to Y × P 1 k . On the other hand, one easily checks the following: Let g 1 , ..., g m be birational transformations of the projective space P n k , and let g be the composition g = g m • g m-1 • . . . • g 1 . Let X be an irreducible hypersurface of P n k . If X is g-exceptional (i.e. g contracts X), then there is an index i, with 1 ≤ i ≤ m, and a g i -exceptional hypersurface X i such that X is birationally equivalent to X i . More precisely, for some index i, g i-1 • . . . • g 1 realizes a birational isomorphism from X to X i , and then g i contracts X i .

Thus, to generate Cr n (k), one needs at least as many families of generators as families of hypersurfaces Remark 3.5. Given f in the Cremona group Cr 3 (k), consider the set of irreducible components {X i } 1≤i≤m of the union of the exceptional loci of f and of its inverse f -1 . Each X i is birationally equivalent to a product P 1 k ×C i , where C i is a smooth irreducible curve. Define g(X i ) as the genus of C i , and the genus of f as the maximum of the g(X i ),

Y ⊂ P n-1 k modulo the equivalence relation "Y Y if and only if Y × P 1 k is birationally equivalent to Y × P 1 k ". But, if
1 ≤ i ≤ m.
Then, the subset of Cr 3 (k) of all birational transformations f of genus at most g 0 is a subgroup of Cr 3 (k): In this way, one obtains a filtration of the Cremona group by an increasing sequence of proper subgroups. See [START_REF] Michael | A filtration in the three-dimensional Cremona group[END_REF][START_REF] Lamy | On the genus of birational maps between threefolds[END_REF] for related ideas and complements.

3.3. Fields which are not algebraically closed. Now, consider the case n = 2, but with a field which is not algebraically closed; for simplicity, take k = Q, the field of rational numbers. Given f in Cr 2 (Q), the indeterminacy locus Ind( f ) of f is a finite subset of P 2 (Q), where Q is a fixed algebraic closure of Q. Fix a number field K, and consider the set of all f ∈ Cr 2 (Q) such that each base point of f and f -1 (including infinitesimally closed points) is defined over K; for instance, if p ∈ P 2 (C) is an indeterminacy point of f -1 , then p = [a 0 : a 1 : a 2 ] with a i in K. This set is a subgroup of Cr 2 (Q); in this way, we get an inductive net of subgroups of Cr 2 (Q). This construction is similar to the filtration obtained in Remark 3.5 (the degree of the extension K/Q plays the same role as the genus).

More generally, fix a field k together with an algebraic closure k of k; denote by k 0 the smallest subfield of k (either Q or F p ). To an element f of Cr 2 (k), one can associate the field k f : The smallest field k 0 ⊂ k f ⊂ k on which f , f -1 and all their base points are defined. With this definition, k f may be smaller than k. Then, the field k f •g is contained in the extension generated by k f and k g . Thus, k f provides a measure for the arithmetic complexity of f , and this measure behaves sub-multiplicatively.( 4) Proposition 3.6. Let k be a field. The Cremona group Cr 2 (k) is not finitely generated.

Proof. Let F be a finite subset of Cr 2 (k). Let k F ⊂ k be the extension of k 0 which is generated by the fields k f , f ∈ F . Let G be the subgroup of Cr 2 (k) generated by F . Then k g ⊂ k F for all elements g of G. Let q(x) be an element of k[x] of degree d, and consider the Jonquières transformation g q which is defined by

g q [x 0 : x 1 : x 2 ] = [x 0 x d 2 : q(x 0 /x 2 )x 1 x d 2 : x d+1 2 ].
Then each root α i of q gives rise to an indeterminacy point [α i : 0 : 1] of g -1 q . Thus, if g q belongs to the group G then all roots of q are contained in k F . If g q is in G for every q, then k F is finitely generated and algebraically closed. No such field exists.

Generating sets and relations for the group Cr 2 (R) have been found in [START_REF] Blanc | Cremona groups of real surfaces[END_REF][START_REF] Iskovskikh | Factorization of birational mappings of rational surfaces from the point of view of Mori theory[END_REF][START_REF] Ronga | Birational diffeomorphisms of the real projective plane[END_REF]140]. For instance, both Cr 2 (R) and Bir ∞ (P 2 (R)) are generated by subsets of Cr 2 (R; ≤ 5); one can even provide presentations of Cr 2 (R) by generators and relations.

In [140], Zimmermann describes a striking application of this circle of ideas. She generates Cr 2 (R) by PGL 3 (R), the group of Jonquières transformations Jonq 2 (R), and a twisted form of it, namely the group Jonq π 2 (R) of birational transformations of the plane that permute the fibers of the rational function

π[x 1 : x 2 : x 3 ] = x 2 2 + (x 1 + x 3 ) 2 x 2 2 + (x 1 -x 3 ) 2 .
This group Jonq π 2 (R) is isomorphic to the semi-direct product A B of the groups A = R * + Z/2Z and B = SO(x 2 + y 2tz 2 ; R(t)). The elements of B preserve each fiber of π, acting as rotations along these circles, with an angle of rotation that depends on the circle. The elements of A permute the circles, the value of the projection π being changed into απ or α/π for some α ∈ R * + . The spinor norm provides a homomorphism from B to the group R(t) * /(R(t) * ) 2 . We may identify R(t) * /(R(t) * ) 2 with the set of polynomial functions g ∈ R[t] with only simple roots; and to such a function g, we associate the function ξ(g)

: [0, π] → Z/2Z
which is defined as follows: for each angle θ ∈ [0, π], ξ(g)(θ) is the number (modulo 2) of roots of g with argument equal to θ (i.e. z = |z|e θ √ -1 ). It turns out that the map g → ξ(g) extends to a homomorphism from Jonq π 2 (R) to the additive group ⊕ [0,π] Z/2Z of functions [0, π] → Z/2Z with finite support. With her explicit presentation of Cr 2 (R), Zimmermann shows that this homomorphism extends to an epimorphism Cr 2 (R) → ⊕ [0,π] Z/2Z, and then she gets the following result. Theorem 3.7 (Zimmermann). The derived subgroup of Cr 2 (R) coincides with the normal closure of PGL 3 (R) in Cr 2 (R) and is a proper subgroup of Cr 2 (R), the abelianization of Cr 2 (R) being isomorphic to the additive group

⊕ [0,π] Z/2Z of functions f : [0, π] → Z/2Z with finite support.
We refer to §7 for a different construction of normal subgroups in Cr 2 (k).

-II-Dimension 2 and hyperbolic geometry

In the forthcoming sections, namely § 4 to 7, we focus on groups of birational transformations of surfaces. The most interesting case is the Cremona group Cr 2 (k) or, what is the same, groups of birational transformations of rational surfaces. Indeed, if X is a projective surface with non-negative Kodaira dimension, then X has a unique minimal model X 0 , and Bir(X) coincides with Aut(X 0 ); if the Kodaira dimension of X is negative and X is not rational, then X is ruled in a unique way, and Bir(X) preserves this ruling. As a consequence, the focus is on the group Cr 2 (k).

An infinite dimensional hyperbolic space

Most recent results on Cr 2 (k) are better understood if one explains how Cr 2 (k) acts by isometries on an infinite dimensional hyperbolic space H ∞ (P 2 k ). This construction is due to Manin and Zariski, but it had not been used much until recently.

Example 4.1. The standard quadratic involution σ 2 maps lines to conics. Thus, it acts by multiplication by 2 on the Picard group of the plane P 2 k (or on the homology group

H 2 (P 2 (C), Z) if k = C). Since σ 2 is an involution, the action of σ 2
2 on that group is the identity, not multiplication by 4. This shows that Cr 2 (k) does not "act" on the Picard group. The forthcoming construction overcomes this difficulty by blowing up all possible indeterminacy points. 4.1.1. General construction. Let X be a smooth, irreducible, projective surface. The Picard group Pic(X) 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 determines a quadratic form on Pic(X), the so-called intersection form

(C, D) → C • D. (6)
The quotient of Pic(X) by the subgroup of divisors E such that E • D = 0 for all divisor classes D is the Néron-Severi group NS(X). It is a free abelian group and its rank, the Picard number ρ(X), is finite; when k = C, NS(X) can be identified to H 1,1 (X; R) ∩ H 2 (X; Z). The Hodge index Theorem asserts that the signature of the intersection form is equal to (1, ρ(X) -1) on NS(X). If π : X → X is a birational morphism, the pull-back map π * is an injective homomorphism from NS(X) to NS(X ) that preserves the intersection form; NS(X ) decomposes as the orthogonal sum of π * NS(X) and the subspace generated by classes of curves contracted by π, on which the intersection form is negative definite.

If π 1 : X 1 → X and π 2 : X 2 → X are two birational morphisms, there is a third birational morphism π 3 : X 3 → X that "covers" π 1 and π 2 , meaning that π 3 • π -1 1 and π 3 • π -1 2 are morphisms; informally, one can obtain X 3 from X by blowing-up all points that are blownup either by π 1 or by π 2 (blowing up more points, one gets several choices for X 3 ). One can therefore define the direct limit of the groups NS(X ), where π : X → X runs over the set of all birational morphisms onto X. This limit

Z(X) := lim π : X →X NS(X ) (7)
is the Picard-Manin space of X. It is an infinite dimensional free abelian group. The intersection forms on NS(X ) determine a quadratic form on Z(X), the signature of which is equal to (1, ∞). By construction, NS(X) embeds naturally as a proper subspace of Z(X),

and the intersection form is negative definite on the infinite dimensional space NS(X) ⊥ .

Example 4.2. The group Pic(P 2 k ) is generated by the class e 0 of a line. Blow-up one point q 1 of the plane, to get a morphism π 1 :

X 1 → P 2 k . Then, Pic(X 1
) is a free abelian group of rank 2, generated by the class e 1 of the exceptional divisor E q 1 , and by the pull-back of e 0 under π 1 (still denoted e 0 in what follows). After n blow-ups

X i → X i-1 of points q i ∈ X i-1 one obtains Pic(X n ) = Ze 0 ⊕ Ze 1 ⊕ . . . ⊕ Ze n (8)
where e 0 (resp. e i ) is the class of the total transform of a line (resp. of the exceptional divisor E q i ) by the composite morphism X n → P 2 k (resp. X n → X i ). The direct sum decomposition ( 8) is orthogonal with respect to the intersection form. More precisely,

e 0 • e 0 = 1, e i • e i = -1 ∀ 1 ≤ i ≤ n, and e i • e j = 0 ∀ 0 ≤ i = j ≤ n. (9)
In particular, Pic(X) = NS(X) for rational surfaces. Taking limits, one sees that the Picard-Manin space Z(P 2 k ) is a direct sum Z(P 2 k ) = Ze 0 ⊕ q Ze q where q runs over all possible points that can be blown-up (including infinitely near points). More precisely, q runs over the so-called bubble space B(X) of X (see [START_REF] Manin | of North-Holland Mathematical Library[END_REF][START_REF] Dolgachev | Classical ALgebraic Geometry: A Modern View[END_REF][START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]).

Minkowski spaces.

This paragraph is a parenthesis on the geometry of Minkowski spaces and their isometries.

Standard Minkowski spaces.-Let H be a real Hilbert space of dimension m + 1 (m can be infinite). Fix a unit vector e 0 of H and a Hilbert basis (e i ) i∈I of the orthogonal complement of e 0 . Define a new scalar product on H by u|u m = a 0 a 0 -∑ i∈I a i a i [START_REF] Bell | The dynamical Mordell-Lang problem for étale maps[END_REF] for every pair u = a 0 e 0 + ∑ i a i e i , u = a 0 e 0 + ∑ i a i e i of vectors. In other words, we just change the sign of the scalar product on e ⊥ 0 . Define H m to be the connected component of the hyperboloid

{u ∈ H | u|u m = 1} (11)
that contains e 0 , and let dist m be the distance on H m defined by (see [START_REF] Benedetti | Lectures on hyperbolic geometry[END_REF][START_REF] Hamal | Teichmüller theory and applications to geometry, topology, and dynamics[END_REF])

cosh(dist m (u, u )) = u|u m . ( 12
)
Figure 1. Three types of isometries (from left to right): Elliptic, parabolic, and loxodromic. Elliptic isometries preserve a point in H m and act as a rotation on the orthogonal complement. Parabolic isometries fix an isotropic vector v; the orthogonal complement of Rv contains it, and is tangent to the isotropic cone. Loxodromic isometries dilate an isotropic line, contract another one, and act as a rotation on the intersection of the planes tangent to the isotropic cone along those lines (see also Figure 2 below).

The metric space (H m , dist m ) is a Riemannian, simply-connected, and complete space of dimension m with constant sectional curvature -1; these properties uniquely characterize it up to isometry. ( 5)

The projection of H m into the projective space P(H ) is one-to-one onto its image. In homogeneous coordinates, its image is the ball a 2 0 > ∑ i a 2 i , and the boundary is the sphere obtained by projection of the isotropic cone

a 2 0 = ∑ i a 2 i .
In what follows, H m is identified with its image in P(H ) and its boundary is denoted by ∂H m ; hence, boundary points correspond to isotropic lines in the space H (for the scalar product

•|• m ).
If h ∈ O + 1,m (R)
is an isometry of H m and v ∈ H is an eigenvector of h with eigenvalue λ, then either |λ| = 1 or v is isotropic. Moreover, since H m is homeomorphic to a ball, h has at least one eigenvector v in H m ∪ ∂H m . Thus, there are three types of isometries [START_REF] Burger | Equivariant embeddings of trees into hyperbolic spaces[END_REF]: Elliptic isometries have a fixed point u in H m ; parabolic isometries have no fixed point in H m but they fix a vector v in the isotropic cone; loxodromic (also called hyperbolic) isometries have an isotropic eigenvector v with eigenvalue λ > 1. They satisfy the following additional properties (see [START_REF] Burger | Equivariant embeddings of trees into hyperbolic spaces[END_REF]).

( (2) An isometry h is parabolic if it is not elliptic and fixes a vector v in the isotropic cone. The line Rv is uniquely determined by the parabolic isometry h. If z is a point of H m , there is an increasing sequence of integers m i such that h m i (z) converges towards the boundary point v.

(3) An isometry h is loxodromic if and only if h has an eigenvector v + h with eigenvalue λ > 1. Such an eigenvector is unique up to scalar multiplication, and there is another, unique, isotropic eigenline Rv - h corresponding to an eigenvalue < 1; this eigenvalue is equal to 1/λ. If u is an element of H m , 1

λ n h n (u) -→ u|v - h m v + h |v - h m v + h
as n goes to +∞, and 1

λ -n h n (u) -→ u|v + h m v + h |v - h m v - h
as n goes to -∞. On the orthogonal complement of Rv + h ⊕ Rv - h , h acts as a rotation with respect to •|• m . The boundary points determined by v + h and v - h are the two fixed points of h in H ∞ ∪ ∂H ∞ : The first one is an attracting fixed point, the second is repulsive.

Moreover, h ∈ Isom(H ∞ ) is loxodromic if and only if its translation length L(h) = inf{dist(x, h(x)) | x ∈ H ∞ } (13)
is positive. In that case, λ = exp(L(h)) is the largest eigenvalue of h and dist(x, h n (x)) grows like nL(h) as n goes to +∞ for every point x in H m . The set of points u with L(h) = dist(u, h(u)) is the geodesic line whose endpoints are the boundary points given by v + h and v - h : By definition, this line is called the axis of h. When h is elliptic or parabolic, the translation length vanishes (there is a point u in H m with L(h) = dist(u, h(u)) if h is elliptic, but no such point exists if h is parabolic).

Remark 4.3. If h is loxodromic and preserves a geodesic subspace W of H m (i.e. the intersection of H m with a vector subspace of H ), then W contains the axis of W (because the attracting fixed points v + h and v - h are automatically contained in the boundary of W ). In particular, the translation length of h on H m is equal to the translation length of h on W . 4.1.3. The hyperbolic space H ∞ (X). Let us come back to the geometry of Z(X), where X is a projective surface. Fix an ample class e 0 in NS(X) ⊂ Z(X). Denote by Z(X,R) and NS(X, R) the tensor products Z(X) ⊗ Z R and NS(X) ⊗ Z R. Elements of Z(X,R) are finite sums u X + ∑ i a i e i where u X is an element of NS(X, R), each e i is the class of an exceptional divisor, and the coefficients a i are real numbers. Allowing infinite sums ∑ i a i e i with ∑ i a 2

i < +∞, one gets a new space Z(X), on which the intersection form extends continuously [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Boucksom | Degree growth of meromorphic surface maps[END_REF].

The set of vectors u in Z(X) such that u • u = 1 is a hyperboloïd. The subset

H ∞ (X) = {u ∈ Z(X) | u • u = 1 and u • e 0 > 0} (14)
Figure 2. For a loxodromic isometry, there are two invariant isotropic lines, one corresponding to the eigenvalue λ > 1, the other to 1/λ. The plane generated by these two lines cuts the hyperbolic space onto a geodesic: This geodesic is the axis of the isometry. The hyperplanes which are tangent to the isotropic cone along these eigenlines are invariant, and the action on their intersection is a rotation, preserving a negative definite quadratic form.

is the sheet of that hyperboloid containing ample classes of NS(X, R). With the distance dist(•, •) defined by cosh dist(u, u

) = u • u , (15)
H ∞ (X) is isometric to a hyperbolic space H ∞ , as described in the previous paragraph (see [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF][START_REF] Benedetti | Lectures on hyperbolic geometry[END_REF][START_REF] Cherix | Groups with the Haagerup property[END_REF]). Thus, starting with any projective surface X, one gets a natural hyperbolic space H ∞ (X) H ∞ . We denote by ∂H ∞ (X) the boundary of H ∞ (X) (viewed as the set of lines in the isotropic cone of Z(X), or as a sphere in P(Z(X))). We denote by Isom(Z(X)) the group of isometries of Z(X) with respect to the intersection form, and by Isom(H ∞ (X)) the subgroup that preserves H ∞ (X).

4.1.4. Action of Bir(X) on Z(X) and H ∞ (X) (following Y. Manin, see [START_REF] Manin | of North-Holland Mathematical Library[END_REF]). Given f ∈ Bir(X), there is a birational morphism π : X → X, obtained by blowing up indeterminacy points of f , such that f lifts to a morphism f : X → X (see [START_REF] Hartshorne | Algebraic geometry[END_REF]). By pull back, the transformation f determines an isometry ( f ) * from Z(X) to Z(X ): Identifying Z(X) to Z(X ) by π * , we obtain an isometry f * of Z(X). Since all points of X have been blown-up to define Z(X), birational transformations behave as regular automorphisms on Z(X), and one can show that the map f → f * = ( f -1 ) * is a homomorphism from Bir(X) to the group Isom(Z(X)); hence, after completion, Bir(X) acts on H ∞ (X) by isometries.

Theorem 4.4 (Manin,[START_REF] Manin | of North-Holland Mathematical Library[END_REF]). Let X be a projective surface defined over an algebraically closed field k. The homomorphism f → f * is an injective homomorphism from Bir(X) to the group of isometries of Z(X) with respect to its intersection form. It preserves H ∞ (X), acting faithfully by isometries on this hyperbolic space.

If k is not algebraically closed, one embeds Bir(X k ) in Bir(X k ) for some algebraic closure k of k, and the theorem applies to Bir(X k ). If k is countable one needs only countably many blow-ups to define Z(X k ); then H ∞ (X k ) is a hypersurface in a separable Hilbert space. A similar phenomenon occurs when one studies a countable subgroup Γ of Bir(X k ), because one only needs to blow-up the base points of the elements of Γ. On the other hand, to apply this construction for the study of Cr 2 (C), one needs uncountably many blow-ups.

4.2.

Types and degree growth. Since Bir(X) acts faithfully on H ∞ (X), there are three types of birational transformations: Elliptic, parabolic, and loxodromic, according to the type of the associated isometry of H ∞ (X). We now describe how each type can be characterized in algebro-geometric terms.

Let h ∈ NS(X, R) be an ample class with self-intersection 1. Define the degree of f with respect to the polarization h by

deg h ( f ) = f * (h) • h = cosh(dist(h, f * h)). (16)
For instance, if f is an element of Bir(P 2 k ), and h = e 0 is the class of a line, then deg h ( f ) is the degree of f , as defined in §1.1. More precisely, if f has degree d, the image of a general line by f is a curve of degree d which goes through the base points q i of f -1 with certain multiplicities a i , and

f * e 0 = de 0 -∑ i a i e i
where e i is the class corresponding to the exceptional divisor that one gets when blowing up the point q i . Then, the intersection f * (e 0 ) • e 0 = deg e 0 ( f ) is equal to d, because e 0 • e i = 0 for i = 0.

If the translation length L( f * ) is positive, we know that the distance dist( f n * (x), x) grows like nL( f * ) for every x ∈ H ∞ (X) (see Section 4.1.2). Since cosh(dist(u, v)) = u • v, this property gives the following lemma. Consequently, λ( f ) does not depend on the polarization and is invariant under conjugacy. In particular, f is loxodromic if and only if λ( f ) > 1, if and only if the sequence deg h ( f n ) grows exponentially fast.

Elliptic and parabolic transformations are also classified in terms of degree growth. Say that a sequence of real numbers (d n ) n≥0 grows linearly (resp. quadratically) if n/c ≤ d n ≤ cn (resp. n 2 /c ≤ d n ≤ cn 2 ) for some c > 0.

Theorem 4.6 (Gizatullin, Cantat, Diller and Favre, see [START_REF] Marat | Rational G-surfaces[END_REF][START_REF] Cantat | Dynamique des automorphismes des surfaces K3[END_REF][START_REF] Cantat | Sur la dynamique du groupe d'automorphismes des surfaces K3[END_REF][START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). Let X be a projective surface, defined over an algebraically closed field k, and h be a polarization of X. Let f be a birational transformation of X.

• f is elliptic if and only if the sequence deg h ( f n ) is bounded. In this case, there exists a birational map φ : Y X and an integer k

≥ 1 such that φ -1 • f • φ is an automorphism of Y and φ -1 • f k • φ is in the connected component of the identity of the group Aut(Y ).
• f is parabolic if and only if the sequence deg h ( f n ) grows linearly or quadratically with n. If f is parabolic, there exists a birational map ψ : Y X and a fibration π : Y → B onto a curve B such that ψ -1 • f •ψ permutes the fibers of π. The fibration is rational if the growth is linear, and elliptic (or quasi-elliptic if char(k) ∈ {2, 3}) if the growth is quadratic.

• f is loxodromic if and only if deg h ( f n ) grows exponentially fast with n: There is a constant b h ( f ) > 0 such that deg h ( f n ) = b h ( f )λ( f ) n + O(1).
We refer to [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF] for a more precise description of the degree growth in the parabolic case.

Remark 4.7. If f is parabolic, the push forward of the fibration π : Y → B by the conjugacy ψ is the unique f -invariant pencil of curves. If the characteristic of k is 0, this pencil is the unique f -invariant (singular) algebraic foliation on X [START_REF] Cantat | Symétries birationnelles des surfaces feuilletées[END_REF]. Apply this construction to the linear transformation (x, y) → (x + y, y) of C 2 : It determines an automorphism f of the abelian surface X = E × E (resp. a birational transformation f of X/m or P 2 C ) such that deg h ( f n ) grows quadratically. Similarly, starting with a linear transformation in GL 2 (Z[ι]) whose spectral radius is α, one gets a birational transformation of the plane whose dynamical degree is α 2 . An example is given by the matrix 1 1 1 0 .

Example 4.8. All transformations (X,Y ) → (X, Q(X)Y ) with Q ∈ k(X)
Its spectral radius is the golden mean (1 + √ 5)/2. One obtains a birational transformation of the plane with dynamical degree (3 + √ 5)/2 (one easily checks that it is not conjugate to the monomial example of Section 1.2.1).

4.3.

Analogy with the mapping class group of a closed, orientable surface. 4.3.1. The mapping class group. Let g ≥ 2 be an integer, and Mod(g) be the mapping class group of the closed orientable surface Σ of genus g. Elements of Mod(g) are isotopy classes of orientation preserving homeomorphisms of Σ.

The three main examples of isotopy classes ϕ ∈ Mod(g) are represented by (1) finite order homeomorphisms, (2) Dehn (multi)-twists, and (3) pseudo-Anosov homeomorphisms. Nielsen-Thurston classification of elements ϕ ∈ Mod(g) tells us that an element which is not pseudo-Anosov has a positive iterate ϕ n that preserves the homotopy class of an essential loop; one can then cut the surface along that loop to reduce the topological complexity of the pair (Σ, ϕ). In a finite number of steps, one ends up with a decomposition of every isotopy class ϕ into pieces of type ( 1), ( 2) and (3) (see [START_REF] Farb | A primer on mapping class groups[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF]).

The mapping class group acts isometrically on the complex of curves and on the Teichmüller space of Σ, and there is a nice analogy between those actions and the action of Cr 2 (k) on H ∞ (P 2 k ). Similarly, given a loxodromic element f in Cr 2 (C), the fixed points of f * on the boundary of H ∞ (X) correspond to closed positive currents which are multiplied by λ( f ) ±1 under the action of f . Those currents are analogous to the invariant foliations of a pseudo-Anosov homeomorphism: They have laminar properties (a weak form of foliated structure). We refer to [START_REF] Bedford | Energy and invariant measures for birational surface maps[END_REF][START_REF] Bedford | Dynamics of rational surface automorphisms: rotation domains[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Cantat | Dynamics of automorphisms of compact complex surfaces[END_REF][START_REF] Diller | Invariant measure and Lyapunov exponents for birational maps of P 2[END_REF][START_REF] Dujardin | Laminar currents and birational dynamics[END_REF][START_REF] Favre | Points périodiques d'applications birationnelles de P 2[END_REF] for this analogy and for dynamical properties of loxodromic birational transformations. 4.3.3. Jonquières, Halphen, and Dehn twists. Recall from Remark 4.7 that a parabolic transformation f of a projective surface X preserves a unique pencil of curves on X; this pencil is birationally equivalent to a rational or a genus 1 fibration on some model X of X. The type of the fiber is related to the degree growth of f : It is rational if the degree growth is linear, and has genus 1 if the growth is quadratic. These two types of parabolic transformations are respectively called Jonquières twists6 and Halphen twists. This is justified by the analogy with Dehn (multi-)twists ϕ ∈ Mod(g) and by the following two facts (they concern the case X = P 2 k , f ∈ Cr 2 (k), and k algebraically closed):

• If the degree-growth is linear, the invariant pencil of f can be transformed into a pencil of lines by an element of Cr 2 (k); hence, after conjugacy, f becomes an element of the Jonquières group Jonq 2 (k).

• If the degree-growth is quadratic, the f -invariant pencil can be transformed in a Halphen pencil [START_REF] Iskovskikh | Algebraic surfaces[END_REF][START_REF] Dolgachev | Rational surfaces with a pencil of elliptic curves (russian)[END_REF]. Halphen pencils are constructed as follows. Start with a smooth cubic curve C ⊂ P 2 k and fix the group law on C with origin at an inflexion point. Choose 9 points on this curve whose sum s is a torsion point of order m for the group law on C. Then, the linear system of curves of degree 3m going through these 9 points with multiplicity m form a pencil of curves of genus 1. Blowing-up these 9 base points, one gets a rational surface with a Halphen fibration.

The Cremona group is thin

In this paragraph, we continue our description of the Cremona group Cr 2 (k) as a group of isometries of an infinite dimensional group H ∞ . One of the leitmotives is to show that this group of isometries is a thin subgroup of the group of all isometries.

5.1. Cremona isometries. Each element f of Cr 2 (k) acts isometrically on H ∞ (P 2 k ).

(1) The isometry f * preserves the "lattice" Z(P 2 k ) of Z(P 2 k ). For instance, the base point e 0 (the class of a line in P 2 k ) is mapped to a finite sum

f * e 0 = de 0 -∑ i a i e i
where each a i is a positive integer, d is the degree of f , and the e i are the classes of the exceptional divisors corresponding to the base points of f -1 .

More precisely, the linear system of all lines in P 2 k is mapped by f to a linear system of curves of degree d = deg( f ); this linear system is, by definition, the homaloidal net of f -1 . Its base points (including infinitely near base points), form a finite set of points q i , with multiplicities a i ; the classes e i in the previous formula are the classes e(q i ) of the blow-ups of the q i . For example, the homaloidal net of the standard quadratic involution σ 2 is the net of conics through the three points q 1 = [1 : 0 : 0], q 2 = [0 : 1 : 0], q 3 = [0 : 0 : 1]. We have (σ 2 ) * e 0 = 2e 0 -e(q 1 ) -e(q 2 ) -e(q 3 ).

Another invariant structure is given by the canonical form. Recall that the canonical class of P 2 k blown up in m points q 1 , . . ., q m is equal to -3e 0 -∑ j e(q j ). Taking intersection products, one gets a linear form ω ∞ : Z(P 2 k ) → Z, defined by ω ∞ : a 0 e 0 -∑ j a j e j → -3a 0 + ∑ j a j .

This form does not extend to the completion Z(P 2 k ) (because there are 2 sequences which are not 1 ).

(2) The isometric action of Cr 2 (k) on Z(P 2 k ) preserves the integral linear form ω ∞ .

The following equalities, which we shall refer to as Noether equalities, follow from the fact that f * is an isometry that preserves ω ∞ : If f * e 0 = de 0 -∑ i a i e i , then

d 2 = 1 + ∑ i a 2 i (17) 3d -3 = ∑ i a i . (18)
These relations impose interesting conditions on the isometries defined by birational transformations of the plane.

Lemma 5.1 (Noether inequality). Let f be an element of Cr 2 (k) of degree d ≥ 2, and let a 1 , . . ., a k be the multiplicities of the base-points of f -1 .

(3) The following equality is satisfied.

(d -1)(a 1 + a 2 + a 3 -(d + 1)) = (a 1 -a 3 )(d -1 -a 1 ) + (a 2 -a 3 )(d -1 -a 2 ) + ∑ 4≤i≤k a i (a 3 -a i ) (3'
) For every pair of indices i, j with 1 ≤ i < j ≤ k, we have a i + a j ≤ d.

(3") Ordering the a i in decreasing order, i.e. a 1 ≥ a 2 ≥ a 3 ≥ a 4 ≥ . . ., we have

a 1 + a 2 + a 3 ≥ d + 1.

Noether Castelnuovo theorem.

One way to state Noether-Castelnuovo theorem, is to say that Cr 2 (k) is generated by the family of standard quadratic involutions, i.e. by the elements g • σ 2 • g -1 with g in Aut(P 2 k ) = PGL 3 (k) (with k algebraically closed). To understand the isometry (σ 2 ) * , denote by q 1 , q 2 , and q 3 the base points of σ 2 , and by X the surface which is obtained by blowing up these three points. On X, σ 2 lifts to an automorphism σ 2 . The Néron-Severi group of X is the lattice of rank 4 generated by the classes e 0 , coming from the class of a line in P 2 k , and the classes e i = e(q i ) given by the three exceptional divisors. The action of σ 2 on NS(X) 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 3 -e 1 ( σ 2 ) * e 3 = e 0 -e 1 -e 2 .
Thus, on NS(X), ( σ 2 ) * coincides with the reflexion with respect to the (-2)-class u = e 0 -e 1 -e 2 -e 3 :

( σ 2 ) * (x) = x + x|u for all x in NS(X). The class u is mapped to its opposite, and the set of fixed points is the hyperplane of vectors x = ∑ i a i e i with a 0 = a 1 + a 2 + a 3 . Note that the class u is not effective, precisely because the three points q 1 , q 2 , and q 3 are not on a line.

Then, blow up all points of X (including infinitely near points) to construct a basis of

Z(P 2 k ), namely Z(P 2 k ) = NS(X) ⊕ p∈B(X) Ze(p)
where B(X) is the set of points that one blows up (see Example 4.2 and [68,[START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). The isometry (σ 2 ) * of Z(P 2 k ) acts on NS(X) as the reflexion ( σ 2 ) * and permutes each vector e(p) with e(σ 2 (p)). Thus, the fixed point set of (σ 2 ) * in H ∞ (P 2 k ) is quite small: It is defined by infinitely many equalities, namely a 0 = a 1 + a 2 + a 3 , and a p = a σ 2 (p) for every p ∈ B(X).

A naive approach to the proof that standard quadratic involutions generate Cr 2 (k) works as follows. Consider an element f in Cr 2 (k), with f * (e 0 ) = de 0 -∑ a i e(q i ). Assume that the multiplicities are decreasing, i.e. a i ≥ a i+1 , and apply Noether inequality to deduce a 1 + a 2 + a 3 ≥ d + 1. Since k is algebraically closed, the base points q i are defined over k. Assume that the base points q 1 , q 2 and q 3 are non-collinear points of P 2 (k) and denote by σ a quadratic involutions with base points q 1 , q 2 and q 3 . Then (σ • f ) * e 0 = (2d -(a 1 + a 2 + a 3 ))e 0 + ... and one sees that the degree e 0 • (σ • f ) * e 0 = 2d -(a 1 + a 2 + a 3 ) is strictly less than d. Thus, in a finite number of steps, one expect to reach a birational transformation of degree 1, i.e. an element of PGL 3 (k). Of course, the difficulty arises from the fact that the dominating base points q 1 , q 2 and q 3 may include infinitesimally near points.

Dynamical degrees, automorphisms, spectral gaps.

Let us come back to the study of birational transformations of arbitrary projective surfaces X. If g is an automorphism of X, g already acts by isometry on NS(X, R) for the intersection form; thus, the dynamical degree λ(g) is equal to the spectral radius of the linear transformation g * : NS(X) → NS(X). This shows that λ(g) is an algebraic number because g * preserves the integral structure of NS(X).

Remark 5.2. As explained in the introduction of Chapter II, a projective surface with non-negative Kodaira dimension has a unique minimal model, on which every birational transformation is an automorphism. On such a surface, all dynamical degrees are algebraic integers, the degree of which is bounded from above by the Picard number of the minimal model. In fact, their degree is bounded by 24 because surfaces with positive Kodaira dimension have no automorphism with dynamical degree > 1 and minimal surfaces with vanishing Kodaira dimension have Picard number at most 24 (see [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]).

A birational transformation of a surface is algebraically stable if the action f * of f on the Néron-Severi group satisfies ( f * ) n = ( f n ) * for all n ≥ 1. This property fails exactly when there is a curve E in the surface X such that f maps E to a point q (i.e. the strict transform is equal to q) and the forward orbit of q contains an indeterminacy point q = f m (q) of f . If this occurs, one can blow up the orbit of q between q and q ; such a modification decreases the number of base points of f . Thus, in a finite number of steps, one reaches a model of X on which f becomes algebraically stable. The precise statement that one gets is the following theorem; it is proved in [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF].

Theorem 5.3 (Diller-Favre, [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF]). Let k be an algebraically closed field. Let X be a projective surface and f be a birational transformation of X, both defined over k. There exists a birational morphism π : Y → X such that f Y := π -1 • f • π is algebraically stable.

For example, if f = σ 2 is the standard quadratic involution, one just needs to blow up its three indeterminacy points. If h is a Hénon automorphism of the affine plane, then h determines an algebraically stable birational transformation of P 2 k . Once f is algebraically stable, the dynamical degree arises as an eigenvalue of f * on the Néron-Severi group and, as such, is an algebraic integer.

A Pisot number is a real algebraic integer α > 1, all of whose conjugates α = α have modulus < 1. A Salem number is a real algebraic integer β > 1 such that 1/β is a conjugate of β, all other conjugates have modulus 1, and there is at least one conjugate β on the unit circle. With such a definition, quadratic units α > 1 are Pisot numbers (and are not Salem numbers). The set of Pisot numbers is countable, closed, and contains accumulation points (the smallest one being the golden mean); the smallest Pisot number is the root λ P 1.3247 of t 3 = t + 1. Salem numbers are not well understood yet; the smallest known Salem number is the Lehmer number λ L 1.1762, a root of t 10 + t 9t 7t 6t 5t 4t 3 + t + 1 = 0, and the existence of Salem numbers between 1 and λ L is an open problem.

The following result, contained in [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF], is a manifestation of Hodge index theorem. Its proof depends deeply on [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF][START_REF] Curtis | Coxeter groups, Salem numbers and the Hilbert metric[END_REF][START_REF] Curtis | Dynamics on blowups of the projective plane[END_REF][START_REF] Curtis | Automorphisms of projective K3 surfaces with minimum entropy[END_REF].

Theorem 5.4. Let X be a projective surface, defined over an algebraically closed field k. Let f be a birational transformation of X with dynamical degree λ( f ) > 1. Then λ( f ) is either a Pisot number or a Salem number and (a) if λ( f ) is a Salem number, then there exists a birational map ψ : Y X which conjugates f to an automorphism of Y ;

(b) if f is conjugate to an automorphism, as in (a), λ( f ) is either a quadratic integer or a Salem number.

Moreover, λ( f ) ≥ λ L , where λ L is the Lehmer number and there are examples of birational transformations of the complex projective plane (resp. of some complex K3 surfaces) such that λ( f ) = λ L .

Define the dynamical spectrum of the surface X by

Λ(X k ) = {λ(h) | h ∈ Bir(X k )}
Theorem 5.4 implies that Λ(X k ) is contained in the union of {1}, the set of Pisot numbers, and the set of Salem numbers. Moreover, there is a spectral gap: Λ(X k ) does not intersect the open interval (1, λ L ). This spectral gap corresponds to an important geometric property of the action of Bir(X) on the hyperbolic space H ∞ (X): If an element f of Bir(X) is loxodromic, its translation length is bounded from below by the uniform constant log(λ L ).

Remark 5.5. Consider a birational transformation g of the plane P 2 k for which λ(g) is a natural integer ≥ 2. The dynamical degree of the Hénon map

(X 1 , X 2 ) → (X 2 + X d 1 , X 2 )
is equal to d. Then, g can not be regularized: There is no birational change of coordinates X P 2 k which conjugates g to a regular automorphism of a projective surface X; this would contradict Assertion (b) in Theorem 5.4. We refer to [START_REF] Brunella | Minimal models of foliated algebraic surfaces[END_REF][START_REF] Chel Tsov | Regularization of birational automorphisms[END_REF] for different arguments leading to birational transformations which are not regularizable.

5.4. Dynamical degrees, well ordered sets of algebraic numbers. Consider a loxodromic element of Cr 2 (k); recall that the degree deg( f ) can be seen as the degree of the homogeneous formulas defining f : P 2 k → P 2 k and as the intersection f * (e 0 ) • e 0 . The inequality λ( f ) ≤ deg( f ) is always satisfied, because the sequence deg( f n ) is submultiplicative, and λ( f ) is the limit of deg( f n ) 1/n . Moreover, λ( f ) is invariant under conjugacy: λ(g f g -1 ) = λ( f ) for all g ∈ Cr 2 (k). Thus, if one defines the minimal degree in the conjugacy class by

mcdeg( f ) = min{deg(g f g -1 ) | g ∈ Cr 2 (k)} one gets the inequality λ( f ) ≤ mcdeg( f )
for all f ∈ Cr 2 (k).

Theorem 5.6 (see [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF]). Let k be an algebraically closed field. Let f be a birational transformation of the plane

P 2 k . If λ( f ) ≥ 10 6 then mcdeg( f ) ≤ 4700λ( f ) 5 . If λ( f ) > 1, then mcdeg( f ) ≤ cosh(110 + 345 log(λ( f ))). In geometric terms, if f is a loxodromic element of Cr 2 (k), one can conjugate f to f in Cr 2 (k) to get L( f ) ≤ dist(e 0 , f * e 0 ) ≤ 110 + 345 L( f ) (19)
(where L( f ), the translation length of f * , is equal to L( f ) and log(λ( f ))). Let us explain the meaning of this statement. Denote by Ax( f ) the axis of f * : By definition, Ax( f ) ⊂ H ∞ (P 2 k ) is the geodesic line whose endpoints are the two fixed points of f * on the boundary ∂H ∞ (P 2 k ); it coincides with the intersection of H ∞ (P 2 k ) with the plane generated by the isotropic lines which are invariant under the action of f * (one is multiplied by λ( f ), the other by 1/λ( f )). Denote by e f the projection of the base point e 0 on Ax( f ); the geodesic segment [e 0 , e f ] is orthogonal to Ax( f ), and its length δ( f ) is the distance from e 0 to Ax( f ). The isometry f * maps e f to a point of Ax( f ) such that dist(e f , f * e f ) = L( f ) = log(λ( f )). The geodesic segment [e 0 , e f ] is mapped to a geodesic segment [ f * e 0 , f * e f ]: It is orthogonal to Ax( f * ), and its length is equal to δ( f ). Thus, dist(e 0 , f * e 0 ) ≤ 2δ( f ) + L( f ).

To get the Inequality [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF], we just need to prove that f can be conjugate to an element f with an explicit bound on δ( f ) (that depends on L( f )). Noether-Castelnuovo theorem leads to a similar problem: Starting with an element f in Cr 2 (k), one tries to compose it with a quadratic involution to decrease dist( f * e 0 , e 0 ), and then to repeat this process and decrease this distance all the way down to 0 (see Section 5.2). Here, one decreases the distance from e 0 to the axis of f by changing f into a conjugate element of Cr 2 (k).

Let us describe a consequence of Theorem 5.6. Given a sequence ( f n ) of loxodromic birational transformations of the plane, one gets a sequence of algebraic numbers λ( f n ).

Assume that this sequence is strictly decreasing, hence bounded. The sequence ( f n ) can be replaced by a sequence ( f n ) such that the degree of f n is uniformly bounded and f n is conjugate to f n for all n. Let d be a degree such that infinitely many of the f n have degree d: One gets a sequence ( f n ) in the algebraic set Cr 2 (k; d). It has been proved by Xie (see [START_REF] Xie | Periodic points of birational transformations on projective surfaces[END_REF]) that the function g → λ(g) is lower semi-continuous with respect to the Zariski topology on Cr 2 (k; d) (resp. along any algebraic family g t of birational transformations). The sequence λ( f n ) decreases with n, and the sets {g ∈ Cr 2 (k; d)| λ(g) ≤ λ( f n )} are Zariski closed; hence, the noetherian property implies that the sequence λ(

f n ) = λ( f n ) is finite.
This argument is detailed in [START_REF] Blanc | Dynamical degrees of birational transformations of projective surfaces[END_REF], and leads to the following result.

Theorem 5.7. Let k be an algebraically closed field. The set of dynamical degrees of all birational transformations of P 2 k is a well ordered set: Every strictly decreasing sequence of dynamical degrees is finite. If k is uncountable, this set is closed.

In particular, given any dynamical degree λ, there is a small interval ]λ, λ + ε] that does not contain any dynamical degree. A similar result holds if one replaces P 2 k by all projective surfaces, all of them taken together, over arbitrary fields, because dynamical degrees are algebraic integers of degree at most 24 on non-rational surfaces (see Remark 5.2).

Finitely generated subgroups

According to Sections 4.1 and 4.2, Cr 2 (k) acts by isometries on an infinite dimensional hyperbolic space, and there is a powerful dictionary between the classification of isometries and the classification of birational maps in terms of degree growth and invariant fibrations. In this section, we explain how this dictionary can be used to describe the algebraic structure of Cr 2 (k) and its subgroups.

Tits Alternative.

A group G satisfies Tits alternative if the following property holds for every finitely generated subgroup Γ of G: Either Γ contains a finite index solvable subgroup or Γ contains a free non-abelian subgroup (i.e. a copy of the free group F r , with r ≥ 2). Tits alternative holds for the linear groups GL n (k) (see [START_REF] Tits | Free subgroups in linear groups[END_REF]), but not for the group of C ∞ -diffeomorphisms of the circle S 1 (see [START_REF] Brin | Groups of piecewise linear homeomorphisms of the real line[END_REF], [START_REF] Ghys | Sur un groupe remarquable de difféomorphismes du cercle[END_REF]). If G satisfies Tits alternative, it does not contain groups with intermediate growth, because solvable groups have either polynomial or exponential growth.

The main technique to prove that a group contains a non-abelian free group is the ping-pong lemma. Let g 1 and g 2 be two bijections of a set S. Assume that S contains two non-empty disjoint subsets S 1 and S 2 such that g m 1 (S 2 ) ⊂ S 1 and g m 2 (S 1 ) ⊂ S 2 for all m ∈ Z \ {0}. Then, according to the ping-pong lemma, the group of transformations of S generated by g 1 and g 2 is a free group on two generators [START_REF] De | Topics in geometric group theory[END_REF]. The proof is as follows. If w = w(a, b) is a reduced word that represents a non-trivial element in the free group F 2 = a, b , one needs to prove that w(g 1 , g 2 ) is a non-trivial transformation of S; for this, one conjugates w with a power of g 1 to assume that w(g 1 , g 2 ) starts and ends with a power of g 1 ; writing

w(g 1 , g 2 ) = g l n 1 g m n 2 . . . g m 1 2 g l 0 1 ,
one checks that g l 0 1 maps S 2 into S 1 , then g m 1 2 g l 0 1 maps S 2 into S 2 , ..., and w maps S 2 into S 1 ; this proves that w(g 1 , g 2 ) is non-trivial because S 2 is disjoint from S 1 . Now, consider a group Γ that acts on a hyperbolic space H ∞ and contains two loxodromic isometries h 1 and h 2 whose fixed points in ∂H ∞ form two disjoint pairs. Take disjoint neighborhoods S i ⊂ H ∞ of the fixed point sets of h i , i = 1, 2. Then, the pingpong lemma applies to sufficiently high powers g 1 = h n 1 and g 2 = h n 2 , and produce a free subgroup of Γ.

This strategy can be used for Bir(X), acting by isometries on H ∞ (X). The difficulty resides in the study of subgroups that do not contain any ping-pong pair of loxodromic isometries; Theorem 4.6 comes in help to deal with this situation, and leads to the following result. Theorem 6.1 ([35]). If X is a projective surface over a field k, the group Bir(X) satisfies Tits alternative.

Moreover, solvable subgroups of Bir(X) which are generated by finitely many elements are well understood: Up to finite index, such a group preserves an algebraic foliation (defined over the algebraic closure of k), or is abelian (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces -version longue. long version with complements[END_REF] and [START_REF] Déserti | On solvable subgroups of the Cremona group[END_REF]). This is analogous to the fact that every solvable subgroup in GL n (k) contains a finite index subgroup that preserves a full flag in k n (if k is algebraically closed).

If M is a projective variety (resp. a compact Kähler manifold), its group of automorphisms satisfies also the Tits alternative [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Dinh | Tits alternative for automorphism groups of compact Kähler manifolds[END_REF].( 7) Question 6.2. Does Cr n (k) satisfy Tits alternative for all n ≥ 3 ? Does Tits alternative holds for Bir(M), for all projective varieties M ? Would the answer be yes, one would obtain a proof of Tits alternative for subgroups of Bir(M): This includes linear groups, mapping class groups of surfaces, and Out(F g ) for all g ≥ 1 (see §1.3.1; see [START_REF] Bestvina | The Tits alternative for Out(F n ). II. A Kolchin type theorem[END_REF] for Tits alternative in this context). The first open case for Question 6.2 concerns the group of polynomial automorphisms of the affine space A 3 k .

Rank one phenomena.

As explained in §2.3, the Cremona group Cr 2 (k) behaves like an algebraic group of rank 2, with a maximal torus given by the group of diagonal matrices in PGL 3 (k). On the other hand, typical elements of degree d ≥ 2 in Cr 2 (C) are loxodromic (not elliptic) and, as such, cannot be conjugate to elements of this maximal torus. This suggests that Cr 2 (k) has rank 1 from the point of view of its typical elements.

The following statement provides a strong version of this principle. [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 k be a field. Let X be a projective surface over k and f be a loxodromic element of Bir(X). Then, the infinite cyclic subgroup of Bir(X) generated by f has finite index in the centralizer {g ∈ Bir(X)

Theorem 6.3 ([
| g • f = f • g}.
Sketch of proof for X = P 2 k (see [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]). If g commutes to f , the isometry g * of H ∞ (P 2 k ) preserves the axis of f * and its two endpoints. Consider the homomorphism which maps the centralizer of f to the group of isometries of Ax( f ); view it as a homomorphism into the group of translations R of the line. Since the translation lengths are bounded from below by log(λ L ) and every discrete subgroup of R is trivial or cyclic, the image of this homomorphism is a cyclic group; its kernel K is made of elliptic elements h fixing all points of Ax( f ) and commuting to f . Let e f be the projection of e 0 on the axis of f ; then dist(h * e 0 , e 0 ) ≤ 2 dist(e 0 , e f ) because K fixes e f . Thus, the group K is a group of birational transformations of bounded degree.

Section 2.2 shows that one can conjugate K to a group of automorphisms of a rational surface Y , and that Aut(Y ) 0 ∩ K becomes a finite index subgroup of K. The Zariski closure of this group in Aut(Y ) 0 is a linear algebraic group G that commutes to f (where f is viewed as a birational transformation of Y ); if this group is infinite, it contains a onedimensional abelian group that commutes to f and whose orbits form a pencil of curves in Y : This contradicts the fact that f does not preserve any pencil of curves (such a pencil would give a fixed point of f * in Z(P 2 k ) with non-negative self intersection, contradicting the loxodromic behaviour of f * ).

Another rank one phenomenum comes from the rigidity of rank 2 subgroups of Cr 2 (k). Let G be a real, almost simple, linear algebraic group and Γ be a lattice in G, i.e. a discrete subgroup such that G/Γ has finite Haar volume. When the R-rank of G is at least 2, Γ inherits its main algebraic properties from G (see [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF] and Section 8.1 below). For instance, Γ has Kazhdan property (T), according to which every representation of Γ by unitary motions on a Hilbert space has a global fixed point. Theorem 6.4 (Déserti, Cantat, [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). Let k be an algebraically closed field and X be a projective surface over k. Let Γ be a countable group with Kazhdan property (T). If ρ : Γ → Bir(X) is a homomorphism with infinite image, then ρ is conjugate to a homomorphism into PGL 3 (k) by a birational map ψ : X P 2 k .

Sketch of proof. The first step is based on a fixed point property: If a group Γ with Kazhdan property (T) acts by isometries on a hyperbolic space H ∞ , then Γ has a fixed point and, as a consequence, all its orbits have bounded diameter (see [START_REF] De | La propriété (T ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger)[END_REF]). Apply this to the action of Bir(X) on H ∞ (X) to deduce that a subgroup of Bir(X) with Kazhdan property (T) has bounded degree (with respect to any given polarization of X). Consequently, there is a birational map π : Y

X that conjugates Γ to a subgroup Γ Y of Aut(Y ) such that Aut(Y ) 0 ∩ Γ Y has finite index in Γ Y .
The last step is based on the classification of algebraic groups of transformations of surfaces, and the fact that every subgroup of SL 2 (k) with Kazhdan property (T) is finite; this leads to the following statement, which concludes the proof: If Aut(Y ) 0 contains an infinite group with Kazhdan property (T), the surface Y must be isomorphic to the projective plane P 2 k (and then Γ Y becomes a subgroup of PGL 3 (k)).

In [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF][START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF][START_REF] Déserti | Le groupe de Cremona est hopfien[END_REF], Déserti draws several algebraic consequences of this result; for instance, she can list all abstract automorphisms of Cr 2 (C) Corollary 6.5. The group of automorphisms of Cr 2 (C) (as an abstract group) is the semidirect product of Cr 2 (C) (acting by conjugacy), and the group Aut(C; +, •) of automorphisms of the field C (acting on the coefficients of the polynomial formulas defining the elements of Cr 2 (C)).

There are now several proofs of this result. It would be interesting to decide whether this statement holds for all algebraically closed fields k (in place of C). Since all proofs depend on Noether-Castelnuovo theorem; they do not extend to higher dimension (see [START_REF] Cantat | Morphisms between Cremona groups, and characterization of rational varieties[END_REF], [START_REF] Kraft | On automorphisms of the affine Cremona group[END_REF][START_REF] Stampfli | A note on automorphisms of the affine Cremona group[END_REF] for partial results).

Small cancellation and normal subgroups

Small cancellation theory is a technique which, starting with a presentation of a group by generators and relations, can be used to prove that the group is large. Assume that the group G is given by a finite symmetric set of generators g i and a finite set of relations R i , each of them being a word in the g i . Enlarge the set of relators in order to satisfy the following property: If R is one of the relators and R ends by the letter g i , then g i Rg -1 i is also an element of our finite set of relators. Under this assumption, a typical small cancellation property assumes that two relators cannot coincide (as words in the g i ) on a piece that occupies at least 1/6 of their length; under such an assumption, the group G is large (it contains a non abelian free group). In particular, in the free group generated by the letters g i , the normal subgroup generated by the R i is rather small. Thus, small cancellation theory can also be seen as a mean to show that a normal subgroup is a proper subgroup.

The first application of this technique to groups of algebraic transformations is due to Danilov (see [START_REF] Danilov | Non-simplicity of the group of unimodular automorphisms of an affine plane[END_REF]). He considered the group Aut 1 (A 2 k ) of polynomial automorphisms of the affine plane with jacobian determinant 1. This group is the amalgamated product of the group of special affine transformations of the plane with the group of elementary auromorphisms (X 1 , X 2 ) → (aX 1 , a -1 X 2 + p(X 1 ))

with a ∈ k * and p(t) ∈ k[t]; the amalgamation is along their intersection. As such, Aut 1 (A 2 k ) acts on a tree by automorphisms (see [START_REF] Serre | Trees. Springer Monographs in Mathematics[END_REF]), and a version of small cancellation theory can be applied to construct many normal subgroups in Aut 1 (A 2 k ). Thus, Aut 1 (A 2 k ) is not simple. Since then, small cancellation theory has made huge progresses, with more geometric, less combinatorial versions. In particular, the work of Gromov, Olshanskii and Delzant on small cancellation and hyperbolic groups led to techniques that can now be applied to the Cremona group Cr 2 (k). We refer to [START_REF] Guirardel | Geometric small cancellation[END_REF] and [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF] for recent geometric viewpoints on small cancellation.

7.1. Rigidity, tightness, axes. Let G be a group of isometries of H ∞ . Let L be a geodesic line in H ∞ . The line L is rigid under the action of G if every element h ∈ G that does not move L too much fixes L. To give a precise definition, one needs to measure the deplacement of L under the action of an isometry h. Say that two geodesic lines L and L are (ε, )-close if the diameter of the set {x ∈ L | dist(x, L ) ≤ ε} is larger than . The precise notion of rigidity is: L is (ε, )-rigid if, for every h ∈ G, h(L) = L if and only if h(L) is (ε, )-close to L; L is rigid if it is (ε, )-rigid for some pair of positive numbers (ε, ) (this pair depends on L and G). In other words, if a geodesic line L is rigid for the action of the group G, the orbit G(L) forms a discrete set in the space of geodesic lines.

Fix a loxodromic element g ∈ G. Consider the stabilizer of its axis:

Stab(Ax(g)) = {h ∈ G | h(Ax(g)) = Ax(g)}.
Say that g is tight if its axis is rigid and every element h of Stab(Ax(g)) satisfies

h • g • h -1 = g or g -1 .
An element of the Cremona group Cr 2 (k) is tight (resp. has a rigid axis) if if it is tight (resp. its axis is rigid) with respect to the action of Cr 2 (k) by isometries on H ∞ (P 2 k ). These notions are directly related to the study of the stabilizer of Ax(g), and the following examples show that this stabilizer may be large. The dynamical degree λ( f M ) is equal to max{|α|, |β|}, where α and β are the eigenvalues of M. Thus, λ( f M ) > 1 if and only if adbc = 1 and |a + d| > 2 or adbc = -1 and a + d = 0. Assuming that f M is loxodromic, we shall prove that f M is not tight. The monomial group GL 2 (Z) normalizes the group of diagonal transformations: If t(x, y) = (ux, vy) then

f M • t • f -1 M (x, y) = (u a v b x, u c v d y) = t (x, y) (20)
where t is obtained from t by the monomial action of

GL 2 (Z) on G m × G m .
The indeterminacy points of monomial transformations are contained in the vertices [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] of the triangle whose edges are the coordinate axis. Blow-up these three points to get a new surface X 1 , and consider the total transform of the triangle: One gets a hexagon of rational curves in X 1 . The group of monomial transformations lifts to a group of birational transformations of X 1 with indeterminacy points located on the 6 vertices of this hexagon. The group of diagonal transformations lifts to a subgroup of Aut(X 1 ) 0 . One can iterate this process, blowing-up the vertices of the hexagon, etc.

The limit of the Néron-Severi groups along this sequence of surfaces X n+1 → X n gives a subspace of Z(P 2 k ) which is invariant under the action of the monomial group GL 2 (Z) and is fixed pointwise by the diagonal group G m (k) × G m (k). Intersect this space with H ∞ (P 2 k ) and denote by H ∞ (toric) its metric completion: One gets a totally geodesic, infinite dimensional subspace of H ∞ (P 2 k ). The axis of every loxodromic element of GL 2 (Z) is contained in H ∞ (toric) and is fixed pointwise under the action of G m (k) × G m (k).

On the other hand, the Equation [START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF]

implies that t • f M •t -1 = t • f M where t (x, y) = (u 1-a v -b x, u -c v 1-d y).
Thus, f M is not tight (as soon as k * contains elements v with v b = 1).

Example 7.2 (see [105,[START_REF] Nicholas | Some effectivity questions for plane cremona transformations[END_REF]). A similar example works for the additive group in place of the multiplicative group when the characteristic p of the field k is positive. For instance, the Hénon mapping h(x, y) = (x py, x) conjugates the translation s(x, y) = (x + u, y

+ v) to h • s • h -1 = (x + u p -v, x + u).
The dynamical degree of h is equal to p, and h normalizes the additive group G a (k) × G a (k) (acting by translations).

The normalizer of the additive group in Cr 2 (k) coincides with the subgroup of elements f ∈ Aut(A 2 k ) which are given in affine coordinates by formulas of type

f (x, y) = (a(x) + b(y), c(x) + d(y))
with a(t), b(t), c(t), d(t) polynomial functions of type ∑ i q i t p i . Another way to state the same result is as follows. Denote by A the ring of linearized polynomials in one variable, i.e. polynomials in the Frobenius endomorphism z → z p of k. This is a noncommutative ring. Then, every 2 by 2 matrix with coefficients in A which is invertible over A determines an algebraic automorphism of G a (k) × G a (k), and every algebraic automorphism of G a (k) × G a (k) is of this type. Thus, GL 2 (A) plays the same role as GL 2 (Z) in the previous example.

Base points of elements of GL 2 (A) are above the line at infinity of the affine plane and are all fixed by G a (k) × G a (k). Thus, again, the group G a (k) × G a (k) acts trivially on a hyperbolic subspace of H ∞ (P 2 k ) that contains the axis of every loxodromic element of GL 2 (A). Theorem 7.3 ([40,105,[START_REF] Nicholas | Some effectivity questions for plane cremona transformations[END_REF]). Let k be an algebraically closed field, and g be a loxodromic element of Cr 2 (k).

(1) The axis of g is rigid.

(2) The cyclic subgroup g Z has finite index in the stabilizer of Ax(g) in Cr 2 (k), if and only if there exists a non-trivial iterate g n of g which is tight.

(3) If the index of g Z in Stab(Ax(g)) is infinite, one of the following possibilities occurs

• g is conjugate to a monomial transformation ;

• g is conjugate to a polynomial automorphism of the affine plane A 2 k that normalizes the group of translations (x, y) → (x + u, y + v), (u, v) ∈ k 2 (this case does not occur if char(k) = 0). Remark 7.4. Let N be a subgroup of Cr 2 (k) that contains at least one loxodromic element. Assume that there exists a short exact sequence 1 → S → N → Q → 1 where S is infinite and contains only elliptic elements. Then N is conjugate to a subgroup of the group GL 2 (Z) G m (k) 2 or GL 2 (A) G a (k) 2 , as in the previous examples. This statement is equivalent to Property (3). (see the appendix of [START_REF] Delzant | Kähler groups, real hyperbolic spaces, and the Cremona group[END_REF] and [START_REF] Nicholas | Some effectivity questions for plane cremona transformations[END_REF]) Remark 7.5. Tightness is equivalent to another property which appeared in the study of the mapping class group Mod(g) of a closed surface (see [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF], and references therein). Consider a group G acting by isometries on H ∞ and a loxodromic element g in G. One says that g is "wpd" (for "weakly properly discontinuous") if ∀D ≥ 0, ∀x ∈ H ∞ , there exists a positive integer N such that the set

S(D, x; N) = {h ∈ G | dist(h(x), x) ≤ D, dist(h(g N (x)), g N (x)) ≤ D}
is finite. To test this property, one can fix the starting point x; for instance, one can fix x on the axis of g.

When one studies the action of the Cremona group Cr 2 (k) on H ∞ (P 2 k ), the axis of every loxodromic element is rigid and the stabilizer of the axis Ax(g) is virtually cyclic if and only if some positive iterate of g is tight (see Theorem 7.3). It follows that for N large, the set S(D, x; N) is contained in the stabilizer of the axis Ax(g), and g is wpd if and only if some positive iterate g m of g is tight. Thus, tightness (for g m , for some m = 0) can be replaced by the wpd property when one studies the Cremona group in 2 variables.

Normal subgroups.

Let us pursue the comparison between groups of birational transformations and groups of diffeomorphisms. If M is a connected compact manifold and Diff ∞ 0 (M) denotes the group of infinitely differentiable diffeomorphisms of M which are isotopic to the identity, then Diff ∞ 0 (M) is a simple group: It does not contain any normal subgroup except {Id M } and the group Diff ∞ 0 (M) itself (see [START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF]). One can show that Cr 2 (C) is "connected" (see [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF]); hence, there is no need to rule out connected components as for diffeomorphisms. Enriques conjectured in 1894 that Cr 2 (C) is a simple group (see [START_REF] Enriques | Conferenze di geometria. Fondamenti di una geometria iperspaziale[END_REF]), and this is indeed true from the point of view of its algebraic subgroups (see § 2.5 and [START_REF] Blanc | Groupes de Cremona, connexité et simplicité[END_REF]). On the other hand, as an abstract group, Cr 2 (k) is far from being simple: Theorem 7.6 (Cantat and Lamy, [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF], Shepherd-Barron [START_REF] Nicholas | Some effectivity questions for plane cremona transformations[END_REF], Lonjou [105]). For every field k, the Cremona group Cr 2 (k) is not simple.

The proof relies on ideas coming from small cancellation theory and the geometry of hyperbolic groups in the sense of Gromov, as in [START_REF] Delzant | Sous-groupes distingués et quotients des groupes hyperboliques[END_REF]; the idea is that, starting with a tight element g in Cr 2 (k), the relations generated by the conjugates of a large iterate g n of g satisfy a small cancellation property, so that the normal subgroup generated by g n is a proper subgroup of Cr 2 (k). We refer to [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF] for a recent survey on this topic. Applied to the action of the Cremona group on the hyperbolic space H ∞ (P 2 k ), the precise result that one obtains is the following. Theorem 7.7 ( [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF][START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF]). Let g be a loxodromic element of Cr 2 (k). If g is a tight element of Cr 2 (k) and n is large enough, the normal subgroup of Cr 2 (k) generated by the n-th iterate g n • does not contain any element h = id whose translation length is less than L(g)

• is a free group.

As a consequence, to prove that Cr 2 (k) is not simple one needs to construct a tight element in Cr 2 (k) (recall that the definition of tightness depends on the group, so that being tight in Cr 2 (k) is not the same as being tight in Cr 2 (k ) if k is an extension of k). According to Theorem 7.3, one just needs to produce an element of Cr 2 (k) which is not conjugate to a monomial transformation or to an automorphism of the affine plane that normalizes the group of translations. This has been done for algebraically closed fields in [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF], for finite fields in [START_REF] Nicholas | Some effectivity questions for plane cremona transformations[END_REF], and for arbitrary fields in The examples of tight elements given by Lonjou are Hénon mappings h(x, y) = (y, y nx), with a degree n which is not divisible by the characteristic of k. By Déserti's theorem, the group of automorphisms of Cr 2 (C) is generated by inner automorphisms and the action of Aut(C, +, •). Since h is defined over Z, the normal subgroup generated by h m is a characteristic subgroup of Cr 2 (C).

Corollary 7.8. The Cremona group Cr 2 (C) contains infinitely many characteristic subgroups.

The same strategy is used in various contexts, as in the recent proof, by Dahmani, Guirardel and Osin, that high powers of pseudo-Anosov elements generate strict, nontrivial, normal subgroups in mapping class groups. Applied to the Cremona group, their techniques lead to the following statement. Theorem 7.9 (Dahmani, Guirardel, and Osin, [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF][START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF]105]). Let k be a field. The Cremona group Cr 2 (k) is sub-quotient universal: Every countable group can be embedded in a quotient group of Cr 2 (k).

Remark 7.10. Being sub-quotient universal, while surprising at first sight, is a common feature of hyperbolic groups [START_REF] Delzant | Sous-groupes distingués et quotients des groupes hyperboliques[END_REF][START_REF] Yu | Ol shanskiȋ. SQ-universality of hyperbolic groups[END_REF]. For instance, SL 2 (Z) is sub-quotient universal [START_REF] Lyndon | Combinatorial group theory[END_REF]. We refer to [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF] for a unified viewpoint on small cancellation theory that includes the study of mapping class groups Mod(g) and the Cremona groups Cr 2 (k).

-III-Higher dimensions, subgroups, and growths Our understanding of groups of birational transformations in dimension ≥ 3 is far less satisfactory than in dimension 2. In this last part, we focus on two open problems: The first one has been solved in many cases, with a wealth of different methods, and we hope that these methods may be useful for other questions; the second one, while much simpler to describe, requires new ideas.

Zimmer program

8.1. Groups of diffeomorphisms. Consider a compact, connected manifold M (of class C ∞ ). Denote by Diff ∞ 0 (M) the group of smooth diffeomorphisms of M which are isotopic to the identity. This group determines M. Indeed, Filipkiewicz proved that every "abstract" isomorphism between Diff ∞ 0 (M) and Diff ∞ 0 (M ) is given by conjugacy with respect to a diffeomorphism ϕ : M → M ; moreover, Hurtado proved that the existence of an embedding Diff ∞ (M) → Diff ∞ (M ) forces the inequality dim(M) ≤ dim(M ) (see [START_REF] Filipkiewicz | Isomorphisms between diffeomorphism groups[END_REF][START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF][START_REF] Hurtado | Continuity of discrete homomorphisms of diffeomorphism groups[END_REF], and the references of these articles).

By the work of Mather, Herman, Thurston and Epstein, the group Diff ∞ 0 (M) is simple (see [START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF]). One way to understand it better is to compare it to classical, (almost) simple, real linear groups, such as SL n (R) or SO p,q (R). Starting with a classical result concerning Lie groups, one may ask to what extent such a result holds in the context of groups of diffeomorphisms.

Recall that the real rank of such a linear group is the dimension of a maximal torus, i.e. the maximal dimension of a closed subgroup which is diagonalizable over R. The real rank rk R (G) is a good measure of the "complexity" of the group G; for instance, rk R (G) ≤ rk R (H) if G embeds in H. This is reflected by actions by diffeomorphisms: If the simple Lie group G acts smoothly and non-trivially on M, then dim(M) ≥ rk R (G) (with equality when M is the projective space of dimension n -1 and G is PSL n (R)).

Lie theory concerns the case of smooth actions of connected Lie groups; Zimmer's program proposes to pursue the comparison between Lie groups and groups of diffeomorphisms by looking at finitely generated subgroups. The following is an emblematic conjecture of this program. This conjecture has been proved in the case when M is the circle [START_REF] Ghys | Actions de réseaux sur le cercle[END_REF][START_REF] Burger | Bounded cohomology of lattices in higher rank Lie groups[END_REF], or when the lattice is not cocompact and the action is by area preserving diffeomorphisms of a compact surface [START_REF] Polterovich | Growth of maps, distortion in groups and symplectic geometry[END_REF]. 8.2. Groups of algebraic transformations. Groups of automorphisms or birational transformations can be compared to groups of diffeomorphisms, like Diff ∞ 0 (M) or linear algebraic groups, like SL n (k). Such comparisons are useful when looking at affine (resp. projective) varieties with a large group of automorphisms (resp. birational transformations); the prototypical example is given by the affine space A n k .

Conjecture 8.2 (Zimmer conjecture for birational transformations). Let G be a simple Lie group and Γ be a lattice in G. If Γ acts faithfully on a projective variety X by birational transformations, then rk R (G) ≤ dim(X).

8.2.1. Regular automorphisms. The same conjecture for actions by regular automorphisms is settled in [START_REF] Cantat | Version kählérienne d'une conjecture de Robert J. Zimmer[END_REF][START_REF] Cantat | Holomorphic actions, Kummer examples, and Zimmer program[END_REF] when one looks at automorphisms of complex projective or compact Kähler manifolds: If Γ is a lattice in a simple Lie group G and Γ acts faithfully by automorphisms of a compact Kähler manifold X, then rk R (G) ≤ dim(X) and in case of equality X is the projective space P n C . The proof works as follows. The group Aut(X) is a complex Lie group; it may have infinitely many connected components, but the connected component of the identity Aut(X) 0 is a Lie group whose Lie algebra is the algebra of holomorphic vector fields on X. The group Aut(X) acts on the cohomology of X, and the kernel K of this action contains Aut(X) 0 as a finite index subgroup (Lieberman's theorem, see [START_REF] David | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds[END_REF]). Let Γ be a lattice in an almost simple Lie group G, and assume that Γ embeds into Aut(X). Assume, moreover, that the rank of G is larger than 1, since otherwise the inequality dim(X) ≥ rk R (G) is obvious. Margulis normal subgoup theorem shows that Γ is almost simple: Every normal subgroup of Γ is finite and central, or co-finite. As a consequence, one can assume (replacing Γ by a finite index subgroup), that (1) Γ embeds into Aut(X) 0 or that (2) the action of Γ on the cohomology of X is faithful. The super-rigidity theorem of Margulis shows that homomorphisms from Γ into Lie groups H are built from homomorphisms of G into H (see [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF][START_REF] Cantat | Version kählérienne d'une conjecture de Robert J. Zimmer[END_REF] for precise statements). In case [START_REF] Banyaga | The structure of classical diffeomorphism groups, volume 400 of Mathematics and its Applications[END_REF], this implies that Aut(X) 0 contains a complex Lie group of rank rk R (G); one can then use Lie theory to conclude that dim(X) ≥ rk R (G). In case (2), one concludes that the action of Γ on the cohomology of H * (X; Z) comes from a linear representation of G into GL(H * (X; R)). But this linear representation preserves the Hodge decomposition, the cup product, the Poincaré duality, etc, because it comes from the original action of Γ by automorphisms. One can then put together Hodge theory (in particular the Hodge index theorem) and the theory of linear representations of (almost) simple Lie groups to get the estimate dim(X) ≥ rk R (G) + 1. (details are given in [START_REF] Cantat | Version kählérienne d'une conjecture de Robert J. Zimmer[END_REF][START_REF] Cantat | Holomorphic actions, Kummer examples, and Zimmer program[END_REF], and similar arguments are used in [START_REF] Dinh | Groupes commutatifs d'automorphismes d'une variété kählérienne compacte[END_REF][START_REF] Zhang | Algebraic varieties with automorphism groups of maximal rank[END_REF]).

It would be nice to adapt such a proof for groups of birational transformations. One way to do it is to consider the limit Z 1 (M) of Néron-Severi groups N 1 (M ) along all birational morphisms M → M (and more generally the limits Z q (M) of all N q (M ), where N q (M ) denotes the space of codimension q cycles modulo numerical equivalence). The intersection determines a multilinear pairing (see [START_REF] Boucksom | Valuations and plurisubharmonic singularities[END_REF], Chapter 4): It provides a geometric structure on these spaces, with nice properties coming from Hodge index theorem and Khovanskii-Teyssier inequalities. In dimension 2, this leads to the construction of the rank one space H ∞ (X); in higher dimension, one expects phenomena of rank dim(M) -1. These properties should provide rich constraints on the action of the group Bir(M) on Z 1 (M), and prevent large rank lattices from acting properly on such spaces (some kind of Mostow-Margulis rigidity for actions on those infinite dimensional spaces).

8.2.2. Birational transformations. Zimmer type problems are harder to study for groups of birational transformations or groups of automorphisms of non-complete varieties, such as the affine space. Nevertheless, a new technique emerged recently in the study of nonlinear analogues of the Skolem-Mahler-Lech theorem. This classical statement says that the indices n for which a linear recursive sequence u n+k = a 1 u n+k-1 + • • • a k u n vanishes form a finite union of arithmetic sequences in Z + . In other words, when one iterates a linear transformation B of A k C , the set of times n such that the orbit B n (x 0 ) of a point x 0 is contained in the hyperplane x k = 0 is a finite union of arithmetic progressions. As shown by Bell and his co-authors, this statement remains true if one replaces B by a polynomial automorphism f of A k C and the hyperplane by any algebraic subvariety of A k C (see [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF][START_REF] Bell | The dynamical Mordell-Lang problem for étale maps[END_REF] for more general statements). These results are based on the following p-adic phenomenon, which we state only in its simpler version.

Theorem 8.3 (Bell, Poonen, see [START_REF] Bell | A generalised Skolem-Mahler-Lech theorem for affine varieties[END_REF][START_REF] Poonen | p-adic interpolation of iterates[END_REF]). Let p be a prime number, with p ≥ 3. Let f be a polynomial automorphism of the affine space A n Q p which is defined by polynomial formulas with coefficients in Z p . Assume that f coincides with the identity map when one reduces all coefficients modulo p. Then, there exists a p-adic analytic action Φ : Z p × (Z p ) n → (Z p ) n of the abelian group (Z p , +) on the polydisk (Z p ) n ⊂ A n (Q p ) such that Φ(m, x) = f m (x) for every m ∈ Z and every x in (Z p ) n .

Here, by p-adic analytic, we mean that Φ(t, x) is given by convergent power series in the variables t and x = (x 1 , . . . , x n ) whose coefficients are in Z p . Thus, if f is the identity map after reduction of its formulas modulo p, then the action of f on the polydisk (Z p ) n is given by the flow, at time t = 1, of an analytic vector field. Theorem 8.3 is a tool to replace a discrete group action (like Z, generated by f ) by the action of a continuous group (like Z p , defined by Φ), at least locally in the p-adic topology. This result turns out to be useful when, instead of a cyclic group Z, one studies a subgroup Γ ⊂ Aut(A n Q p ) whose pro-p completion is small (for instance a p-adic Lie group).

When Γ is a subgroup of Aut(A n C ) which is generated by a finite number of elements f i ∈ Γ, one can replace the field C by the field generated by the coefficients of the formulas defining the f i ; such a finitely generated fields embeds (in many ways) in p-adic fields Q p . Thus, p-adic methods can be used to study groups of automorphisms and birational transformations of complex algebraic varieties.

This argument turns out to be quite powerful, and leads to the following statement (see [START_REF] Cantat | Algebraic actions of discrete groups: the p-adic method[END_REF]). Theorem 8.4. Let X be an irreducible complex projective variety. Let S < GL n be an almost simple linear algebraic group over the field of rational numbers Q. Assume that S(Z) is not co-compact. If a finite index subgroup of S(Z) embeds into Bir(X), then dim(X) ≥ rk R (S). If dim(X) = rk R (S) ≥ 2, then S(R) is isogeneous to SL dim(X)+1 (R).

In other words, Zimmer conjecture holds for birational actions of lattices which are not co-compact. For instance, one can take S = SL n in this theorem. Unfortunately, cocompact lattices are not handled by this theorem, and Conjecture 8.2 is still open for co-compact lattices. 8.3. Residual finiteness. In the same spirit -comparing groups of rational transformations to groups of linear transformations -the most basic question that has not been answered yet is the following one, which parodies Malcev's and Selberg's theorems. Question 8.5. Are finitely generated subgroups of Cr n (k) residually finite ? Does every finitely generated subgroup of Cr n (k) contain a torsion free subgroup of finite index ?

Bass and Lubotzky obtained a positive answer to this question when Cr n (k) is replaced by the group of regular automorphisms of an algebraic variety, for instance by Aut(A n k ) (see [START_REF] Bass | Automorphisms of groups and of schemes of finite type[END_REF]). The article [START_REF] Cantat | Algebraic actions of discrete groups: the p-adic method[END_REF] provides a positive answer for groups with Kazhdan property (T). 9. Growths 9.1. Degree growth. Consider a birational transformation f of a smooth projective variety X, over a field k. Fix a polarization H of X, and defines the degree of f with respect to H by deg 1

H ( f ) = ( f * H) • (H n-1
), where f * (H) is the total transform of H, U • V is the intersection form, and n is the dimension of X. This degree controles the complexity of the operator f * acting on algebraic hypersurfaces of X. Similarly, for every co-dimension 1 ≤ k ≤ n, one defines a degree in co-dimension k by

deg k H ( f ) = ( f * H k ) • (H n-k
). The degrees behave submultiplicatively: There is a constant A(X, H) > 0, which depends only on X and its polarization, such that

deg k H ( f • g) ≤ A(X; H)deg k H ( f )deg k H (g)
for every pair of birational transformations f , g in Bir(X); moreover, up to a uniform multiplicative constant, deg k H (•) does not depend on H:

deg k H ( f ) ≤ A (X, H, H )deg k H ( f )
for all f in Bir(X). This has been proved by Dinh and Sibony for fields of characteristic 0, and then by Truong in positive characteristic (see [START_REF] Dinh | Une borne supérieure pour l'entropie topologique d'une application rationnelle[END_REF], [START_REF] Trung | Relative dynamical degrees of correspondences over a field of arbitrary characteristic[END_REF], and also [START_REF] Nguyen-Bac | Degrees of iterates of rational maps on normal projective varieties[END_REF]). Thus, given a birational transformation f of a projective variety X, one gets dim(X) sequences m → deg k H ( f m ) which, up to multiplicative constants, do not depend on H and are invariant under conjugacy. One can show that there are only countably many possible sequences of the form (deg k H ( f m )) m≥0 (see [START_REF] Bonifant | Growth of degree for iterates of rational maps in several variables[END_REF][START_REF] Urech | Remarks on the degree growth of birational transformations[END_REF]). Moreover, the sequences (deg k H ( f m )) are linked together: For instance, the dynamical degrees

λ k ( f ) = lim sup m→+∞ deg k H ( f m ) 1/m
determine a concave sequence k → log(λ k ( f )) and, in particular, one of the λ k ( f ), with k > 0, is larger than 1 if and only if all of them are (see the survey [START_REF] Guedj | Propriétés ergodiques des applications rationnelles[END_REF]). When X is a surface, there are only 4 possibilities for the sequence deg 1 H ( f m ): It is bounded, or it grows linearly or quadratically, or it grows exponentially fast. Moreover, the first three cases have a geometric meaning (see Theorem 4.6). Nothing like that is known in dimension ≥ 3. Does there exist a polynomial automorphism g of the affine space 9.2. Divisibility and distorsion. Questions related to degree growths are connected to algebraic properties of (subgroups of) Bir(X). An element f in a group G is distorted if there is a subgroup Γ of G such that (1) Γ is generated by a finite subset S, (2) f is an element of Γ, and (3) f m can be written as a word of length (m) in the elements of S with lim( (m)/m) = 0. If a birational transformation f : X X is distorted in Bir(X), then λ k ( f ) = 1 for all 0 ≤ k ≤ dim(X). It would be great to classify, or at least to get geometric constraints on distorted elements in Aut(A n C ) and Bir(P n C ). One says that an element f of the group G is divisible, if for every m > 0 there is an element g m in G such that (g m ) m = f . Can we classify divisible elements of Aut(A 3 C ) ? When Γ is the cyclic group generated by S = { f , f -1 }, D Γ,S (m) is the maximum of deg( f l ) for l in between -m and m. Our former questions on the sequence (deg 1 H ( f m )) can now be stated for the sequence (D Γ,S (m)). Again, there are only countably many such sequences, and one would like to know their possible growth types. This is related to the growth type of Γ, viewed as an abstract, finitely geneated group, i.e. to the growth of the function counting the number of elements of the ball of radius m in the Cayley graph of Γ (with respect to S). If a group contains a non abelian free group, then Vol S (m) grows exponentially fast; if the growth is bounded by m d for some d > 0, then Γ contains a finite index nilpotent subgroup [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF]; if Γ is solvable, the growth is either polyomial or exponential. In particular, if G satisfies the Tits alternative, the growth of every finitely generated subgroup of G is either polynomial or exponential. But there are many groups with intermediate growth, in between polynomial and exponential (see [START_REF] Rotislav | Degrees of growth of finitely generated groups and the theory of invariant means[END_REF][START_REF] De | Topics in geometric group theory[END_REF][START_REF] Bartholdi | Groups of given intermediate word growth[END_REF] for instance). Question 9.2. Does Bir(P n C ) contain finitely generated subgroups with intermediate growth ?

and the inverse map g → g - 1 are

 1 homeomorphisms of Cr n (k) with respect to the Zariski topology. Define Cr n (k; d) ⊂ Cr n (k) to be the subset of all birational transformations of degree d: An element of Cr n (k; d) is defined by homogeneous formulas of degree d in the variables [x 0 : . . . : x n ] without common factor of positive degree. Let Cr n (k; ≤ d) be the union of these sets for all degrees d ≤ d. Consider the projective space Pol n (k; d) of dimension r(n, d) = (n + 1) n + d d -1

  2 (C) up to conjugacy. One in PGL 2 (C), one in PGL 3 (C), and one in the group of automorphisms of the del Pezzo surface which is obtained by blowing up P 2 C at the points [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1], and [1 : 1 : 1].

3. 1 .

 1 Dimension 2. Recall from Example 2.3 that the Jonquières group Jonq 2 (k) is the group of birational transformations of P 1 k × P 1 k that permute the fibers of the first projection; we may identify it to the group of birational transformations of P 2 k preserving the pencil of lines through the point [1 : 0 : 0].

  Y and Y are general hypersurfaces of degree ≥ n + 2, then Y and Y have general type, and the relation Y Y implies that Y and Y are isomorphic.

4. 1 .

 1 The Picard-Manin space.

  Isometries.-Denote by O 1,m (R) the group of linear transformations of H preserving the scalar product •|• m . The group of isometries Isom(H m ) coincides with the index 2 subgroup O + 1,m (R) of O(H ) that preserves the chosen sheet H m of the hyperboloid {u ∈ H | u|u m = 1}. This group acts transitively on H m , and on its unit tangent bundle.

)

  An isometry h is elliptic if and only if it fixes a point u in H m . Since •|• m is negative definite on the orthogonal complement u ⊥ , the linear transformation h fixes pointwise the line Ru and acts by rotation on u ⊥ with respect to •|• m .

Lemma 4 . 5 .

 45 The sequence deg h ( f n ) 1/n converges towards a real number λ( f ) ≥ 1, called the dynamical degree of f ; its logarithm log(λ( f )) is the translation length L( f * ) of the isometry f * .

  of degree ≥ 1 provide parabolic transformations of P 2 k with linear degree growth. Example 4.9. Assume k = C. Let ι be a square root of -1 (resp. a non-trivial cubic root of 1) and E be the elliptic curve C/Z[ι]. The linear action of the group GL 2 (Z[ι]) on the complex plane C 2 preserves the lattice Z[ι] × Z[ι]; this leads to an action of GL 2 (Z[ι]) by regular automorphisms on the abelian surface X = E × E. This action commutes to m(x, y) = (ιx, ιy); this provides a homomorphism from PGL 2 (Z[ι]) to the group of automorphisms of X/m. Since X/m is a rational surface, one gets an embedding of PGL 2 (Z[ι]) into the Cremona group Cr 2 (C).

4. 3 . 2 .

 32 Pseudo-Anosov, versus loxodromic. To a pseudo-Anosov mapping class, one associates a dilatation factor λ(ϕ): Given any pair of non-trivial homotopy classes of simple closed curves c and c on Σ, the intersection numbers ϕ n (c) • c grow like c st λ(ϕ) n as n goes to +∞ (here c • c is the minimum number of intersection points of curves C and C in the homotopy classes c and c ). A similar property is satisfied by every loxodromic element f of Bir(X): If e and e are points on H ∞ (X) which are determined in NS(X) by e = C/ √ C •C, e = C / √ C •C , for two curves C and C with positive self-intersection, then f n * (e) • e grows like c st λ( f ) n . Also, every pseudo-Anosov class ϕ is represented by a pseudo-Anosov homeomorphism Φ : Σ → Σ; such a homeomorphism preserves two singular foliations on Σ, one being uniformly contracted, the other uniformly dilated. Those foliations are geometric objects which, in Thurston compactification, correspond to fixed points of ϕ on the boundary of the Teichmüller space.

Example 7 . 1 .

 71 Consider the group of monomial transformations in two variables; this group is isomorphic to GL 2 (Z). To a matrix M = a b c d with det(M) = ±1 corresponds a monomial transformation f M : In affine coordinates, f M (x, y) = (x a y b , x c y d ).

  [105].(8) For instance, very general elements of Cr 2 (C; 2) are tight; this implies that Cr 2 (C) contains uncountably many distinct normal subgroups.

Conjecture 8 . 1 (

 81 Zimmer conjecture). Let G be a simple Lie group and Γ be a lattice in G. If Γ acts faithfully on a compact connected manifold M by diffeomorphisms, then rk R (G) ≤ dim(M).

Question 9 . 1 .

 91 What type of sequences do we get under this process ? In particular, what can be said on the growth type of m → deg 1 H ( f m ) ?

A 3 C

 3 for which deg 1 H (g m ) grows like exp( √ m) ? Do the results of Lo Bianco in [104] hold for birational transformations of P 3 C ?

9. 3 .

 3 Groups. Given a group Γ in Bir(X), one gets a degree function on Γ, namelyf ∈ Γ → deg( f ),where, for simplicity, deg( f ) stands for deg 1 H ( f ). Assume that Γ is generated by a finite symmetric set S, and denote by D Γ,S (m) the maximum of deg( f ) for f in the ball of radius m in the Cayley graph of Γ:D Γ,S (m) = max{deg( f ) | ∃l ≤ m, ∃s 1 , . . . , s l ∈ S, f = s 1 • s 2 • • • • • s l }.

  Vol S : m → Vol S (m) = Card{ f ∈ Γ | ∃l ≤ m, ∃s 1 , . . . , s l ∈ S, f = s 1 • s 2 • • • • • s l }

  . , x n ) of degree d, modulo multiplication by a non-zero common scalar factor; denote by Form

n (k; d) the subset of Pol n (k; d) made of formulas for birational maps, i.e. n-tuples of polynomial functions (h i ) 0≤i≤n such that [x 0 : . . . :

x n ] → [h 0 : . . . : h n ]

is a birational transformation of the projective space (of degree ≤ d). The set Form n (k; d) is locally closed in Pol n (k; d) for the Zariski topology, and there is a natural projection

π n : Form n (k; d) → Cr n (k; ≤ d). One can then show that • If f : B → Cr n (k) is a morphism, its

image is contained in Cr n (k; ≤ d) for some degree d and it can be locally lifted, on affine open subsets B i ⊂ B, to morphisms B → Form n (k; d i ) for some d i ≥ d; • a subset S of Cr n (k) is closed if and only if its π n -preimage in Form n (k; d) is closed for all d ≥ 1; • for every d ≥ 1, Cr n (k; ≤ d) is closed in Cr n (k); • the projection Form n (k; d) → Cr n (k; ≤ d) is surjective, continuous, and closed for every d ≥ 1 (it is a topological quotient mapping);

Such a lift would be given byf a,b,c (x 0 , x 1 , x 2 ) = (x 0 (a 2 x 2 + b 2 x 0 ), x 1 (a 2 x 2 + b(a + b)x 0 ), x 2 (a 2 x 2 + b 2 x 0 ))modulo multiplication by a function of (a, b, c), but this expression does not correspond to a birational map when (a, b, c) is the singular point (0, 0, 0) of C. Details are given in[START_REF] Blanc | Topologies and structures of the Cremona groups[END_REF].

or the "de Jonquières" group

This sub-section follows from a discussion with Jérémy Blanc and Christian Urech, during which Blanc explained the proof of Proposition 3.6.

The Riemannian structure is defined as follows. If u is an element of H m , the tangent space T u H m is the affine space through u that is parallel to u ⊥ , where u ⊥ is the orthogonal complement of Ru with respect to •|• m ; since u|u m = 1, the form •|• m is negative definite on u ⊥ , and its opposite defines a positive scalar product on T u H m ; this family of scalar products determines a Riemannian metric, and the associated distance coincides with dist m (see[START_REF] Benedetti | Lectures on hyperbolic geometry[END_REF]).

or "de Jonquières" twists

The proof is simple: The action of Aut(M) on the cohomology of M is a linear representation, and Tits theorem can be applied to its image; its kernel is a Lie group with finitely many components, and Tits theorem can again be applied to it. There is a mistake in the proof of Lemma 6.1 of[START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]; this has been corrected in[START_REF] Dinh | Tits alternative for automorphism groups of compact Kähler manifolds[END_REF] and[START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces -version longue. long version with complements[END_REF].

Note that Theorem 7.3 has been proved several years after[START_REF] Cantat | Normal subgroups in the Cremona group[END_REF], so that the existence of tight elements in Cr 2 (k) could not rely on it.