J Ér

Julie D Éserti
email: julie.deserti@unibas.ch

EMBEDDINGS OF SL(2, Z) INTO THE CREMONA GROUP

Keywords: Subject Classification. -14E07 (primary), 14L30, 15B36 (secondary)

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings that preserve the type (i.e. that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also shown.

In particular, a group G of automorphisms of a smooth surface S obtained by blowingup 10 points of the complex projective plane is given. The group G is isomorphic to SL(2,Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.

INTRODUCTION

Our article is motivated by the following result on the embeddings of the groups SL(n, Z) into the group Bir(P 2 ) of birational maps of P 2 (C): the group SL(n, Z) does not embed into Bir(P 2 ) for n ≥ 4 and SL(3, Z) only embeds linearly (i.e. in Aut(P 2 ) = PGL(3, C)) into Bir(P 2 ) up to conjugacy [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF]Theorem 1.4].

It is thus natural to look at the embeddings of SL(2, Z) into Bir(P 2 ). As SL(2, Z) has almost a structure of free group, it admits many embeddings of different type into Bir(P 2 ), and it is not reasonable to look for a classification of all embeddings. We thus focus on embeddings having some geometric properties; among them the most natural ones are the embeddings which preserve the type evoked by Favre in [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF]Question 4].

The elements of SL(2, Z) are classified into elliptic, parabolic and hyperbolic elements, with respect to their action on the hyperbolic upper-plane (or similarly to their trace, see §2.1). The Cremona group Bir(P 2 ) naturally acts on a hyperbolic space of infinite dimension (see [START_REF] Yu | Cubic forms, volume 4 of North-Holland Mathematical Library[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]), so there is a notion of elliptic, parabolic and hyperbolic elements in this group; this classification can also be deduced from the growth rate of degrees of iterates (see [DiFa] and §2.3). Note that some authors prefer the term of loxodromic elements instead of hyperbolic elements (see for example [And,Proposition 2.16]). A morphism from SL(2, Z) to Bir(P 2 ) preserves the type if it sends elliptic, parabolic and hyperbolic elements of SL(2, Z) on elements of Bir(P 2 ) of the same type. Up to now, the only known example is the classical embedding θ s : SL(2, Z) → Bir(P 2 ), which associates to a matrix M = a b c d the birational map θ s (M), given in affine coordinates by (x, y) (x a y b , x c y d ) (or written simply (x a y b , x c y d )). In this article, we provide infinitely many non-conjugate embeddings that preserve the type (Theorem 1 below):

Recall that the group SL(2, Z) is generated by the elements R and S given by R = 1 1 0 1 and S = 0 1 -1 0 .

Theorem 1. (see §3.1) Let ε be a real positive number, and set θ ε (S) = (y, -x) , θ ε (R) =

x + εy ε + xy , εy .

Then θ ε is an embedding of SL(2, Z) into the Cremona group that preserves the type. Furthermore, if ε and ε ′ are two real positive numbers such that εε ′ = 1, then θ ε (SL(2, Z)) and θ ε ′ (SL(2, Z)) are not conjugate in Bir(P 2 ).

The standard embedding θ s is conjugate to θ 1 .

This family of embeddings is a first step in the classification of all embeddings of SL(2, Z) preserving the type. We do not know if other embeddings exist (except one special embedding θ -described in §3.1 which is a "twist" of the standard embedding θ s defined by: θ -(S) = θ s (S) = (y, 1

x ) and θ -(R) = (xy, -y) = θ s (R) = (xy, y)), in particular if it is possible to find an embedding where the parabolic elements act by preserving elliptic fibrations.

Question 1.1. Does there exist an embedding of SL(2, Z) into Bir(P 2 ) that preserves the type and which is not conjugate to θ -or to some θ ε ?

The last two assertions of Theorem 1 yield to the following question: Question 1.2. Is the embedding θ -rigid? i.e. not extendable to a one parameter family of non conjugate embeddings ? Note that some morphisms SL(2, Z) → Bir(P 2 ) preserving the type have been described ( [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF]page 9], [CaLo] and [Gol]), but that these ones are not embedding, the central involution acting trivially. See §3.1 for more details.

One can also consider elliptic, parabolic and hyperbolic embeddings of SL(2, Z) into Bir (P 2 ). An embedding θ of SL(2, Z) into the Cremona group is said to be elliptic if each element of im θ is elliptic; θ is parabolic (respectively hyperbolic) if each element of infinite order of im θ is parabolic (respectively hyperbolic).

In Sections 3.2, 3.3 and 3.4, we prove the existence of an infinite number of nonconjugate elliptic, parabolic and hyperbolic embeddings (see Propositions 3.7,3.8,3.9 and Corollary 3.11). It is possible to find many other such embeddings; we only give a simple way to construct infinitely many of each family.

One can then ask if it is possible to find an embedding of SL(2, Z) into the Cremona group which is regularisable, i.e. which comes from an embedding into the group of automorphisms of a projective rational surface. It is easy to construct elliptic embeddings which are regularisable (see Section 3.2). In Section 4, we give a way to construct infinitely many hyperbolic embeddings of SL(2, Z) into the Cremona group which are regularisable, and each of the groups constructed moreover preserves an elliptic curve (one fixing it pointwise). The existence of regularisable embeddings which preserve the type is still open (and should contain parabolic elements with quadratic growth of degree).

Note that the existence of hyperbolic automorphisms preserving an elliptic curve was not clear. In [START_REF] Pan | Sur le degré dynamique des transformations de Cremona du plan qui stabilisent une courbe irrationnelle non-elliptique[END_REF]Theorem 1.1], it was proved that a curve preserved by an hyperbolic element of Bir(P 2 ) has geometric genus 0 or 1; examples of genus 0 (easy to create by blowing-up) were provided, and the existence of genus 1 curves invariant was raised (see [START_REF] Pan | Sur le degré dynamique des transformations de Cremona du plan qui stabilisent une courbe irrationnelle non-elliptique[END_REF]page 443]). The related question of the existence of curves of arithmetic genus 1 preserved by hyperbolic automorphisms of rational surfaces was also raised two years after in [DFS, page 2987]. In [McM], the author constructs hyperbolic automorphisms of rational surfaces which correspond to Coxeter elements (any hyperbolic automorphism of a rational surface corresponds to an element of the Weyl group associated to the surface), that preserve a cuspidal (resp. nodal) curve. However, a general automorphism of a rational surface corresponding to a Coxeter element is hyperbolic but does not preserve any curve ( [BeKi]).

The following statement yields existence of a group of automorphisms preserving a (smooth) elliptic curve such that every non-periodic element is hyperbolic. This is also possible with free groups (see [START_REF] Cantat | Invariant hypersurfaces in holomorphic dynamics[END_REF]Remark 3.2] and [START_REF] Blanc | On the inertia group of elliptic curves in the Cremona group of the plane[END_REF]), but the construction is harder with more complicated groups like SL(2, Z). The method that we describe in Section 4 should be useful to create other groups generated by elements of finite order.

Theorem 2. There exist hyperbolic embeddings θ h,1 , θ h,2 , θ h,3 of SL(2, Z) into Bir(P 2 ) such that:

• for each i, the group θ h,i preserves a smooth cubic curve Γ ⊂ P 2 ;

• the action of θ h,1 on Γ is trivial, the action of θ h,2 on Γ is generated by a translation of order 3 and the action of θ h,3 on Γ is generated by an automorphism of order 3 with fixed points; • for i = 1, 2, 3, the blow-up X i → P 2 of respectively 12, 10, 10 points of Γ conjugates θ h,i (SL(2, Z)) to a subgroup of automorphisms of X i . The strict transform Γ of Γ on X i is the only invariant curve; in particular the orbit of any element of X i \ Γ is either finite or dense in the Zariski topology.

Moreover, in cases i = 1, 2, we can choose Γ to be any smooth cubic curve, and this yields infinitely many hyperbolic embeddings of SL(2, Z) into Bir(P 2 ), up to conjugacy.

Remark 1.3. In θ h,1 , θ h,2 , θ h,3 , the letter h is no parameter but only means "hyperbolic", to distinguish them from the other embeddings θ s , θ -and {θ ε } ε∈R , defined above.

It could be interesting to study more precisely the orbits of the action of the above groups, in particular to answer the following questions: Question 1.4. Are the typical orbits of θ h,i dense in the transcendental topology ? Question 1.5. Are there some finite orbits in X i \ Γ?

We finish this introduction by mentioning related results.

The statement of [START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF]Theorem 1.4] for SL(3, Z) was generalised in [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF], where it is proven that any finitely generated group having Kazhdan's property (T) only embeds linearly into Bir(P 2 ) (up to conjugation).

Let us also mention [CaLa, Theorem A] which says that if a lattice Γ of a simple Lie group G embeds into the group Aut(C 2 ), then G is isomorphic to PSO(1, n) or PSU(1, n) for some n. If the embedding is not conjugate to a subgroup of the affine group, the only possibility is G ≃ PSO(1, 2) ≃ PSL(2, R), this latter case being intensively studied in [CaLa].

Note that our techniques heavily use the special structure of SL(2, Z), and one could ask similar questions for any lattice of GL(2, R) or PGL(2, R); the behaviour and results could be very different.

Acknowledgements. The authors would like to thank Charles Favre for interesting comments and suggestions, and Pierre de la Harpe for interesting discussions. Thanks also to the referees for their helpful remarks and corrections.

2. SOME REMINDERS ON SL(2, Z) AND Bir(P 2 ) 2.1. About SL(2, Z). Division algorithm implies that the group SL(2, Z) is generated by the elements R and S given by

R = 1 1 0 1 and S = 0 1 -1 0 .
Remark that R is of infinite order and S of order 4. The square of S generates the center of SL(2, Z).

Moreover RS = -1 1 -1 0 and SR = 0 1 -1 -1
are conjugate by S and both have order 3.

A presentation of SL(2, Z) is given by

R, S | S 4 = (RS) 3 = 1, S 2 (RS) = (RS)S 2
(see for example [START_REF] Newman | Integral matrices[END_REF]Chapter 8]). This implies that the quotient of SL(2, Z) by its center is a free product of Z/2Z and Z/3Z generated by the classes [S] of S and

[RS] of RS PSL(2, Z) = [S], [RS] | [S] 2 = [RS] 3 = 1 .
2.2. Dynamic of elements of SL(2, Z). Recall that the group SL(2, R) acts on the upper half plane

H = {x + iy ∈ C | x, y ∈ R, y > 0}
by Möbius transformations:

SL(2, R) × H → H, a b c d , z → az + b cz + d .
The hyperbolic structure of H being preserved, this yields to a natural notion of elliptic, parabolic, and hyperbolic elements of SL(2, R), and thus to elements of SL(2, Z) (as in [START_REF] Iversen | Hyperbolic geometry[END_REF]II.8]).

If M is an element of SL(2, Z), we can be more precise and check the following easy observations:

• M is elliptic if and only if M has finite order;

• M is parabolic (respectively hyperbolic) if and only if M has infinite order and its trace is ±2 (respectively = ±2). Up to conjugacy the elliptic elements of SL(2, Z) are

-1 0 0 -1 , 0 1 -1 -1 , 0 1 -1 0 , 0 -1 1 0 , 0 -1 1 1 ;
in particular an element of finite order is of order 2, 3, 4 or 6.

A parabolic element of SL(2, Z) is up to conjugacy one of the following

1 a 0 1 , -1 a 0 -1 , a ∈ Z.
2.3. Cremona group and dynamic of its elements. Let us recall the following classical definitions.

Definitions 2.1. A rational map of the projective plane into itself is a map of the following type

f : P 2 (C) P 2 (C), (x : y : z) ( f 0 (x, y, z) : f 1 (x, y, z) : f 2 (x, y, z)),
where the f i 's are homogeneous polynomials of the same degree without common factor.

The degree of f is by definition: deg f = deg f i . A birational map f is a rational map that admits a rational inverse. We denote by Bir(P 2 ) the group of birational maps of the projective plane into itself; Bir(P 2 ) is also called the Cremona group.

The degree is not a birational invariant; if f and g are in Bir(P 2 ), then in general deg(g f g -1 ) = deg f . Nevertheless there exist two strictly positive constants a, b ∈ R such that for all n the following holds

a deg f n ≤ deg(g f n g -1 ) ≤ b deg f n .
In other words the degree growth is a birational invariant; so we introduce the following notion ( [Fri, RuSh]).

Definition 2.2. Let f be a birational map. The first dynamical degree of f is defined by

λ( f ) = lim(deg f n ) 1/n .
There is a classification of birational maps of P 2 up to birational conjugation.

Theorem 2.3 ( [Giz, DiFa]). Let f be an element of Bir(P 2 ). Up to birational conjugation, exactly one of the following holds.

• The sequence (deg f n ) n∈N is bounded, f is an automorphism on some projective rational surface and an iterate of f is an automorphism isotopic to the identity; • the sequence (deg f n ) n∈N grows linearly, and f preserves a rational fibration. In this case f cannot be conjugate to an automorphism of a projective surface; • the sequence (deg f n ) n∈N grows quadratically, and f is conjugate to an automorphism preserving an elliptic fibration; • the sequence (deg f n ) n∈N grows exponentially.

In the second and third case, the invariant fibration is unique. In the first three cases λ( f ) is equal to 1, in the last case λ( f ) is strictly greater than 1.

Definitions 2.4. Let f be a birational map of P 2 .

If the sequence (deg f n ) n∈N is bounded, f is said to be elliptic. When (deg f n ) n∈N grows linearly or quadratically, we say that f is parabolic.

If λ( f ) > 1, then f is an hyperbolic map.
As we said the Cremona group acts naturally on a hyperbolic space of infinite dimension ( [START_REF] Yu | Cubic forms, volume 4 of North-Holland Mathematical Library[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]); we can say that a birational map is elliptic, resp. parabolic, resp. hyperbolic, if the corresponding isometry is elliptic, resp. parabolic, resp. hyperbolic ( [START_REF] Ghys | Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF]Chapter 8,§2]). This definition coincides with the previous one [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]).

Examples 2.5. Any automorphism of P 2 or of an Hirzebruch surface F n and any birational map of finite order is elliptic.

The map (x : y : z)

(xy : yz : z 2 ) is parabolic. A Hénon map (automorphism of C 2 ) (x, y) → (y, P(y) -δx), δ ∈ C * , P ∈ C[y], deg P ≥ 2
extends to a hyperbolic birational map of P 2 , of dynamical degree deg P.

Definitions 2.6. Let θ : SL(2, Z) → Bir(P 2 ) be an embedding of SL(2, Z) into the Cremona group. We say that θ preserves the type if θ sends elliptic (respectively parabolic, respectively hyperbolic) element onto elliptic (respectively parabolic, respectively hyperbolic) map.

We say that θ is elliptic if each element of im θ is elliptic. The morphism θ is parabolic (respectively hyperbolic) if each element of infinite order of im θ is parabolic (respectively hyperbolic).

2.4. The central involution of SL(2, Z) and its image into Bir(P 2 ). The element S 2 ∈ SL(2, Z) is an involution; therefore its image by any embedding θ : SL(2, Z) → Bir(P 2 ) is a birational involution. As it was proved by Bertini, we have the following classification: Theorem 2.7 ( [Ber]). An element of order 2 of the Cremona group is up to conjugacy one of the following

• an automorphism of P 2 ; • a de Jonquières involution ι dJ of degree ν ≥ 2;

• a Bertini involution ι B ; • a Geiser involution ι G .
Bayle and Beauville showed that the conjugacy classes of involutions in Bir(P 2 ) are determined by the birational type of the curves of fixed points of positive genus ( [BaBe]). More precisely the set of conjugacy classes is parametrised by a disconnected algebraic variety whose connected components are respectively

• the moduli spaces of hyperelliptic curves of genus g (de Jonquières involutions);

• the moduli space of canonical curves of genus 3 (Geiser involutions);

• the moduli space of canonical curves of genus 4 with vanishing theta characteristic, isomorphic to a non singular intersection of a cubic surface and a quadratic cone in P 3 (C) (Bertini involutions). The image of S 2 can neither be a Geiser involution, nor a Bertini involution; more precisely, we have the following: Lemma 2.8. Let θ be an embedding of SL(2, Z) into the Cremona group. Up to birational conjugation, one of the following holds.

• The involution θ(S 2 ) is an automorphism of P 2 ;

• the map θ(S 2 ) is a de Jonquières involution of degree 3 fixing (pointwise) an elliptic curve.

Remark 2.9.

→ G ′ → G → H → 1
where H is a subgroup of Aut(Γ), G ′ contains θ(S 2 ) and fixes Γ. Since the genus of Γ is positive H cannot be equal to G/ θ(S 2 ) , free product of Z/2Z and Z/3Z. This implies that the normal subgroup G ′ of G strictly contains θ(S 2 ) and thus that it is infinite and not abelian. In particular the group of birational maps fixing (pointwise) Γ is infinite, and not abelian, thus Γ is of genus 1 (see [START_REF] Blanc | Sur un théorème de Castelnuovo[END_REF], Theorem 1.5).

EMBEDDINGS PRESERVING THE TYPE AND ELLIPTIC, PARABOLIC EMBEDDINGS

3.1. Embeddings preserving the type. Henceforth we will often denote by ( f 1 (x, y, z) :

f 2 (x, y, z) : f 3 (x, y, z)) the map (x : y : z) ( f 1 (x, y, z) : f 2 (x, y, z) : f 3 (x, y, z))
and by (p(x, y), q(x, y)) the birational map

(x, y) (p(x, y), q(x, y)) of C 2 .
Let us begin this section by a property satisfied by all embeddings of SL(2, Z) → Bir(P 2 ) that preserve the type.

Lemma 3.1. Let θ : SL(2, Z) → Bir(P 2 ) be an embedding that preserves the type. Either for all parabolic matrices M, θ(M) preserves a unique rational fibration, or for all parabolic matrices M, θ(M) preserves a unique elliptic fibration.

Proof. Let us recall that a parabolic element of SL(2, Z) is up to conjugacy one of the following

T + a = 1 a 0 1 , T - a = -1 a 0 -1 , a ∈ Z.
For any a = 0, the image θ(T + a ) of T + a preserves a unique fibration on P 2 . Denote by F the fibration preserved by T + 1 , given by F : P 2 P 1 . For any a = 0, T + a and T - a commute with T + 1 so the θ(T + a )'s and the θ(T - a )'s preserve the fibration F and F is the only fibration invariant by these elements.

Let M be a parabolic matrix. On the one hand M is conjugate to T + a or T - a for some a via a matrix N M and on the other hand parabolic maps preserve a unique fibration; thus θ(M) preserves the fibration given by Fθ(N M ) -1 . In particular if F defines a rational (respectively elliptic) fibration, then Fθ(N M ) -1 defines a rational (respectively elliptic) one.

The standard embedding θ s . The classical embedding

θ s : SL(2, Z) → Bir(P 2 ), a b c d → (x a y b , x c y d )
preserves the type (see for example [START_REF] Lin | Algebraic stability and degree growth of monomial maps and polynomial maps[END_REF]Theorem 7.1]).

For any

M ∈ SL(2, Z), if M is elliptic, θ s (M)
is, up to conjugacy, one of the following birational maps of finite order

1 x , 1 y , y, 1 xy , y, 1 x , 1 y , x , 1 x , xy . If M is parabolic, θ s (M n ) is, up to conjugacy, (xy na , y), or (y na /x, 1/y) with a in Z so θ s (M) is parabolic. If M is hyperbolic, M has two real eigenvalues µ and µ -1 such that |µ| -1 < 1 < |µ| and λ(θ s (M)) = |µ| > 1 and θ s (M) is hyperbolic.
In [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF]page 9], a construction of a morphism SL(2, Z) → Bir(P 2 ) preserving the type was given, inspired from [CaLo] and [Gol]: the quotient of P 1 × P 1 by the involution (x, y) → 1

x , 1 y is a rational (singular) cubic surface C ⊂ P 3 , called Cayley cubic surface. Explicitly, we can assume (by a good choice of coordinates) that

C = {(W : X : Y : Z) ∈ P 3 | XY Z + WY Z + W XZ + W XY = 0}
and that the quotient is given by

P 1 × P 1 → C, (x, y) → (x -1)(x -y)(1 + y) : (y -1)(y -x)(1 + x) : (xy + 1)(x + 1)(y + 1) : (x -1)(y -1)(xy + 1) .
The involution (x, y) → 1

x , 1 y being the center of θ s (SL(2, Z)), the quotient provides a morphism θ ′ s : SL(2, Z) → Bir(C) ≃ Bir(P 2 ) whose kernel is generated by S 2 . The morphism preserves the type, but is not an embedding. It is also possible to deform the construction in order to have similar actions on other cubic surfaces (see [CaLo]).

One first twisting of θ s . We can "twist" the standard embedding θ s in the following way.

Let

θ -(S) = θ s (S) = (y, 1 x ) and θ -(R) = (xy, -y) = θ s (R) = (xy, y). The map θ -(RS) = θ -(R)θ -(S) = ( y x , -1 x ) has order 3. Since θ -(R) commutes with θ -(S 2
), the relations of SL(2, Z) are satisfied and θ -is a morphism from SL(2, Z) into Bir(P 2 ).

Proposition 3.2. The map θ -: SL(2, Z) → Bir(P 2 ) is an embedding that preserves the type.

The groups θ s (SL(2, Z)) and θ -(SL(2, Z)) are not conjugate in the Cremona group.

Proof.

For each M ∈ SL(2, Z), one has θ -(M) = α M • θ s (M)
where α M = (±x, ±y), and in particular θ -(M) and θ s (M) have the same degree. This observation implies that θ -is an embedding, and that it preserves the type, since θ s does.

We now prove the second assertion. Suppose, for contradiction, that

θ s (SL(2, Z)) is conjugate to θ -(SL(2, Z)); then θ s (R) = (xy, y) is conjugate to some parabolic element of θ -(SL(2, Z)), which has no root in the group. This implies that θ s (R) = (xy, y) or its inverse is conjugate to θ -(R) = (xy, -y) or θ -(RS 2 ) = 1
xy , -1 y in Bir(P 2 ). All these elements are parabolic elements of the Cremona group, each of them preserves a unique rational fibration, which is (x, y) → y. Since θ s (R) preserves any fibre and both θ

-(R), θ -(RS 2 ) permute the fibres, neither θ s (R) nor θ s (R -1 ) is conjugate to θ -(R) or θ -(RS 2 ) in Bir(P 2 ).
The map θ -yields a "new" embedding of SL(2, Z) preserving the type. However, this map is not very far from the first one, and remains in (C * , C * ) ⋊ SL(2, Z). We construct now new ones, more interesting. Conjugating the elements θ s (S) = (y, 1

x ) and θ s (R) = (xy, y) by the birational map x-1 x+1 , y-1 y+1 , we get respectively (y, -x) and x+y xy+1 , y . More generally, we choose any ε ∈ C * , and set

θ ε (S) = (y, -x) , θ ε (R) = x + εy ε + xy , εy .
The map θ ε (R) commutes with θ ε (S 2 ) = (-x, -y), and

θ ε (RS) = y -εx ε -xy , -εx
is of order 3, so θ ε gives an homomorphism from SL(2, Z) to Bir(P 2 ). The map θ 1 being conjugate to the standard embedding, we can view this family as a deformation of the standard embedding. We prove now some technical results to show that the family consists of embedding preserving the type when ε is a positive real number.

Lemma 3.3. We view these maps on P 1 × P 1 , via the embedding (x, y) → ((x : 1), (y : 1)).

(i) Writing R 1 = 1 1 0 1 , R 2 = 1 0 1 1 , both maps θ ε (R 1 ) = x + εy ε + xy , εy and θ ε (R 2 ) = x ε , ε(x + εy) ε + xy
have exactly two base-points both belonging to P 1 × P 1 (no infinitely near point), and being p 1 = (ε, -1) and p 2 = (-ε, 1) (or ((ε : 1), (-1 : 1)) and ((-ε : 1), (1 : 1))).

(ii) Both maps

θ ε (R 1 ) -1 = ε(εx -y) ε -xy , y ε and θ ε (R 2 ) -1 = εx, y -εx ε -xy have exactly two base-points, being q 1 = (1, ε) and q 2 = (-1, -ε). (iii) If ε is a positive real number and M = R i k . . . R i 1 , for i 1 , . . . , i k ∈ {1, 2}
, the following hold:

• the points q 1 and q 2 are not base-points of θ ε (M), and

θ ε (M)({q 1 , q 2 })∩{p 1 , p 2 } = / 0. • the points p 1 and p 2 are not base-points of θ ε (M -1 ), and θ ε (M -1 )({p 1 , p 2 }) ∩ {q 1 , q 2 } = / 0.
Proof. Parts (i) and (ii) follow from an easy calculation, it remains to prove (iii).

Let U + ⊂ R 2 ⊂ P 1 × P 1 (resp. U -⊂ R 2 ⊂ P 1 × P 1 ) be the subset of points (x, y) with x, y ∈ R, xy > 0 (resp. xy < 0). When ε is a positive real number, {p 1 , p 2 } ⊂ U - and {q 1 , q 2 } ⊂ U + , which implies that θ ε (R i ) (resp. θ ε (R -1 i )) is defined at any point of U + (resp. of U -), since U + ∩U -= / 0.
Moreover, the explicit form of the four maps given in (i), (ii)

shows that θ ε (R i )(U + ) ⊂ U + and θ ε (R -1 i )(U -) ⊂ U -for i = 1, 2.
This yields the result. Recall that Pic(P 1 × P 1 ) = Z f 1 ⊕ Z f 2 , where f i is the fibre of the projection on the ith factor. In particular, any curve on P 1 × P 1 has a bidegree (d 1 , d 2 ) and any element of Bir(P 1 × P 1 ) has a quadridegree, which is given by the two bidegrees of the pull-backs of f 1 and f 2 , or equivalently by the two bidegrees of the polynomials which define the map.

Remark that the dynamical degree of a birational map ϕ of P 1 × P 1 is uniquely determined by the sequence of quadridegrees of ϕ n . Proposition 3.4. If ε is a positive real number, the following hold:

(i) For any M = a b c d ∈ SL(2, Z)
, the maps θ ε (M) and θ s (M) have the same quadridegree as birational maps of P 1 × P 1 , which is (|a|, |b|, |c|, |d|).

(ii) The homomorphism θ ε is an embedding of SL(2, Z) into the Cremona group that preserves the type.

Proof. Observe first that (i) implies that the kernel of θ ε is trivial (since θ ε (S 2 ) = (-x, -y) is not trivial) so that θ ε is an embedding, and also implies that the dynamical degree of θ ε (M) and θ s (M) are the same for any M. This shows that (i) implies (ii).

We now prove assertion (i). Since θ s (S) = y, 1 x and θ ε (S) = (y, -x) are automorphisms of P 1 × P 1 having the same action on Pic(P 1 × P 1 ), θ ε (M) and θ s (M) have the same quadridegree if and only if θ ε (MS) and θ s (MS) have the same quadridegree. The same holds when we multiply on the left: θ ε (M) and θ s (M) have the same quadridegree if and only if θ ε (SM) and θ s (SM) have the same quadridegree.

Recall that SL(2, Z) has the presentation R, RS | S 4 = (RS) 3 = 1, S 2 (RS) = (RS)S 2 . It suffices thus to prove that θ ε (M) and θ s (M) have the same quadridegree when M = (RS) i k . . . S(RS) i 2 S(RS) i 1 S, for some i 1 , . . . , i k ∈ {±1}. For any index i j equal to 1, we replace the S immediately after by S -1 (since S 2 commutes with all matrices), and obtain now a product of non-negative powers of (RS)S -1 = R and (RS) 2 S. We will write R 1 = R and R 2 = (RS) 2 S, and have

R 1 = 1 1 0 1 , R 2 = 1 0 1 1 .
It is thus sufficient to prove the following assertion:

(⋆) if M = a b c d = R i k R i k-1 . . . R i 1 , for some i 1 , . . . , i k ∈ {1, 2},
then a, b, c, d ≥ 0, and θ s (M), θ ε (M) have both quadridegree (a, b, c, d).

We proceed now by induction on k. For k = 1, Assertion (⋆) can be directly checked:

Both θ s (R 1 ) = (xy, y) and θ ε (R 1 ) = x+εy ε+xy , εy have quadridegree (1, 1, 0, 1). Both θ s (R 2 ) = (x, xy) and θ ε (R 2 ) = x ε , ε(x+εy) ε+xy have quadridegree (1, 0, 1, 1). Now, assume that (⋆) is true for M = a b c d , and let us prove it for R 1 M = a + c b + d c d and R 2 M = a b a + c b + d
. By induction hypothesis one has θ ε (M) = ((x 1 : x 2 ), (y 1 : y 2 )) ((P 1 : P 2 ), (P 3 : P 4 )),

where P 1 , P 2 , P 3 , We have thus

P 4 ∈ C[x 1 , x 2 ,
θ ε (R 1 )θ ε (M) = θ ε (R 1 M) = ((x 1 :
x 2 ), (y 1 : y 2 )) ((P 1 P 4 + εP 2 P 3 : εP 2 P 4 + P 1 P 3 ), (εP 3 :

P 4 )), θ ε (R 2 )θ ε (M) = θ ε (R 2 M) = ((x 1 :
x 2 ), (y 1 : y 2 )) ((P 1 : εP 2 ), (ε(P 1 P 4 + εP 2 P 3 ) : εP 2 P 4 + P 1 P 3 )).

To prove (⋆) for R 1 M and R 2 M, it suffices to show that the polynomials P 1 P 4 + εP 2 P 3 and εP 2 P 4 + P 1 P 3 have no common component. Suppose the converse for contradiction, and denote by h

∈ C[x 1 , x 2 , y 1 , y 2 ] the common component. The polynomial h corresponds to a curve of P 1 × P 1 that is contracted by θ ε (M) onto a base-point of θ ε (R 1 ) or θ ε (R 2 ), i.e. onto p 1 = (ε, -1) or p 2 = (-ε, 1) (Lemma 3.3). But this condition means that (θ ε (M)) -1
has a base-point at p 1 or p 2 . We proved in Lemma 3.3 that this is impossible when ε is a positive real number.

We now show that this construction yields infinitely many conjugacy classes of embeddings of SL(2, Z) into the Cremona group that preserve the type. Proposition 3.5. If ε and ε ′ are two real positive numbers with εε ′ = 1, the two groups θ ε (SL(2, Z)) and θ ε ′ (SL(2, Z)) are not conjugate in the Cremona group.

The standard embedding θ s is conjugate to

θ 1 , but θ -(SL(2, Z)) is not conjugate to θ ε (SL(2, Z)) for any positive ε ∈ R.
Proof. The proof is similar to the one of Proposition 3.2. Assume, for contradiction, that θ ε (SL(2, Z)) is conjugate to θ ε ′ (SL(2, Z)); then θ ε (R) = x+εy ε+xy , εy is conjugate to some parabolic element of θ ε ′ (SL(2, Z)), which has no root in the group. This implies that θ ε (R) = x+εy ε+xy , εy or its inverse is conjugate to

θ ε ′ (R) = x+ε ′ y ε ′ +xy , ε ′ y or to θ ε ′ (RS 2 ) = -x-ε ′ y ε ′ +xy , -ε ′ y in Bir(P 2 ).
These elements are parabolic elements of the Cremona group, each of them preserves a unique rational fibration, which is (x, y) → y. The action on the basis being different up to conjugacy (since

εε ′ = ±1), neither θ ε (R) nor its inverse is conjugate to θ ε ′ (R) or θ ε ′ (RS 2 ) in Bir(P 2 ). It remains to show that θ -(SL(2, Z)) is not conjugate to θ ε (SL(2, Z)) for any positive ε ∈ R. Every parabolic element of θ -(SL(2, Z)) without root is conjugate to θ -(R) = (xy, -y), θ -(RS 2 ) = ( 1 xy , -1 y )
or their inverses, and acts thus non-trivially on the basis of the unique fibration preserved, with an action of order 2. We get the result by observing that θ ε (SL(2, Z)) contains θ ε (R) = x+εy ε+xy , εy , which is parabolic, without root and acting on the basis with an action which has not order 2.

Note that in all our examples of embeddings preserving the type, the parabolic elements have a linear degree growth. One can then ask the following question (which could yield a positive answer to Question 1.1). Question 3.6. Does there exist an embedding of SL(2, Z) into Bir(P 2 ) that preserves the type and such that the degree growth of parabolic elements is quadratic? 3.2. Elliptic embeddings. The simplest elliptic embedding is given by θ e : SL(2, Z) → Bir(P 2 ), a b c d → (ax + by : cx + dy : z).

We now generalise this embedding. Choose n ∈ N and let χ : SL(2, Z) → C * be a character such that χ -1 0 0 -1 = (-1) n . For simplicity, we choose χ such that χ(RS) = 1, and such that χ(S) is equal to 1 if n is odd and to i if n is even. Then we define θ n : SL(2, Z) → Bir(P 2 ) by

M = a b c d → ax + b cx + d , χ(M)y (cx + d) n .
The action on the first component and the fact that θ n (S 2 ) = 1 imply that θ n is an embedding. The degree of all elements being bounded, the embeddings are elliptic.

Proposition 3.7. For any n ∈ N, the group θ n (SL(2, Z)) is conjugate to a subgroup of Aut(F n ), where F n is the n-th Hirzebruch surface.

The groups θ m (SL(2, Z)) and θ n (SL(2, Z)) are conjugate in the Cremona group if and only if m = n.

Proof. If n = 0, the embedding (x, y) → ((x : 1), (y : 1)) of C 2 into P 1 × P 1 = F 0 conjugates θ 0 (SL(2, Z)) to a subgroup of Aut(F 0 ).

For n ≥ 1, recall that the weighted projective space P(1, 1, n) is equal to

P(1, 1, n) = (x 1 , x 2 , z) ∈ C 3 \{0} (x 1 , x 2 , z) ∼ (µx 1 , µx 2 , µ n z), µ ∈ C * .
The surface P(1, 1, 1) is equal to P 2 , and the surfaces P(1, 1, n) for n ≥ 2 have one singular point, which is (0 : 0 : 1).

For any n ≥ 1, the embedding (x, y) → (x : y : 1) of C 2 into P(1, 1, n) conjugates θ n (SL(2, Z)) to a subgroup of Aut(P(1, 1, n)) that fixes the point (0 : 0 : 1). The blow-up of this fixed point gives the Hirzebruch surface F n , and conjugates thus θ n (SL(2, Z)) to a subgroup of Aut(F n ).

In all cases n ≥ 0, the group preserves the fibration F n → P 1 corresponding to (x, y) → x. The action on the basis of the fibration corresponds to the standard homomorphism SL(2, Z) → PSL(2, Z) ⊂ PGL(2, C) = Aut(P 1 ). This action has no orbit of finite size on P 1 . In particular, there is no orbit of finite size on F n . This shows that the subgroup of Aut(F n ) corresponding to θ n (SL(2, Z)) is birationally rigid for n = 1, i.e. that it is not conjugate to any group of automorphisms of any other smooth projective surface. This shows that θ m (SL(2, Z)) and θ n (SL(2, Z)) are conjugate in the Cremona group only when m = n.

θ 0 (S) = - 1 x , iy and θ 0 (RS) = x -1 x , y .
We will "twist" θ 0 in order to construct parabolic embeddings. Recall that SL(2, Z)

acts via θ 0 on the projective line; the element a b c d acts as x ax+b cx+d . The group is countable so a very general point of the line has no isotropy. Let P ∈ C(x) be a rational function with m simple poles and m simple zeroes, where m > 0, and such that the 2m corresponding points of C are all on different orbits under the action of SL(2, Z) and have no isotropy. We denote by ϕ P = (x, y • P(x)) the associated birational map; it preserves the fibration and commutes with θ 0 (S 2 ) = (x, -y).

We choose

θ P (S) = θ 0 (S) = - 1 x , iy and θ P (RS) = ϕ P • θ 0 (RS) • ϕ -1 P , therefore θ P (S) = - 1 x , iy and θ P (RS) = x -1 x , y • P( x-1 x ) P(x)
.

The maps ϕ P and θ P (S 2 ) commute so θ P (RS) and θ P (S 2 ) commute too. Then, by definition of θ P (S) and θ P (RS) there is a unique morphism θ P : SL(2, Z) → Bir(C 2 ).

Proposition 3.8. The morphism θ P is a parabolic embedding for any P ∈ C(x).

Proof. The action on the basis of the fibration and the fact that θ P (S 2 ) = id imply that θ P is an embedding. It remains to show that any element of infinite order is sent onto a parabolic element.

Writing α = θ P (RS) and β = θ P (S), it suffices to show that h or hβ 2 is parabolic, where

h = βα i n β . . . α i 2 βα i 1 , n ≥ 1 and i 1 , . . . , i n ∈ {-1, 1}.
We view our maps acting on P 1 × P 1 . The fibration given by the projection on the first factor is preserved by h, which is thus either parabolic or elliptic. The first possibility occurs if the sequence of number of base-points of h k grows linearly and the second if the sequence is bounded.

Let p ∈ C be a pole or a zero of P. Let F 0 ⊂ P 1 × P 1 be the fibre of (p : 1) and let Σ ⊂ P 1 × P 1 be the (countable) union of fibres of points that belong to the orbit of (p : 1) under the action of SL(2, Z).

Recall that θ 0 (RS) is an automorphism of P 1 × P 1 . Set F 1 = θ 0 (RS)(F 0 ) and F 2 = θ 0 (RS)(F 1 ); remark that F 0 = θ 0 (RS)(F 2 ). Then ϕ P and its inverse contract F 0 on a point of F 0 but send isomorphically F 1 and F 2 onto themselves. The map α is the conjugate of θ 0 (RS) by ϕ P , so it contracts F 0 and F 2 on points lying respectively on F 1 and F 0 , but sends isomorphically F 1 onto F 2 and doesn't contract any other fibre contained in Σ. Similarly α -1 contracts F 0 and F 1 on points lying on F 2 and F 0 and neither contracts F 2 nor any other fibre of Σ.

Each fibre is preserved by β 2 , but β and β 3 send F 0 , F 1 , F 2 onto three other fibres contained in Σ. Then α ±1 β and α ±1 β 3 send isomorphically F 0 onto a fibre contained in Σ \ {F i }. By induction on n, we obtain that for any k < 0, h k and (hβ 2 ) k send isomorphically F 0 onto a curve in Σ \ {F i }.

Then we note that α and α -1 contract F 0 on a point contained in one of the F i , point sent by β onto an other point not contained in the F i 's. So, by induction on n, for any k > 0 both h k and (hβ 2 ) k contract F 0 on a point not contained in the F i 's and for which the fibre belongs to Σ.

For each integer k > 0, the fibre F 0 is contracted by h k and by (hβ 2 ) k = h k (β 2k ) on a point of Σ. Moreover, for each integer k < 0, F 0 is sent isomorphically by h k onto a fibre contained in Σ. Set F ′ i = h -i (F 0 ) for all i > 0; we obtain that h k and (hβ 2 ) k contract F 0 and F ′ 1 , . . . , F ′ k for each integer k > 0. This means that the number of base-points of h k and (hβ 2 ) k is at least equal to k. As h and hβ 2 preserve the fibration, they are parabolic. Proposition 3.9. When P varies, we obtain infinitely many parabolic embeddings.

Proof. Let P, Q ∈ C(x), and suppose that θ P (SL(2, Z)) is conjugate to θ Q (SL(2, Z)) by some birational map ϕ of P 1 × P 1 . Then ϕ preserves the fibration (x, y) → x, which is the unique fibration preserved by the two groups. Its action on the basis of the fibration is an element ψ ∈ PGL(2, C) that normalises PSL(2, Z) ⊂ PSL(2, C) = PGL(2, C). This means that ψ ∈ PSL(2, Z). Replacing ϕ by its product with an element of θ Q (SL(2, Z)), we can thus assume that ϕ acts trivially on the basis.

This means that ϕ is equal to x, a(x)y+b(x) c(x)y+d(x) for some a, b, c, d ∈ C(x), adbc = 0. Since ϕ conjugates θ P (S) = θ Q (S) = (-1

x , iy) to itself or its inverse, the map ϕ is equal to (x, a(x)y ±1 ) where a ∈ C(x), a(-1

x ) = ±a(x). The map ϕ conjugates θ P (RS) = x-1

x , y

• P( x-1 x ) P(x) to θ Q (RS) = x-1 x , y • Q( x-1 x ) Q(x) or to θ Q (RS 3 ) = x-1 x , -y • Q( x-1 x ) Q(x)
in Bir(P 1 × P 1 ). Assume that ϕ = (x, a(x)y)

where a ∈ C(x), a(-

1 x ) = a(x); then ϕθ P (RS)ϕ -1 = x-1 x , y • a( x-1 x )P( x-1 x ) a(x)P(x)
. Thus ϕθ P (RS)ϕ

-1 = θ Q (RS), resp. θ Q (RS 3 )
if and only if

a x-1 x a(x) = P(x)Q( x-1 x ) Q(x)P x-1 x , resp. a x-1 x a(x) = - P(x)Q( x-1 x ) Q(x)P x-1 x since a(x) is invariant under the homography x → -1
x , the same holds for

P(x)Q( x-1 x ) Q(x)P( x-1 x )
. This implies, in both cases, the following condition on P and Q

P(x)P(1 + x) P -1 x P x-1 x = Q(x)Q(1 + x) Q -1 x Q x-1 x .
We get the same formula when ϕ is equal to (x, a(x)y -1 ) where a ∈ C(x), a(-1 x ) = -a(x). When P varies, we thus obtain infinitely many parabolic embeddings.

3.4. Hyperbolic embeddings. In this section, we "twist" the standard elliptic embedding θ e defined in §3.2 to get many hyperbolic embeddings of SL(2, Z) into Bir(P 2 ). Recall that θ e is given by θ e : SL(2, Z) → Bir(P 2 ), a b c d → (ax + by : cx + dy : z).

The group θ e (SL(2, Z)) preserves the line L z of equation z = 0, and acts on it via the natural maps SL(2, Z)

→ PSL(2, Z) ⊂ PSL(2, C) = Aut(L z ).
We choose µ ∈ C * such that the point p = (µ : 1 : 0) ∈ L z has a trivial isotropy group under the action of PSL(2, Z), fix an even integer k > 0, and then define a morphism θ k : SL(2, Z) → Bir(P 2 ) by the following way:

θ k (S) = θ e (S) = (y : -x : z) θ k (RS) = ψθ e (RS)ψ -1
where ψ is the conjugation of ψ ′ = (x k : yx k-1 + z k : zx k-1 ) by (x + µy : y : z).

Note that ψ ′ restricts to an automorphism of the affine plane where x = 0, commutes with θ e (S 2 ) = (x : y : -z) and acts trivially on L z . Since ψ commutes with θ e (S 2 ) = θ k (S 2 ), the element θ k (RS) commutes with θ k (S 2 ), and θ k is thus a well-defined morphism. The fact that ψ preserves L z and acts trivially on it implies that the action of θ e and θ k on L z are the same, so θ k is an embedding.

Lemma 3.10. Let m be a positive integer, and let a

1 , . . . , a m , b 1 , . . . , b m ∈ {±1}. The bira- tional map θ k (S b m (RS) a m • • • S b 1 (RS) a 1 )
has degree k 2m and exactly 2m proper base-points, all lying on L z , which are p, ((RS) a 1 ) -1 (p), (S b 1 (RS) a 1 ) -1 (p), ((RS)

a 2 S b 1 (RS) a 1 ) -1 (p), . . . , ((RS) a m • • • S b 1 (RS) a 1 ) -1 (p), (S b m (RS) a m • • • S b 1 (RS) a 1 ) -1 (p),
where the action of R, RS ∈ SL(2, Z) on L z is here the action via θ e or θ k .

Proof. The birational map ψ has degree k and has an unique proper base-point which is p = (µ : 1 : 0) ∈ L z ; the same is true for ψ -1 . Moreover both maps fix any other point of L z .

Since θ e (RS) a 1 is an automorphism of P 2 that moves the point p onto an other point of L z , the map θ k ((RS) a 1 ) = ψθ e (RS) a 1 ψ -1 has degree k 2 and exactly two proper basepoints, which are p and ψθ e (RS) -a 1 (p) = ((RS) a 1 ) -1 (p). The map θ k (S) being an automorphism of P 2 , θ k (S b 1 (RS) a 1 ) has also degree k 2 and two proper base-points, which are p and ((RS) a 1 ) -1 (p). This gives the result for m = 1.

Proceeding by induction for m > 1, we assume that θ k (S b m (RS) a m • • • S b 2 (RS) a 2 ) has degree k 2m-2 and exactly 2m -2 proper base-points, all lying on L z , which are

p, ((RS) a 2 ) -1 (p), (S b 2 (RS) a 2 ) -1 (p), . . . , (S b m (RS) a m • • • S b 2 (RS) a 2 ) -1 (p).
The map θ k (S b 1 (RS) a 1 ) -1 = θ k ((RS) -a 1 )θ k (S -b 1 ) has degree k 2 and two proper basepoints, which are S b 1 (p) and S b 1 (RS) a 1 (p). These two points being distinct from the 2m -2 points above, the map

θ k (S b m (RS) a m • • • S b 1 (RS) a 1 ) has degree k 2 • k 2m-2 = k 2m ,
and its proper base-points are the 2 proper base-points of θ k (S b 1 (RS) a 1 ) and the image by

(S b 1 (RS) a 1 ) -1 of the base-points of θ k (S b m (RS) a m • • • S b 2 (RS) a 2
). This gives the result.

As a corollary, we get infinitely many hyperbolic embeddings of SL(2, Z) into the Cremona group.

Corollary 3.11. Let m be a positive integer, and let a

1 , . . . , a m , b 1 , . . . , b m ∈ {±1}. The birational map θ k (S b m (RS) a m • • • S b 1 (RS) a 1 )
has dynamical degree k 2m . In particular, the map θ k is an hyperbolic embedding and the set of all dynamical degrees of

θ k (SL(2, Z)) is {1, k 2 , k 4 , k 6 , . . . }.
Proof. Any element of infinite order of SL(2, Z) is conjugate to g = S b m (RS) a m • • • S b 1 (RS) a 1 for some a 1 , . . . , a m , b 1 , . . . , b m ∈ {±1}. Lemma 3.10 implies that the degree of θ k (g r ) is equal to k 2mr . The dynamical degree of θ k (g) is therefore equal to k 2m .

DESCRIPTION OF HYPERBOLIC EMBEDDINGS FOR WHICH THE CENTRAL ELEMENT FIXES (POINTWISE) AN ELLIPTIC CURVE

4.1. Outline of the construction and notation. In this section, we give a general way of constructing embeddings of SL(2, Z) into the Cremona group where the central involution fixes pointwise an elliptic curve. Recall that all conjugacy classes of elements of order 4 or 6 in Bir(P 2 ) have been classified (see [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]). Many of them can act on del Pezzo surfaces of degree 1, 2, 3 or 4.

In order to create our embedding, we will define del Pezzo surfaces X, Y of degree ≤ 4, and automorphisms α ∈ Aut(X), β ∈ Aut(Y ) of order respectively 6 and 4, so that α 3 and β 2 fix pointwise an elliptic curve, and that Pic(X) α , Pic(Y ) β have both rank 1. Note that we say that a curve is fixed by a birational map if it is pointwise fixed, and say that it is invariant or preserved if the map induces a birational action (trivial or not) on the curve. Contracting (-1)-curves invariant by these involutions (but not by α, β, which act minimally on X and Y ), we obtain birational morphisms X → X 4 and Y → Y 4 , where X 4 , Y 4 are del Pezzo surfaces on which α 3 and β 2 act minimally. Lemma 4.1 below shows that X 4 and Y 4 are del Pezzo surfaces of degree 4 and both Pic(X 4 ) α 3 and Pic(Y 4 ) β 2 have rank 2 and are generated by the fibres of two conic bundles on X 4 and Y 4 . Choosing a birational map X 4 Y 4 conjugating α 3 to β 2 (which exists if and only if the elliptic curves are isomorphic), which is general enough, we should obtain an embedding of SL(2, Z) such that any element of infinite order is hyperbolic.

In order to prove that there is no more relation in the group generated by α and β and that all elements of infinite order are hyperbolic, we describe the morphisms X → X 4 and Y → Y 4 and the action of α and β on Pic(X) α 3 and Pic(Y ) β 2 (which are generated by the fibres of the two conic bundles on X 4 , and Y 4 and by the exceptional curves obtained by blowing-up points on the elliptic curves fixed), and then observe that the composition of the elements does what is expected. 4.2. Technical results on automorphisms of del Pezzo surfaces of degree 4. Recall some classical facts about del Pezzo surfaces, that the reader can find in [Dem] (see also [Man]). A del Pezzo surface is a smooth projective surface Z such that the anti-canonical divisor -K Z is ample. These are P 1 × P 1 , P 2 or P 2 blown-up at 1 ≤ r ≤ 8 points in general position (no 3 collinear, no 6 on the same conic, no 8 on the same cubic singular at one of the 8 points). The degree of a del Pezzo surface Z is (K Z ) 2 , which is 8 for P 1 × P 1 , 9 for P 2 and 9r for the blow-up of P 2 at r points.

Any del Pezzo surface Z contains a finite number of (-1)-curves (smooth curves isomorphic to P 1 and of self-intersection -1), each of these can be contracted to obtain another del Pezzo surface of degree (K Z ) 2 +1. These are moreover the only irreducible curves of Z of negative self-intersection. If Z is not P 2 , there is a finite number of conic bundles Z → P 1 (up to automorphism of P 1 ), and each of them has exactly 8 -(K Z ) 2 singular fibres. This latter fact can be find by contracting one component in each singular fibre, which is the union of two (-1)-curves, obtaining a line bundle on a del Pezzo surface, isomorphic to P 1 × P 1 or F 1 and having degree 8. Lemma 4.1. Let Z be a del Pezzo surface, and let σ ∈ Aut(Z) be an involution that fixes (pointwise) an elliptic curve. Denote by η : Z → Z 4 any < σ >-invariant birational morphism such that the action on Z 4 is minimal.

Then, Z 4 is a del Pezzo surface of degree 4, and Pic(Z 4 ) σ = Z f 1 ⊕ Z f 2 , where f 1 , f 2 correspond to the fibres of the two conic bundles π 1 , π 2 : Z 4 → P 1 (defined up to automorphism of P 1 ) that are invariant by σ. Moreover

f 1 + f 2 = -K Z 4 , f 1 • f 2 = 2 and Pic(Z) σ = Zη * ( f 1 ) ⊕ Zη * ( f 2 ) ⊕ ZE 1 ⊕ • • • ⊕ ZE r
where E 1 , . . . , E r are the r irreducible curves contracted by η (in particular, η only contracts invariant (-1)-curves).

Proof. Since Z is a del Pezzo surface, Z 4 is also a del Pezzo surface. As σ acts minimally on Z 4 and fixes an elliptic curve, we have the following situation ( [START_REF] Bayle | Birational involutions of P 2[END_REF]Theorem 1.4]): there exists a conic bundle π 1 : Z 4 → P 1 such that π 1 σ = π 1 , σ induces a non-trivial involution on each smooth fibre of π 1 , and exchanges the two components of each singular fibre, which meet at one point. The restriction of π 1 to the elliptic curve is a double covering ramified over 4 points, which implies that there are four singular fibres. The surface Z 4 is thus the blow-up of four points on F 1 or P 1 × P 1 , and has therefore degree 4. The fact that there are exactly two conic bundles π 1 , π 2 : Z 4 → P 2 invariant by σ, that Pic(Z 4 ) σ is generated by the two fibres, that f 1 + f 2 = -K Z 4 and that f 1 • f 2 = 2 can be checked in [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF]Lemma 9.11].

It remains to observe that all points blown-up by η are fixed by σ. If η blows-up an orbit of at least two points of Z 4 invariant by σ, the points would be on the same fibre of π 1 . The transform of this fibre on Z would then contain a curve isomorphic to P 1 and having self-intersection ≤ -2; this is impossible on a del Pezzo surface. Lemma 4.2. For i = 1, 2, let X i be a projective smooth surface, with K 2 X i = 4, and let σ i ∈ Aut(X i ) be an involution which fixes an elliptic curve Γ i ⊂ X i . Let π i : X i → P 1 be a conic bundle such that π i σ i = π i and let F i , G i ⊂ X i be two sections of π i of self-intersection -1, intersecting transversally into one point.

Then, X 1 , X 2 are del Pezzo surfaces of degree 4 and the following assertions are equivalent:

(1) There exists an isomorphism ϕ : X 1 → X 2 which conjugates σ 1 to σ 2 , which sends F 1 , G 1 onto F 2 and G 2 respectively and such that π 2 ϕ = π 1 ;

(2) The points of P 1 whose fibres by π i are singular are the same for i = 1, 2, and

π 1 (F 1 ∩ G 1 ) = π 2 (F 2 ∩ G 2 ).
Proof. For i = 1, 2, we denote by η i : X i → F 1 the birational morphism that contracts, in each singular fibre of π i , the (-1)-curve that does not intersect F i . The curve η i (F i ) is equal to the exceptional section E of the line bundle π :

F 1 → P 1 , with π = π i η -1 i . Since η i (G i
) intersects E into exactly one point, it is a section of self-intersection 3. In particular, the four points blown-up by η i lie on η i (G i ). Contracting E onto a point of P 2 , η i (G i ) becomes a conic of P 2 passing through the five points blown-up by the birational morphism X i → P 2 ; this implies that no 3 are collinear and thus that X i is a del Pezzo surface of degree 4.

It is clear that the first assertion implies the second one. It remains to prove the converse. The second assertion implies that η 1 (G 1 ) ∩ E = η 2 (G 2 ) ∩ E, and this yields the existence of an automorphism of F 1 that sends η 1 (G 1 ) onto η 2 (G 2 ) and that preserves any fibre of π. We can thus assume that η 1 (G 1 ) = η 2 (G 2 ), which implies that the four points blown-up by η 1 and η 2 are the same. The isomorphism ϕ can be chosen as ϕ = η -1 2 • η 1 . The map ϕ conjugates σ 1 to σ 2 because, for each i, σ i is the unique involution that preserves any fibre of π i and exchanges the two components of each singular fibre (see for example [START_REF] Blanc | Linearisation of finite abelian subgroups of the Cremona group of the plane[END_REF]Lemma 9.11]). 4.3. Actions on the Picard groups of α and β. We now describe the actions of α and β on Pic(X) and Pic(Y ).

Proposition 4.3. Let X be a del Pezzo surface of degree (K X ) 2 < 4, and let α ∈ Aut(X) be an automorphism of order 6 such that Pic(X) α = ZK X and such that α 3 fixes pointwise an elliptic curve. Let η X : X → X 4 be a birational morphism, so that α 3 acts minimally on X 4 , and let f 1 , f 2 ∈ Pic(X) be the divisors corresponding to the two conic bundles on X 4 which are invariant by α 3 (see Lemma 4.1). Then, one of the following occurs:

(i) (K X ) 2 = 3, η X contracts a curve E 1 , and α, α 2 act on Pic(X)

α 3 as   1 1 1 1 0 0 -2 0 -1   and   0 1 0 1 1 1 0 -2 -1   relatively to the basis ( f 1 , f 2 , E 1 ) (up to an exchange of f 1 , f 2 ). (ii) (K X ) 2 = 1, η X contracts E 1 , E 2 , E 3 , and α, α 2 act on Pic(X) α 3 as       1 3 1 1 1 3 4 2 2 2 -2 -4 -2 -2 -1 -2 -4 -1 -2 -2 -2 -4 -2 -1 -2       and       4 3 2 2 2 3 1 1 1 1 -4 -2 -2 -1 -2 -4 -2 -2 -2 -1 -4 -2 -1 -2 -2       relatively to the basis ( f 1 , f 2 , E 1 , E 2 , E 3 ) (up to a good choice of E 1 , E 2 , E 3 and an exchange of f 1 , f 2 ).
Proof. Let E ⊂ X be any (-1)-curve invariant by α 3 . The divisor E + α(E) + α 2 (E) is invariant by α and thus equivalent to sK X for some integer s. Computing the intersection with K X and the self-intersection, we obtain -3 = s(K X ) 2 and -3 + 6(E •α(E)) = s 2 (K X ) 2 . This gives two possibilities:

(i) (K X ) 2 = 3, s = -1, E • α(E) = 1 (ii) (K X ) 2 = 1, s = -3, E • α(E) = 2
In case (i), η X is given by the choice of one (-1)-curve E 1 invariant by α 3 . Since E 1 • α(E 1 ) = 1, the divisor E 1 + α(E 1 ) corresponds to a conic bundle on X and X 4 . Up to renumbering, we can say that

f 1 = E 1 + α(E 1 ) and f 2 = E 1 + α 2 (E 1 ). This means that α(E 1 ) = f 1 -E 1 , α 2 (E 1 ) = f 2 -E 1 , α( f 1 ) = f 1 + f 2 -2E 1 and α( f 2 ) = f 1 .
In case (ii), there are three curves E 1 , E 2 , E 3 contracted by η X . We first choose E 1 , and then choose

E ′ 2 = ι B (α(E 1 )) = -2K X -α(E 1 ) (where ι B is the Bertini involution of the surface). Since E ′ 2 does not intersect E 1 , we can contract E 1 , E ′ 2 ,
and another curve E 3 to obtain an α 3 -equivariant birational morphism X → X ′ 4 , where X ′ 4 is a del Pezzo surface of degree 4. This choice gives us two conic bundles f ′ 1 , f ′ 2 on X ′ 4 , which we also see on X 4 , invariant by α 3 . We now compute α(E 3 ). We have

α(E 3 ) • E 3 = 2, α(E 3 ) • E 1 = E 3 • α 2 (E 1 ) = E 3 • (-3K X -E 1 -α(E 1 )) = E 3 • (-K X -E 1 + E ′ 2 ) = 1, α(E 3 ) • E ′ 2 = E 3 • α 2 (E ′ 2 ) = E 3 • (-2K X -E 1 ) = 2. This implies that α(E 3 ) = a f ′ 1 + b f ′ 2 -E 1 -2E ′ 2 -2E 3 , for some integers a, b. Computing the intersection with -K X we find 1 = 2a + 2b -1 -2 -2 = 2(a + b) -5, which means that a + b = 3. Computing the self-intersection, we obtain that -1 = 2ab -1 -4 -4 = 4ab -9, so ab = 2. Up to an exchange of f ′ 1 , f ′ 2 , we can assume that a = 1, b = 2, and obtain that α(E 3 ) = f ′ 1 + 2 f ′ 2 -E 1 -2E ′ 2 -2E 3 = -2K X -( f ′ 1 -E 1 ). We now call E 2 the (-1)-curve f ′ 1 -E ′ 2 , which does not intersect E 1 or E 3 . We take f 1 = f ′ 1 and f 2 = f ′ 1 + f ′ 2 -2E ′ 2
, so that f 1 , f 2 are conic bundles, with intersection 2, and

-K X = f 1 + f 2 -E 1 -E 2 -E 3 . The contraction of E 1 , E 2 , E 3 is a α 3 -
equivariant birational morphism X → X 4 and f 1 , f 2 correspond to the two conic bundles of X 4 invariant by α 3 . With this choice, we can compute

α(E 1 ) = ι B (E ′ 2 ) = ι B ( f 1 -E 2 ) = -2K X -( f 1 -E 2 ), α 2 (E 1 ) = -3K X -α(E 1 ) -E 1 = -K X -( f 1 -E 2 ) -E 1 = -2K X -( f 2 -E 3 ), α(E 3 ) = -2K X -( f ′ 1 -E 1 ) = -2K X -( f 1 -E 1 ), α 2 (E 3 ) = -3K X -α(E 3 ) -E 3 = -K X -( f 1 -E 1 ) -E 3 = -2K X -( f 1 -E 2 ).

This yields the equalities f

1 = -2K X + E 1 -α(E 3 ) and f 2 = -2K X + E 3 -α 2 (E 1 ), E 2 = α 2 (E 3 )+2K X -f 1 ,
and a straightforward computation gives, with the four equations above, α i ( f j ) and α i (E 2 ) for i, j = 1, 2. Proposition 4.4. Let Y be a del Pezzo surface of degree (K Y ) 2 < 4, and let β ∈ Aut(Y ) be an automorphism of order 4 such that Pic(Y ) β = ZK Y and that β 2 fixes pointwise an elliptic curve. Let η Y : Y → Y 4 be a birational morphism, so that β 2 acts minimally on Y 4 , and let f 1 , f 2 ∈ Pic(Y ) be the divisors corresponding to the two conic bundles on Y 4 that are invariant by β 2 (see Lemma 4.1). Then, one of the following occurs:

(i) (K Y ) 2 = 2, η Y contracts two curves E 1 , E 2 and β acts on Pic(Y ) β 2 as     1 2 1 1 2 1 1 1 -2 -2 -2 -1 -2 -2 -1 -2     relatively to the basis ( f 1 , f 2 , E 1 , E 2 ). (ii) (K Y ) 2 = 1, η Y contracts E 1 , E 2 , E 3 , and β acts on Pic(Y ) β 2 as       3 4 2 2 2 4 3 2 2 2 -3 -3 -3 -2 -2 -3 -3 -2 -3 -2 -3 -3 -2 -2 -3       relatively to the basis ( f 1 , f 2 , E 1 , E 2 , E 3 ).
Remark 4.5. The second case, numerically possible, does not exist (see [DoIs] or [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]).

Proof. Let E ⊂ Y be any (-1)-curve invariant by β 2 . The divisor E + β(E) is invariant by β and thus equivalent to sK Y for some integer s. Computing the intersection with K Y and the self-intersection, we obtain -2 = s(K Y ) 2 and -2 + 2(E • β(E)) = s 2 (K Y ) 2 . This gives two possibilities:

(i) (K Y ) 2 = 2, s = -1, E • β(E) = 2 (ii) (K Y ) 2 = 1, s = -2, E • β(E) = 3
In case (i), there are two curves E 1 , E 2 contracted by η Y , and

β(E i ) = -K Y -E i for i = 1, 2. Moreover f i -E 1 is also a (-1)-curve for i = 1, 2, so β( f i ) = β(E 1 ) + β( f i -E 1 ) = -K Y -E 1 -K Y -( f i -E 1 ) = -2K Y -f i .
In case (ii), there are three curves E 1 , E 2 , E 3 contracted by η Y , and

β(E i ) = -2K Y -E i for i = 1, 2, 3. As before, we find β( f i ) = -4K Y -f i .
4.4. Automorphisms of del Pezzo surfaces of order 6, resp. 4 -description of α and β.

Automorphisms of del Pezzo surfaces of order 6. We now give explicit possibilities the automorphism α ∈ Aut(X) of order 6. Case I X = (w : x : y : z) ∈ P(3, 1, 1, 2) w 2 = z 3 + µxz 4 + x 6 + y 6 α((w : x : y : z)) = (w : x : -ωy : z)

for some general µ ∈ C so that the surface is smooth and where ω = e 2iπ/3 . The surface is a del Pezzo surface of degree 1, and α fixes pointwise the elliptic curve given by y = 0. When µ varies, all possible elliptic curves are obtained. The rank of Pic(X) α is 1 (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]Corollary 6.11]).

Case II X = (w : x : y : z) ∈ P 3 wx 2 + w 3 + y 3 + z 3 + µwyz = 0 , α((w : x : y : z)) = (w : -x : ωy :

ω 2 z),
where µ ∈ C is such that the cubic surface is smooth. The surface is a del Pezzo surface of degree 3, α 3 fixes pointwise the elliptic curve given by x = 0, and α acts on this via a translation of order 3. When µ varies, all possible elliptic curves are obtained. The rank of Pic(X) α is 1 (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]Page 79]).

Case III X = (w : x : y : z) ∈ P 3 w 3 + x 3 + y 3 + (x + µy)z 2 = 0 , α((w : x : y : z)) = (ωw : x : y : -z),

where µ ∈ C is such that the cubic surface is smooth. The surface is a del Pezzo surface of degree 3, α 3 fixes pointwise the elliptic curve given by z = 0, and α acts on it via an automorphism of order 3 with 3 fixed points. When µ varies the birational class of α changes (because the isomorphism class of the curve fixed by α 2 changes) but not the isomorphism class of the elliptic curve fixed by α 3 . The rank of Pic(X) α is 1 (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]Page 79]). where µ ∈ C\{0, 1}. The surface is a del Pezzo surface of degree 2 and β fixes pointwise the elliptic curve given by x = 0. When µ varies, all possible elliptic curves are obtained. The rank of Pic(Y ) β is 1 (see [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]last line of page 67] or [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]).

Automorphisms of del

There are other possibilities of automorphisms β of order 4 of rational surfaces Y such that β 2 fixes an elliptic curve, but none for which the rank of Pic(Y ) β is 1 (see [START_REF] Blanc | Elements and cyclic subgroups of finite order of the Cremona group[END_REF]).

The map X 4

Y 4 that conjugates α 3 to β 2 . We now fix α ∈ Aut(X), β ∈ Aut(Y ), automorphisms of order 6 and 4 respectively, which act minimally on del Pezzo surfaces X and Y , so that α 3 and β 2 fix (pointwise) elliptic curves Γ X ⊂ X and Γ Y ⊂ Y , which are isomorphic (as abstract curves).

We denote by η X : X → X 4 and η Y : Y → Y 4 two birational morphisms to del Pezzo surfaces of degree 4, so that α 3 and β 2 act minimally on X 4 and Y 4 respectively. We denote by f 1 , f 2 ∈ Pic(X 4 ) ⊂ Pic(X), respectively by f ′ 1 , f ′ 2 ∈ Pic(Y 4 ) ⊂ Pic(Y ), the two divisors corresponding to the two conic bundles invariant by α 3 , respectively by β 2 .

We will choose two points q 1 , q 2 ∈ η X (Γ X ) ⊂ X 4 , and denote by τ : Z 4 → X 4 the blow-up of these two points.

Lemma 4.6. For some good choice of q 1 , q 2 , there exists a birational morphism τ ′ : Z 4 → Y 4 satisfying the following properties:

(1) the morphism τ ′ is the contraction of the strict transforms of the two irreducible curves equivalent to f 1 passing through q 1 and q 2 onto two points q ′ 1 , q ′ 2 ∈ η Y (Γ Y ); (2) the map ϕ = τ ′ τ -1 conjugates α 3 to β 2 (i.e. ϕα 3 = β 2 ϕ);

(3) neither q 1 nor q 2 is blown-up by η X , and neither q ′ 1 nor q ′ 2 is blown-up by η Y ;

(4) identifying f 1 , f 2 with τ * ( f 1 ), τ * ( f 2 ) ∈ Pic(Z 4 ) and f ′ 1 , f ′ 2 with τ ′ * ( f ′ 1 ), τ ′ * ( f ′ 2 ) ∈ Pic(Z 4
), we have the following relations in Pic(Z 4 ):

f 1 = f ′ 1 , f ′ 1 = f 1 , f 2 = f ′ 2 + 2 f ′ 1 -2E τ ′ , f ′ 2 = f 2 + 2 f 1 -2E τ , E τ = 2 f ′ 1 -E τ ′ , E τ ′ = 2 f 1 -E τ ,
where E τ , E τ ′ ∈ Pic(Z 4 ) correspond to the exceptional divisors of τ and τ ′ respectively, which are the sum of two exceptional curves.

Proof. Denote by π : X 4 → P 1 and π ′ : Y 4 → P 1 the morphisms whose fibres are f 1 and f ′ 1 respectively. As it was already observed in the proof of Lemma 4.1, both π, π ′ are conic bundles, with four singular fibres, and the four singular fibres correspond to the four branch points of the double coverings π : η X (Γ X ) → P 1 and π ′ : η Y (Γ Y ) → P 1 . Since Γ X and Γ Y are isomorphic elliptic curves, we can assume that the four points are the same for both morphisms. Denote by ∆ ⊂ P 1 the union of the image by π of the points blown-up by η X , the image by π ′ of the points blown-up by η Y , and the points corresponding to singular fibres of π (or π ′ ). We define a closed subset V ⊂ Γ X × Γ X consisting of pairs (q 1 , q 2 ) that we "do not want", and denote by U its complement. The closed subset V is the union of the pairs (q 1 , q 2 ) such that π(q 1 ) or π(q 2 ) belongs to ∆. Observe that V is a finite union of curves of Γ X × Γ X (of bidegree (0, 1) or (1, 0)).

Choosing (q 1 , q 2 ) ∈ U, such that q 1 , q 2 are on distinct fibres of π, we can define a birational morphism τ ′ : Z 4 → W which contracts the strict transforms of the fibres of π which pass through q 1 and q 2 . The map ϕ = τ ′ τ -1 conjugates α 3 to a biregular automorphism of W , which preserves any fibre of the conic bundle π W = πϕ -1 . In fact, ϕ is a sequence of two elementary links of conic bundles. It remains to show that for a good choice of (q 1 , q 2 ) ∈ U, the triplet (W, π W , ϕα 3 ϕ -1 ) is isomorphic to (Y, π ′ , β 2 ), using Lemma 4.2.

Let E 1 ⊂ X 4 be a (-1)-curve which is a section of π; we fix a birational morphism µ X : X 4 → P 2 which contracts E 1 and all (-1)-curves lying on fibres of π that do not intersect E 1 , which we call E 2 , . . . , E 5 . The fibres of π correspond to lines of P 2 passing through the point p 1 = µ X (E 1 ), the curves equivalent to f 2 correspond to conics passing through p 2 = µ X (E 2 ), . . . , p 5 = µ X (E 5 ). For any pair (q 1 , q 2 ), we denote by C ⊂ X 4 (respectively D ⊂ X 4 ) the strict transform of the conic of P 2 passing through p 1 , p 2 , p 3 , q 1 , q 2 (respectively p 1 , p 4 , p 5 , q 1 , q 2 ), and denote by C ′ , D ′ ⊂ W their strict transforms by ϕ. The curves C, D are sections of π and intersect into three points: q 1 , q 2 , r ∈ X 4 . The curves C ′ , D ′ are sections of π W of self-intersection -1, and intersect into one point, which is ϕ(r) ∈ W . The isomorphism class of the triplet (W, π W , ϕα 3 ϕ -1 ) is given by π W (ϕ(r)) ∈ P 1 (Lemma 4.2), equal to π(r) ∈ P 1 . Fixing q 1 , and choosing one of the two possibilities for r, on the fibre given by the isomorphism class of (Y, π Y , β 2 ), the curves C, D can be chosen as the conics passing respectively through p 1 , p 2 , p 3 , q 1 , r and p 1 , p 4 , p 5 , q 1 , r, so q 2 is uniquely defined. This gives us two irreducible curves

V 1 ,V 2 of bidegree (1, 1) in Γ X × Γ X , which are thus not contained in V . Choosing a general point of V 1 ∩U, the triplet (W, π W , ϕα 3 ϕ -1 ) is isomorphic to (Y, π Y , β 2 ).
The fact that η X does not blow-up q 1 or q 2 and that η Y does not blow-up q ′ 1 or q ′ 2 is given by the fact that π(q i ) = π ′ (q ′ i ) / ∈ ∆ for i = 1, 2. It remains to show the relations in Pic(Z 4 ). The equalities f 1 = f ′ 1 and E τ + E τ ′ = 2 f 1 are given by the construction of τ, τ ′ . The adjunction formula, and the fact that -K

X 4 = f 1 + f 2 , -K Y 4 = f ′ 1 + f ′ 2 yields -K Z 4 = f 1 + f 2 -E τ = f ′ 1 + f ′ 2 -E τ ′
and the remaining equalities.

4.6. The hyperbolic embeddings. Now we have the map ϕ : X 4 Y 4 constructed in §4.5 above, which conjugates α 3 to β 2 , the group generated by α and β is a subgroup of the Cremona group, which is isomorphic to SL(2, Z) if and only if there is no other relation than the obvious 1 = α 6 = β 4 = α 3 β 2 which arise by construction. We compute the action of α, β on Pic(X), Pic(Y ), and on a surface Z which dominates X, Y , where both α, β act. This surface exists if the group generated by the action of both maps on the elliptic curve fixed by α 3 and β 2 is a finite subgroup of automorphisms of the curve (which is true for example when either α or β fixes the curve), and if it does not exist, we can also compute the action on the limit of the Picard groups obtained.

(1) Z

π X }} } } } } } } π Y A A A A A A A A X η X Z 4 τ }} } } } } } τ ′ @ @ @ @ @ @ @ Y η Y X 4 ϕ / / _ _ _ _ _ _ _ Y 4
Proposition 4.7. For j = 1, 2, 3, choose α ∈ Aut(X) as an automorphism of order 6 of a del Pezzo surface X, which is respectively given in case I, II or III of §4.4, such that α 3 fixes pointwise an elliptic curve Γ X , and choose β as an automorphism of order 4 of a del Pezzo surface Y of degree 2, which fixes pointwise an elliptic curve Γ Y isomorphic to Γ X , (which implies that α 3 and β 2 are conjugate). This yields, with the above construction, a hyperbolic embedding θ h, j : SL(2, Z) ⊂ Aut(Z) ⊂ Bir(Z) ≃ Bir(P 2 ) which preserves an elliptic curve Γ isomorphic to Γ X and Γ Y .

The surface Z is obtained by blowing-up respectively 12, 10 and 10 points on a smooth cubic curve of P 2 isomorphic to Γ, and the action of θ h,i (SL(2, Z)) on Γ is respectively the identity, a translation of order 3 and an automorphism of order 3 with fixed point. There is no curve of Z distinct from Γ which is invariant by θ h,i (SL(2, Z)). The curve Γ can be chosen to be any elliptic curve for j = 1, 2.

Proof. In case j = 1, we take ( f 1 , f 2 , E 1 , E 2 , E 3 ) as a basis of Pic(X) α 3 , where E 1 , E 2 , E 3 are the three curves contracted by η X , and f 1 , f 2 correspond to the fibres of the two conic bundles invariant by α 3 on X 4 . Applying Proposition 4.3, α preserves the submodule generated by f 1 , f 2 , E, where E = E 1 + E 2 + E 3 is the divisor contracted by η X , and its action relatively to this basis is

  1 3 3 3 4 6 -2 -4 -5   .
In cases j = 2, 3, we take ( f 1 , f 2 , E) as a basis of Pic(X) α 3 , where E = E 1 is the (irreducible) divisor contracted by η X , and f 1 , f 2 correspond to the fibres of the two conic bundles invariant by α 3 on X 4 . Applying Proposition 4.3, the action of α on Pic(X) α 3 relatively to this basis is

  0 1 0 1 1 1 0 -2 -1   (for a good choice of f 1 , f 2 , E).
In each of the three cases, we take

( f ′ 1 , f ′ 2 , E ′ 1 , E ′ 2 ) as a basis of Pic(Y ) β 2
, where E ′ 1 , E ′ 2 are the divisors contracted by η Y , and f ′ 1 , f ′ 2 correspond to the fibres of the two conic bundles invariant by β 2 on Y 4 . Applying Proposition 4.4, β preserves the submodule generated by

f ′ 1 , f ′ 2 , E ′ , where E ′ = E ′ 1 + E ′ 2
is the divisor contracted by η Y and its action relatively to this basis is

  1 2 2 2 1 2 -2 -2 -3   .
We denote by π X : Z → X the blow-up of the points corresponding to the points blownup by τ and η Y (see Diagram (1)), and denote again their exceptional divisors by E τ and E ′ . Similarly, we denote by π Y : Z → Y the blow-up of the points corresponding to the two points blown-up by τ ′ and η X , and denote again their exceptional divisors by E τ ′ and E. Since X 4 and Y 4 are del Pezzo surfaces of degree 4, they are obtained by blowing-up 5 points of P 2 , all lying on the smooth cubic being the image of Γ X or Γ Y . This implies that Z is the blow-up of 12 points of P 2 if i = 1 and of 10 points of P 2 if i = 2, 3, all points belonging to the smooth cubic curve. Moreover, both α and β lift to automorphisms of Z.

We denote by the same name the pull-backs of the divisors f 1 , f 2 , E, E ′ , E τ on Z. Recall that E τ is the sum of two (-1)-curves. The action of α in case j = 1, α in case j ∈ {2, 3} and β in each case on the subvectorspace W of Pic(Z) ⊗ R generated by ( f 1 , f 2 , E, E ′ , E τ ) are respectively relatively to this basis. The first two matrices are obtained because α fixes the curve Γ X , and because E ′ , E τ correspond to points of Γ X which are not blown-up by η X (Lemma 4.6). The second matrix is obtained applying again Lemma 4.6, which yields the equations

f 1 = f ′ 1 , f 2 = f ′ 2 + 2 f ′ 1 -2E τ ′ , E τ = 2 f ′ 1 -E τ ′ .
One easily checks that the only elements of W which are fixed by α and β are the multiples of the canonical divisor, corresponding to [1, 1, -1, -1, -1]. This implies that any curve C ⊂ Z invariant by the group is a multiple of the elliptic curve Γ Z ⊂ Z (strict transform of Γ X and Γ Y ). This curve having negative self-intersection, C has to be equal to Γ Z .

By construction, we have α 6 = β 4 = 1 and β 2 = α 3 . We have to prove that no other relation holds, and that any element of infinite order corresponds to a hyperbolic element of Aut(Z). Writing ρ 1 = αβ and ρ 2 = α 2 β, this corresponds to show that for any sequence (i 1 , . . . , i n ) with i k ∈ {1, 2}, the element ρ i n • • • • • ρ i 1 is a hyperbolic element of Aut(Z).

To show this, we look at the action of α, β on the orthogonal W 0 = K ⊥ of the canonical divisor K ∈ W ⊂ Pic(Z) in W . We choose a basis of W 0 , made of orthogonal eigenvectors of β.

If j = 1, the basis is < [1, 0, 0, -1, 0], [2, 1, 0, -1, -2], [3, 1, -2, -1, -2], [4, 2, -2, -2, -3] >, which has signature < -2, -2, -2, 2 > and the actions of α, α 2 , β relatively to it are respectively

    0 -1 -2 -2 -2 -2 -3 -4 -1 0 -2 -2 2 2 4 5     ,     0 -2 -1 -2 -1 -2 0 -2 -2 -3 -2 -4 2 4 2 5     ,     -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1     .
We denote by H the fourth basis vector, which is the only one with positive square, and compute by induction on n the vector

H n = ρ i n • • • • • ρ i 1 (H)
for n ≥ 0 (with H 0 = H).

Writing

H n =     -a n -b n -c n ℓ n    
, we prove by induction on n the following inequalities:

(2)

a n , b n , c n , ℓ n ≥ 0 ℓ n > 6 5 c n ℓ n > 2a n ℓ n ≥ ( 53 ) n , where the last one will yield the result, implying that ρ i k • • • • • ρ i 1 is a hyperbolic element of Aut(Z) of dynamical degree ≥ ( 5 3 ) k . Note that (2) is easily checked for n = 0, since ℓ 0 = 1, a 0 = b 0 = c 0 = 0. We assume the result true for n and prove it for n + 1. We have

H n+1 = ρ i n+1 (H n ) = α i n+1 β(H n ), which is equal to     -b n + 2c n -2ℓ n -2a n -2b n + 3c n -4ℓ n -a n + 2c n -2ℓ n 2a n + 2b n -4c n + 5ℓ n     or     -2b n + c n -2ℓ n -a n -2b n -2ℓ n -2a n -3b n + 2c n -4ℓ n 2a n + 4b n -2c n + 5ℓ n     .
We deduce the inequalities a n+1 , b n+1 , c n+1 , ℓ n+1 ≥ 0 directly from a n , b n ≥ 0 and ℓ n ≥ c n ≥ 0. Computing ℓ n+1 -2a n+1 = ℓ n + 2a 2n , we obtain ℓ n+1 > 2a n+1 . We compute then 5ℓ n+1 -6c n+1 to see that it is positive, and obtain either 13ℓ n -8c n + 4a n + 10b n > (13 -8 • 5 6 )ℓ n + 4a n + 10b n > 0 or ℓ n + 2c n + 2b n -2a n > 0. To get (2), it remains to see that ℓ n+1 ≥ 5ℓ n -4c n = 5 3 ℓ n + 4( 5 6 ℓ nc n ) > 5 3 ℓ n ≥ ( 5 3 ) n+1 .

For j = 2, 3, the situation is similar, with other data. The basis is now < [1, 0, 0, -1, 0], [2, 1, 0, -1, -2], [8, 2, -2, -2, -5], [9, 3, -2, -3, -6] >, which has signature < -2, -2, -6, 6 > and the actions of α, α We again denote by H the fourth basis vector, which is the only one with positive square, and compute by induction on n the vector

H n = ρ i n • • • • • ρ i 1 (H) for n ≥ 0 (with H 0 = H).
Writing

H n =     -a n -b n -c n ℓ n    
, we prove by induction on n the following inequalities:

(3)

a n , b n , c n , ℓ n ≥ 0 ℓ n > c n ℓ n ≥ 10 n ,
where the last one will yield the result, implying that ρ i k • • • • • ρ i 1 is a hyperbolic element of Aut(Z) of dynamical degree ≥ 10 k . Again, (3) is easily checked for n = 0, since ℓ 0 = 1, a 0 = b 0 = c 0 = 0. We assume the result true for n and prove it for n + 1. We have

H n+1 = ρ i n+1 (H n ) = α i n+1 β(H n ), which is equal to     -2a n -9b n + 18c n -24ℓ n -6a n -20b n + 36c n -51ℓ n -6a n -18b n + 35c n -48ℓ n 7a n + 22b n -42c n + 58ℓ n     or     -2a n -6b n + 18c n -21ℓ n -9a n -20b n + 54c n -66ℓ n -6a n -12b n + 35c n -42ℓ n 8a n + 17b n -48c n + 58ℓ n     .
We deduce the inequalities a n+1 , b n+1 , c n+1 , ℓ n+1 ≥ 0 directly from a n , b n ≥ 0 and ℓ n ≥ c n ≥ 0. Since ℓ n+1c n+1 is either equal to a n + 4b n -7b n + 10ℓ n or to 2a n + 5b n -13c n + 16ℓ n , it is positive. To get (3), it remains to see that

ℓ n+1 ≥ 58ℓ n -48c n = 10ℓ n + 48(ℓ n -c n ) ≥ 10ℓ n ≥ (10) n+1 .

  y 1 , y 2 ] are bihomogeneous polynomials, of bidegree (a, b), (a, b), (c, d), (c, d).

3. 3 .

 3 Parabolic embeddings. Recall that the morphism θ 0 defined in §3.2 can also be viewed as follow: M = a b c d → ax+b cx+d , χ(M)y ; it preserves the fibration (x, y) → x. Remembering that χ(S) = i and χ(RS) = 1 we have

  Pezzo surfaces of order 4. We now give explicit possibilities for the automorphism β ∈ Aut(Y ) of order 4. Y = (w : x : y : z) ∈ P(2, 1, 1, 1) w 2x 4 = 4 ∏ i=1 yz(y + z)(y + µz) = 0 β((w : x : y : z)) = (w : ix : y : z),

  The first case is satisfied by the examples of §3.1, §3.2, and §3.3. The second case is also possible, for any elliptic curve (see §4).

	Proof. Since S 2 commutes with SL(2, Z) the group G = θ(SL(2, Z)) is contained in the centraliser of the involution S 2 . If θ(S 2 ) is a Bertini or Geiser involution, the centraliser of θ(S 2 ) is finite ([BPV2], Corollary 2.3.6); as a consequence θ(S 2 ) is a de Jonquières
	involution. Assume that θ(S 2 ) is not linearisable; then θ(S 2 ) fixes (pointwise) a unique irreducible curve Γ of genus ≥ 1. The group G preserves Γ and the action of G on Γ gives the exact
	sequence
	1

Both authors supported by the Swiss National Science Foundation grant no PP00P2 128422 /1.


Julie Déserti

REPRESENTATIONS OF SOME LATTICES INTO THE GROUP OF ANALYTIC DIFFEOMORPHISMS OF THE SPHERE S 2 by

it is proved that any morphism from a subgroup of finite index of SL(n, Z) to the group of analytic diffeomorphisms of S 2 has a finite image as soon as n ≥ 5. The case n = 4 is also claimed to follow along the same arguments; in fact this is not straightforward and this case indeed needs a modification of the argument. In this paper we recall the strategy for n ≥ 5 and then focus on the case n = 4. 2010 Mathematics Subject Classification. -58D05, 58B25

1. The stabilizer of every point is trivial, i.e. the action of a non trivial element of f , g has no fixed point.

Introduction

After the works of Margulis ([14,[START_REF] Vinberg | Discrete subgroups of Lie groups[END_REF]) on the linear representations of lattices of simple, real Lie groups with R-rank larger than 1, some authors, like Zimmer, suggest to study the actions of lattices on compact manifolds ( [START_REF] Zimmer | Kazhdan groups acting on compact manifolds[END_REF][START_REF] Zimmer | On connection-preserving actions of discrete linear groups[END_REF][START_REF] Zimmer | Actions of semisimple groups and discrete subgroups[END_REF][START_REF] Zimmer | Lattices in semisimple groups and invariant geometric structures on compact manifolds[END_REF]). One of the main conjectures of this program is the following: let us consider a connected, simple, real Lie group G and let Γ be a lattice of G of R-rank larger than 1. If there exists a morphism of infinite image from Γ to the group of diffeomorphisms of a compact manifold M, then the R-rank of G is bounded by the dimension of M. There are a lot of contributions in that direction ( [3,[START_REF] Cantat | Version kählérienne d'une conjecture de Robert J. Zimmer[END_REF][START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF][START_REF] Déserti | Groupe de Cremona et dynamique complexe: une approche de la conjecture de Zimmer[END_REF][START_REF] Farb | Real-analytic actions of lattices[END_REF][START_REF] Franks | Area preserving group actions on surfaces[END_REF][START_REF] Ghys | Sur les groupes engendrés par des difféomorphismes proches de l'identité[END_REF][START_REF] Ghys | Actions de réseaux sur le cercle[END_REF][START_REF] Navas | Actions de groupes de Kazhdan sur le cercle[END_REF][START_REF] Polterovich | Growth of maps, distortion in groups and symplectic geometry[END_REF]). In this article we will focus on the embeddings of subgroups of finite index of SL(n, Z) into the group Diff ω (S 2 ) of real analytic diffeomorphisms of S 2 (see [START_REF] Ghys | Sur les groupes engendrés par des difféomorphismes proches de l'identité[END_REF]).

The article is organized as follows. First of all we will recall the strategy of [START_REF] Ghys | Sur les groupes engendrés par des difféomorphismes proches de l'identité[END_REF]: the study of the nilpotent subgroups of Diff ω (S 2 ) implies that such subgroups are metabelian. But subgroups of finite index of SL(n, Z), for n ≥ 5, contain nilpotent subgroups of length n -1 of finite index which are not metabelian; as a consequence Ghys gets the following statement.

Theorem A ([10]

). -Let Γ be a subgroup of finite index of SL(n, Z). As soon as n ≥ 5 there is no embedding of Γ into Diff ω (S 2 ).

To study nilpotent subgroups of Diff ω (S 2 ) one has to study nilpotent subgroups of Diff ω + (S 1 ) (see §2) and then nilpotent subgroups of the group of formal diffeomorphisms of C 2 (see §3). The last section is devoted to establish the following result.

Theorem B.

-Let Γ be a subgroup of finite index of SL(n, Z). As soon as n ≥ 4 there is no embedding of Γ into Diff ω (S 2 ).

The proof relies on the characterization, up to isomorphism, of nilpotent subalgebras of length 3 of the algebra of formal vector fields of C 2 which vanish at the origin.

Acknowledgements. -The author would like to thank Dominique Cerveau and Étienne Ghys for interesting discussions and advices.

Nilpotent subgroups of the group of analytic diffeomorphisms of S 1

Let G be a group; let us set

G (0) = G & G (i) = [G, G (i-1) ] ∀ i ≥ 1.
The group G is nilpotent if there exists an integer n such that G

(n) = {id}; the length of nilpotence of G is the smallest integer k such that G (k) = {id}. Set G (0) = G & G (i) = [G (i-1) , G (i-1) ] ∀ i ≥ 1.
The group G is solvable if G (n) = {id} for a certain n; the length of solvability of G is the smallest integer k such that G (k) = {id}.

We say that the group G (resp. algebra g) is metabelian if [G, G] (resp. [g, g]) is abelian.

Proposition 2.1 ([10]

). -Any nilpotent subgroup of Diff ω + (S 1 ) is abelian.

Proof. -Let G be a nilpotent subgroup of Diff ω + (S 1 ). Assume that G is not abelian; it thus contains a Heisenberg group

f , g, h | [ f , g] = h, [ f , h] = [g, h] = id .
The application "rotation number"

Diff ω + (S 1 ) → R/Z, ψ → lim n→+∞ ψ n (x) -x n
is not a morphism but its restriction to a solvable subgroup is ( [START_REF] Bavard | Longueur stable des commutateurs[END_REF]). Thus the rotation number of h is zero and the set Fix(h) of fixed points of h is non-empty and finite. Considering some iterates of f and g instead of f and g one can assume that f and g fix any point of Fix(h). The set of fixed points of a non trivial element of f , g is finite and invariant by h so the action of f , g is free (1) on each component of S 1 \ Fix(h). But the action of a free group on R is abelian: contradiction.

Nilpotent subgroups of the group of formal diffeomorphisms of C 2

Let us denote Diff(C 2 , 0) the group of formal diffeomorphisms of C 2 , i.e. the formal completion of the group of germs of holomorphic diffeomorphisms at 0. For any i let Diff i be the quotient of Diff(C 2 , 0) by the normal subgroups of formal diffeomorphisms tangent to the identity with multiplicity i; it can be viewed as the set of jets of diffeomorphisms at order i with the law of composition with truncation at order i. Note that Diff i is a complex linear algebraic group. One can see Diff(C 2 , 0) as the projective limit of the Diff i 's: Diff(C 2 , 0) = lim ← Diff i . Let us denote by χ(C 2 , 0) the algebra of formal vector fields in C 2 vanishing at 0. One can define the set χ i of the i-th jets of vector fields; one has lim

← χ i = χ(C 2 , 0).
Let O(C 2 ) be the ring of formal series in two variables and let K(C 2 ) be its fraction field; O i is the set of elements of O(C 2 ) truncated at order i.

The family exp i : χ i → Diff i i is filtered, i.e. compatible with the truncation. We then define the exponential application as follows: exp = lim

← exp i : χ(C 2 , 0) → Diff(C 2 , 0).
As in the classical case, if X belongs to χ(C 2 , 0), then exp(X ) can be seen as the "flow at time 1" of X .

Indeed an element X i of χ i can be seen as a derivation of O i ; so it can be written S i + N i where S i and N i are two semi-simple, resp. nilpotent derivations which commute. Taking the limit, one gets X = S + N where S is a semi-simple vector field and N a nilpotent one and [S, N] = id (see [START_REF] Martinet | Normalisation des champs de vecteurs holomorphes (d'après A.-D Brjuno)[END_REF]). A semi-simple vector field is a formal vector field conjugate to a diagonal linear vector field which is complete. A vector field is nilpotent if and only if its linear part is; let us remark that the usual flow ϕ t of a nilpotent vector field is polynomial in t

ϕ t (x) = ∑ I P I (t)x I , P I ∈ (C[t]) 2
so ϕ 1 (x) is well defined. As a consequence exp(tX ) = exp(tS) exp(tN) is well defined for t = 1. Note that the Jordan decomposition is purely formal: if X is holomorphic, S and N are not necessary holomorphic.

Proposition 3.1 ([10]

). -Any nilpotent subalgebra of χ(C 2 , 0) is metabelian.

Proof. -Let l be a nilpotent subalgebra of χ(C 2 , 0) and let Z(l) be its center. Since χ(C 2 , 0) ⊗ K(C 2 ) is a vector space of dimension 2 over K(C 2 ) one has the following alternative:

• the dimension of the subspace generated by

Z(l) in χ(C 2 , 0) ⊗ K(C 2 ) is 1;
• the dimension of the subspace generated by

Z(l) in χ(C 2 , 0) ⊗ K(C 2 ) is 2.
Let us study these different cases. Under the first assumption there exists an element X of Z(l) having the following property: any vector field of Z(l) can be written uX with u in K(C 2 ). Let us consider the subalgebra g of l given by

g = X ∈ l | ∃ u ∈ K(C 2 ), X = uX .
Since X belongs to Z(l), the algebra g is abelian; it is also an ideal of l. Let us assume that l is not abelian: let Y be an element of l whose projection on l/g is non trivial and central. Any vector field of l can be written as uX + vY with u, v in K(C 2 ). As X belongs to Z(l) and Y is central modulo g one has

X (u) = X (v) = Y (v) = 0.
The vector fields ∂ ∂x and ∂ ∂y being some linear combinations of X and Y with coefficients in K(C 2 , 0), the partial derivatives of v are zero so v is a constant. Therefore [l, l] ⊂ g; but g is abelian so l is metabelian.

In the second case Z(l) has two elements X and Y which are linearly independent on K(C 2 ). Any vector field of l can be written as uX + vY with u and v in K(C 2 ). Since X and Y belong to Z(l) one has

X (u) = X (v) = Y (u) = Y (v) = 0.
As a consequence u and v are constant, i.e. l ⊂ {uX + vY | u, v ∈ C}; in particular l is abelian.

Proposition 3.2 ([10]). -Any nilpotent subgroup of Diff(C 2 , 0) is metabelian. Proof. -Let G be a nilpotent subgroup of Diff(C 2 , 0) of length k. Let us denote G i the projection of G on Diff i . The Zariski closure G i of G i in Diff i is an algebraic nilpotent subgroup of length k. It is sufficient to prove that G i is metabelian.
Since G i is a complex algebraic subgroup it is the direct product of the subgroup G i,u of its unipotent elements and the subgroup G i,s of its semi-simple elements (see for example [START_REF] Borel | Linear algebraic groups[END_REF]).

An element of Diff i is unipotent if and only if its linear part, which is in GL(2, C), is; so G i,s projects injectively onto a nilpotent subgroup of GL(2, C). Therefore G i,s is abelian.

Let us now consider G i,u ; this group is the exponential of a nilpotent Lie algebra l i of χ i of length k. Taking the limit one thus obtains the existence of a nilpotent subalgebra l of χ(C 2 , 0) of length k such that exp(l) projects onto G i,u for any i. According to Proposition 3.1 the subalgebra l and thus G i,u are metabelian. Proof. -Let G be a nilpotent subgroup of Diff ω (S 2 ); up to finite index one can assume that the elements of G preserve the orientation. Let φ be a non trivial element of G which commutes with G. Let Fix(φ) be the set of fixed points of φ; it is a non empty analytic subspace of S 2 invariant by G. If p is an isolated fixed point of φ, then the orbit of p under the action of G is finite. So it is sufficient to study the case where Fix(φ) only contains curves; there are thus two possibilities:

Nilpotent subgroups of the group of analytic diffeomorphisms of S 2

• Fix(φ) is a singular analytic curve whose set of singular points is a finite orbit for G;

• Fix(φ) is a smooth analytic curve, not necessary connected. One of the connected component of S 2 \ Fix(φ) is a disk denoted D. Any subgroup Γ of finite index of G which contains φ fixes D. Let us consider an element γ of Γ and a fixed point m of γ which is in D. By construction φ has no fixed point in D so according to the Brouwer Theorem (φ k (m)) k has a limit point on the boundary ∂D of D. Therefore γ has at least one fixed point on ∂D. The group Γ thus acts on the circle ∂D and any of its elements has a fixed point on D. Then Γ has a fixed point on ∂D (Proposition 2.1).

Theorem 4.2 ([10]

). -Any nilpotent subgroup of Diff ω (S 2 ) is metabelian.

Proof. -Let G be a nilpotent subgroup of Diff ω (S 2 ) and let Γ be a subgroup of finite index of G having a fixed point m (such a subgroup exists according to Proposition 4.1). One can embed Γ into Diff(R 2 , 0), and so into Diff(C 2 , 0), by considering the jets of infinite order of elements of Γ in m. According to Proposition 3.2 the group Γ is metabelian. One can assume that G is a finitely generated group. Let us first assume that G has no element of finite order. Then G is a cocompact lattice of the nilpotent, simply connected Lie group G ⊗ R (see [START_REF] Raghunathan | Discrete subgroups of Lie groups[END_REF]). The group G is metabelian if and only if G ⊗ R is; but Γ is metabelian so G ⊗ R also.

Finally let us consider the case where G has at least one element of finite order. The set of such elements is a normal subgroup of G which thus intersects non trivially the center Z(G) of G. Let us consider a non trivial element φ of Z(G) which has finite order. Let us recall that a finite group of diffeomorphisms of the sphere is conjugate to a group of isometries. Denote by G + the subgroup of elements of G which preserve the orientation. It is thus sufficient to prove that G + is metabelian; indeed if φ does not preserve the orientation, φ has order 2 and G = Z/2Z × G + . So let us assume that φ preserves the orientation; φ is conjugate to a direct isometry of S 2 and has exactly two fixed points on the sphere. The group G has thus an invariant set of two elements. By considering germs in the neighborhood of these two points, one gets that G can be embedded into 2 • Diff(R 2 , 0) (2) and thus into 2 • Diff(C 2 , 0):

1 -→ Diff(C 2 , 0) -→ 2 • Diff(C 2 , 0) -→ Z/2Z -→ 0.
Let us remark that 2 • Diff(C 2 , 0) is the projective limit of the algebraic groups 2 • Diff i . The end of the proof is thus the same as the proof of Proposition 3.2 except that the subgroup of the semi-simple elements of 2 • Diff i embeds now in 2 • GL(2, C); it is metabelian because it contains an abelian subgroup of index 2.

Let Γ be a subgroup of finite index of SL(n, Z) for n ≥ 5. Since Γ contains nilpotent subgroups of finite index of length n -1 (for example the group of upper triangular unipotent matrices) which are not metabelian one gets the following statement.

Corollary 4.3 ([10]

). -Let Γ be a subgroup of finite index of SL(n, Z); as soon as n ≥ 5 there is no embedding of Γ into Diff ω (S 2 ).

Nilpotent subgroups of length 3 of the group of analytic diffeomorphisms of S 2

Let us precise Proposition 3.1 for nilpotent subalgebras of length 3 of χ(C 2 , 0). Let l be such an algebra. The center Z(l) of l generates a subspace of dimension at most 1 of χ(C 2 , 0) ⊗ K(C 2 ), for else l would be abelian (Proposition 3.1) and this is impossible under our assumptions. So let us assume that the dimension of the subspace generated by Z(l) in χ(C 2 , 0) ⊗ K(C 2 ) is 1. There exists an element X in Z(l) with the following property: any element of Z(l) can be written uX with u in K(C 2 ). Let g denote the abelian ideal of l defined by

g = X ∈ l ∃ u ∈ K(C 2 ), X = uX .
By hypothesis l is not abelian. Let Y be in l; assume that its projection onto l/g is a non trivial element of Z(l/g). Any vector field of l can be written

uX + vY, u, v ∈ K(C 2 ).
Since X , resp. Y belongs to Z(l), resp. Z(l/g) and since the length of l is 3, one has

X (u) = Y 3 (u) = X (v) = Y (v) = 0. ( 5.1)
If X and Y are non singular, one can choose formal coordinates x and y such that X = ∂ ∂x and Y = ∂ ∂y . The previous conditions can be thus translated as follows: v is a constant and u is a polynomial in y of degree 2. We will see that we have a similar property without assumption on X and Y .

2. Let G be a group and let q be a positive integer; q • G denotes the semi-direct product of G q by Z/qZ under the action of the cyclic permutation of the factors.

Lemma 5.1. -Let X and Y be two vector fields of χ(C 2 , 0) that commute and are not colinear. One can assume that (X ,Y ) = ∂ ∂ x , ∂ ∂ ỹ where x and y are two independent variables in a Liouvillian extension of K(C 2 , 0).

Proof. -Since X and Y are non colinear, there exist two 1-forms α, β with coefficients in K(C 2 ) such that

α(X ) = 1, α(Y ) = 0, β(X ) = 0, β(X ) = 1.
The vector fields X and Y commute if and only if α and β are closed (this statement of linear algebra is true for convergent meromorphic vector fields and is also true in the completion). The 1-form α is closed so according to [START_REF] Cerveau | Formes intégrables holomorphes singulières[END_REF] one has

α = r ∑ i=1 λ i d φ i φ i + d ψ 1 ψ 2 = d r ∑ i=1 λ i log φ i + ψ 1 ψ 2
where ψ 1 , ψ 2 and the φ i denote some formal series and the λ i some complex numbers. One has a similar expression for β. So there exists a Liouvillian extension κ of K(C 2 ) having two elements x and y with α = d x and β = d y. One thus has

X ( x) = 1, X ( y) = 0, Y ( x) = 0, Y ( y) = 1.
From (5.1) one gets: v is a constant and u is a polynomial in y of degree 2; so one proves the following statement.

Proposition 5.2. -Let l be a nilpotent subalgebra of χ(C 2 , 0) of length 3. Then l is isomorphic to a subalgebra of

n = P( y) ∂ ∂ x + α ∂ ∂ y α ∈ C, P ∈ C[ y], deg P = 2 .
Remark 5.3. -We use a real version of this statement whose proof is an adaptation of the previous one: a nilpotent subalgebra l of length 3 of χ(R 2 , 0) is isomorphic to a subalgebra of

n = P( y) ∂ ∂ x + α ∂ ∂ ỹ α ∈ R, P ∈ R[
(ρ(U(4, Z))) is a subgroup of λ t 0 λ λ ∈ R * , t ∈ R .
Up to index 2 one can thus assume that j 1 • ρ takes values in the connected, simply connected group T defined by

T = λ t 0 λ λ, t ∈ R, λ > 0 .
Let us set Diff i (T) = f ∈ Diff i | j 1 ( f ) ∈ T ; the group Diff i (T) is a connected, simply connected, nilpotent and algebraic group. The morphism ρ i : U(4, Z) → Diff i can be extended to a unique continuous morphism ρ i : U(4, R) → Diff i (T) (see [START_REF] Malcev | On a class of homogeneous spaces[END_REF][START_REF] Malcev | On a class of homogeneous spaces[END_REF]) so to an algebraic morphism (3) . Let us note that ρ i (U(4, Z)) is an algebraic subgroup of Diff i (T) which contains ρ i (U(4, Z)); in particular H i = ρ i (U(4, Z)) ⊂ ρ i (U(4, R)). By construction the family (H i ) i is filtered; since the extension is unique, the family ( ρ i ) i is also filtered. Therefore K = lim ← H i is well defined. Since ρ is injective, H is a nilpotent subgroup of length 3; as H ⊂ K and as any H i is nilpotent of length at most 3 the group K is nilpotent of length at most 3. For i sufficiently large ρ i (U(4, R)) is nilpotent of length 3; this group is connected so its Lie algebra is also nilpotent of length 3. Therefore the image of

D ρ := lim ← D ρ i : u(4, R) → χ(R 2 , 0)
is isomorphic to n (Proposition 5.2). So there exists a surjective map ψ from u(4, R) onto n. The kernel of ψ is an ideal of u(4, R) of dimension 2; hence ker ψ = δ 14 , aδ 13 + bδ 24 where the δ i j denote the Kronecker matrices. One concludes by remarking that dim Z(u(4, R)/ ker ψ) = 2 whereas dim Z(n) = 1.

Corollary 5.5. -The image of a morphism from a subgroup of SL(n, Z) of finite index to Diff ω (S 2 ) is finite as soon as n ≥ 4.

Proposition 4 . 1 (

 41 [START_REF] Ghys | Sur les groupes engendrés par des difféomorphismes proches de l'identité[END_REF]). -Any nilpotent subgroup of Diff ω (S 2 ) has a finite orbit.

  Proof. -Let U(4, Z) (resp. U(4, R)) be the subgroup of unipotent upper triangular matrices of SL(4, Z) (resp. SL(4, R)); it is a nilpotent subgroup of length 3. Assume that there exists an embedding from a subgroup Γ of finite index of SL(4, Z) into Diff ω (S 2 ). Up to finite index Γ contains U(4, Z). Let us set H = ρ(U(4, Z)). Up to finite index H has a fixed point (Proposition 4.1). One can thus see H as a subgroup of Diff(R 2 , 0) ⊂ Diff(R 2 , 0) up to finite index.Let us denote j 1 the morphism from Diff(R 2 , 0) to Diff i . Up to conjugation j 1

y], deg P = 2 . Theorem 5.4. -Let Γ be a subgroup of finite index of SL(n, Z); as soon as n ≥ 4 there is no embedding of Γ into Diff ω (S 2 ).

The author is supported by the Swiss National Science Foundation grant no PP00P2_128422 /1 and the A.N.R. project "BirPol".