Julie Déserti

ON SOLVABLE SUBGROUPS OF THE CREMONA GROUP

Keywords:

The Cremona group Bir(P 2 C ) is the group of birational self-maps of P 2 C . Using the action of Bir(P 2 C ) on the Picard-Manin space of P 2 C we characterize its solvable subgroups. If G ⊂ Bir(P 2 C

) is solvable, non virtually abelian, and infinite, then up to finite index: either any element of G is of finite order or conjugate to an automorphism of P 2 C , or G preserves a unique fibration that is rational or elliptic, or G is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.

We also give some corollaries.

INTRODUCTION

We know properties on finite subgroups ( [START_REF] Dolgachev | Finite subgroups of the plane Cremona group[END_REF]), finitely generated subgroups ( [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]), uncountable maximal abelian subgroups ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF]), nilpotent subgroups ( [START_REF] Déserti | Sur les sous-groupes nilpotents du groupe de Cremona[END_REF]) of the Cremona group. In this article we focus on solvable subgroups of the Cremona group.

Let G be a group. Recall that [g, h] = ghg -1 h -1 denotes the commutator of g and h. If Γ 1 and Γ 2 are two subgroups of G, then [Γ 1 , Γ 2 ] is the subgroup of G generated by the elements of the form [g, h] with g ∈ Γ 1 and h ∈ Γ 2 . We define the derived series of G by setting

G (0) = G G (n+1) = [G (n) , G (n) ] ∀ n ≥ 0.
The soluble length ℓ(G) of G is defined by

ℓ(G) = min k ∈ N ∪ {0} | G (k) = {id}
with the convention: min / 0 = ∞. We say that G is solvable if ℓ(G) < ∞. The study of solvable groups started a long time ago, and any linear solvable subgroup is up to finite index triangularizable (Lie-Kolchin theorem, [START_REF] Kargapolov | Fundamentals of the theory of groups, volume 62 of Graduate Texts in Mathematics[END_REF]Theorem 21.1.5]). The assumption "up to finite index" is essential: for instance the subgroup of PGL(2, C) generated by 1 0 1 -1 and -1 1 0 1 is isomorphic to S 3 so is solvable but is not triangularizable.

Theorem A. Let G be an infinite, solvable, non virtually abelian subgroup of Bir(P 2 C ). Then, up to finite index, one of the following holds: [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF] any element of G is either of finite order, or conjugate to an automorphism of P 2 C ; (2) G preserves a unique fibration that is rational, in particular G is, up to conjugacy, a subgroup of PGL(2, C(y)) ⋊ PGL(2, C);

(3) G preserves a unique fibration that is elliptic;

(4) G is, up to birational conjugacy, contained in the group generated by (x p y q , x r y s ), (αx, βy) | α, β ∈ C * where M = p q r s denotes an element of GL(2, Z) with spectral radius

> 1.
The group G preserves the two holomorphic foliations defined by the 1forms α 1 xdy+β 1 ydx and α 2 xdy+β 2 ydx where (α 1 , β 1 ) and (α 2 , β 2 ) denote the eigenvectors of t M.

Furthermore if G is uncountable, case 3. does not hold.

Examples.

• Denote by S 3 the group generated by 1 0 1 -1 and -1 1 0 1 .

As we recall before S 3 ≃ S 3 . Consider now the subgroup G of Bir(P 2 C ) whose elements are the monomial maps (x p y q , x r y s ) with p q r s ∈ S 3 . Then any element of G has finite order, and G is solvable; it gives an example of case 1.

• Other examples that illustrate case 1. are the following groups

(αx + βy + γ, δy + ε) | α, δ ∈ C * , β, γ, ε ∈ C ⊂ Aut(P 2 C )
, and

E = (αx + P(y), βy + γ) | α, β ∈ C * , γ ∈ C, P ∈ C[y] ⊂ Aut(C 2 ).
• The centralizer of a birational map of P 2 C that preserves a unique fibration that is rational is virtually solvable ([9, Corollary C]); this example falls in case 2 (see §3.2).

• In [10, Proposition 2.2] Cornulier proved that the group

(x + 1, y), (x, y + 1), (x, xy)
is solvable of length 3, and is not linear over any field; this example falls in case 2. The invariant fibration is given by x = cst.

Remark.

In case 1. if there exists an integer d such that degφ ≤ d for any φ in G, then there exist a smooth projective variety M and a birational map ψ : M P 2

C

such that ψ -1 Gψ is a solvable subgroup of Aut(M) (see §3.3). But there is some solvable subgroups G with only elliptic elements that do not satisfy this property:

the group E introduced in Examples 1. Let us mention an other example: Wright constructs abelian subgroups H of Aut(C 2 ) such that any element of H is of finite order, H is unbounded and does not preserve any fibration ( [START_REF] Wright | Abelian subgroups of Aut k (k[X, Y ]) and applications to actions on the affine plane[END_REF]).

In §3 we prove Theorem A: we first assume that our solvable, infinite and non virtually abelian subgroup G contains a hyperbolic map, then that it contains a twist and no hyperbolic map, and finally that all elements of G are elliptic. In the last section ( §4) we also

• recover the following fact: if G is an infinite nilpotent subgroup of Bir(P 2 C ), then G does not contain a hyperbolic map;

• remark that we can bound the soluble length of a nilpotent subgroup of Bir(P 2 C ) by the dimension of P 2 C as Epstein and Thurston did in the context of Lie algebras of rational vector fields on a connected complex manifold; • give a negative answer to the following question of Favre: does any solvable and finitely generated subgroup G of Bir(P 2 C ) contain a subgroup of finite index whose commutator subgroup is nilpotent ? if we assume that [G, G] is not a torsion group;

• give a description of the embeddings of the solvable Baumslag-Solitar groups into the Cremona group.

Acknowledgments. I would like to thank D. Cerveau for always having time to listen to me, for his precious advices, and for pointing out [START_REF] Martelo | Derived length of solvable groups of local diffeomorphisms[END_REF]. Thanks to S. Cantat for helpful remarks and comments, and also to J. Ribon for the example of §4.3. I would like also thank P. Sad for his permanent invitation at IMPA where I found the quietness to put my ideas in order. Thanks to the referee whose comments and suggestions help me to improve the presentation.

SOME PROPERTIES OF THE BIRATIONAL MAPS

First definitions. Let S be a projective surface. We will denote by Bir(S ) the group of birational self-maps of S; in the particular case of the complex projective plane the group Bir(P 2 C ) is called Cremona group. Take φ in Bir(S ), we will denote by Indφ the set of points of indeterminacy of φ; the codimension of Ind φ is ≥ 2.

A birational map from P 2 C into itself can be written

(x : y : z) φ 0 (x, y, z) : φ 1 (x, y, z) : φ 2 (x, y, z)
where the φ i 's denote some homogeneous polynomials of the same degree and without common factors of positive degree. The degree of φ is equal to the degree of the φ i 's. Let φ be a birational map of P 2 C . One can define the dynamical degree of φ as

λ(φ) = lim n→+∞ (deg φ n ) 1/n .
More generally let S be a projective surface, and φ : S S be a birational map.

Take any norm || • || on the Néron-Severi real vector space N 1 (S ). If φ * is the induced action by φ on N 1 (S ), we can define

λ(φ) = lim n→+∞ ||(φ n ) * || 1/n .
Remark that this quantity is a birational invariant: if ψ : S S ′ is a birational map, then λ(ψφψ -1 ) = λ(φ).

Classification of birational maps. The algebraic degree is not a birational invariant, but the first dynamical degree is; more precisely one has a classification of birational maps based on the degree growth. Before stating it, let us first introduce the following definitions. Let φ be an element of Bir(P ([15, 20, 3]). Let φ be an element of Bir(P 2 C ). Then one and only one of the following cases holds

• φ is elliptic, furthermore if φ is of infinite order, then φ is up to birational conjugacy an automorphism of P 2 C , • φ is a Jonquières twist, φ preserves a unique fibration that is rational and every conjugate of φ is not an automorphism of a projective surface, • φ is a Halphen twist, φ preserves a unique fibration that is elliptic and φ is conjugate to an automorphism of a projective surface, • φ is a hyperbolic map. In the three first cases, λ(φ) = 1, in the last one λ(φ) > 1.

The Picard-Manin and bubble spaces. Let S, and S i be complex projective surfaces. If π : S 1 → S is a birational morphism, one gets π * : N 1 (S ) → N 1 (S 1 ) an embedding of Néron-Severi groups. Take two birational morphisms π 1 : S 1 → S and π 2 : S 2 → S; the morphism π 2 is above π 1 if π -1 1 π 2 is regular. Starting with two birational morphisms one can always find a third one that covers the two first. Therefore the inductive limit of all groups N 1 (S i ) for all surfaces S i above S is well-defined; it is the Picard-Manin space Z S of S. For any birational map π, π * preserves the intersection form and maps nef classes to nef classes hence the limit space Z S is endowed with an intersection form of signature (1, ∞) and a nef cone.

Let S be a complex projective surface. Consider all complex and projective surfaces S i above S, that is all birational morphisms π i : S i → S. If p (resp. q) is a point of a complex projective surface S 1 (resp. S 2 ), and if

π 1 : S 1 → S (resp. π 2 : S 2 → S) is a birational morphism, then p is identified with q if π -1 1 π 2 is a local isomorphism
in a neighborhood of q that maps q onto p. The bubble space B(S) is the union of all points of all surfaces above S modulo the equivalence relation induced by this identification. If p belongs to B(S) represented by a point p on a surface S i → S,

denote by E p the exceptional divisor of the blow-up of p and by e p its divisor class viewed as a point in Z S . The following properties are satisfied

e p • e q = 0 if p = q e p • e p = -1
Hyperbolic space H S . Embed N 1 (S ) as a subgroup of Z S ; this finite dimensional lattice is orthogonal to e p for any p ∈ B(S), and

Z S = D + ∑ p∈B(S ) a p e p | D ∈ N 1 (S ), a p ∈ R .
The completed Picard-Manin space Z S of S is the L 2 -completion of Z S ; in other words

Z S = D + ∑ p∈B(S ) a p e p | D ∈ N 1 (S ), a p ∈ R, ∑ a 2 p < +∞ .
The intersection form extends as an intersection form with signature (1, ∞) on Z S .

Let

Z + S = d ∈ Z S | d • c ≥ 0 ∀ c ∈ Z S } be the nef cone of Z S and LZ S = d ∈ Z S | d • d = 0 be the light cone of Z S .
The hyperbolic space H S of S is then defined by

H S = d ∈ Z + S | d • d = 1 .
Let us remark that H S is an infinite dimensional analogue of the classical hyperbolic space H n . The distance on H S is defined by

cosh(dist(d, d ′ )) = d • d ′ ∀ d, d ′ ∈ H S .
The geodesics are intersections of H S with planes. The projection of H S onto P(Z S ) is one-to-one, and the boundary of its image is the projection of the cone of isotropic vectors of Z S . Hence

∂H S = R + d | d ∈ Z + S , d • d = 0 .
Isometries of H S . If π : S ′ → S is a birational morphism, we get a canonical isometry π * (and not only an embedding) between H S and H S ′ . This allows to define an action of Bir(S ) on H S . Consider a birational map φ on a complex projective surface S.

There exists a surface S ′ , and π 1 : S ′ → S, π 2 : S ′ → S two morphisms such that

φ = π 2 π -1 1 . One can define φ • by φ • = (π * 2 ) -1 π * 1 ;
in fact, one gets a faithful representation of Bir(S ) into the group of isometries of H S (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]).

The isometries of H S are classified in three types ( [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF][START_REF] Ghys | Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF]). The translation length of an isometry φ • of H S is defined by

L(φ • ) = inf dist(p, φ • (p)) | p ∈ H S If
(φ • ) in Z S such that φ • (ω(φ • )) = λ(φ) ω(φ • ) φ • (α(φ • )) = 1 λ(φ) α(φ • )
The axis Ax(φ • ) of φ • is the intersection of H S with the plane containing ω(φ • ) and α(φ • ); furthermore φ • acts as a translation of length L(φ [START_REF] Cantat | Normal subgroups in the Cremona group[END_REF]Remark 4.5]). For all p in H S one has

• ) = log λ(φ) along Ax(φ • ) (see
lim k→+∞ φ -k • (p) λ(φ) = α(φ • ) lim k→+∞ φ k • (p) λ(φ) = ω(φ • ).
When the infimum is not realized, L(φ • ) = 0 and φ • is parabolic: φ • fixes a unique line in LZ S ; this line is fixed pointwise, and all orbits φ n • (p) in H S accumulate to the corresponding boundary point when n goes to ±∞.

There is a strong relationship between this classification and the classification of birational maps of the complex projective plane ([6, Theorem 3.6]): if φ is an element of Bir(P 2 C ), then • φ • is an elliptic isometry if and only if φ is an elliptic map;

• φ • is a parabolic isometry if and only if φ is a twist;

• φ • is a hyperbolic isometry if and only if φ is a hyperbolic map. Tits alternative. Cantat proved the Tits alternative for the Cremona group ([6, Theorem C]): let G be a finitely generated subgroup of Bir(P 2 C ), then • either G contains a free non abelian subgroup,

• or G contains a subgroup of finite index that is solvable.

As a consequence he studied finitely generated and solvable subgroups of Bir(P 2 C ) without torsion ([6, Theorem 7.3]): let G be such a group, there exists a subgroup G 0 of G of finite index such that • either G 0 is abelian,

• or G 0 preserves a foliation.

PROOF OF THEOREM A

3.1. Solvable groups of birational maps containing a hyperbolic map. Let us recall the following criterion (for its proof see for example [START_REF] De La Harpe | Topics in geometric group theory[END_REF]) used on many occasions by Klein, and also by Tits ( [START_REF] Tits | Free subgroups in linear groups[END_REF]) known as Ping-Pong Lemma: let H be a group acting on a set X, let Γ 1 , Γ 2 be two subgroups of H, and let Γ be the subgroup generated by Γ 1 and Γ 2 . Assume that Γ 1 contains at least three elements, and Γ 2 at least two elements. Suppose that there exist two non-empty subsets X 1 , X 2 of X such that

X 2 ⊂ X 1 , γ(X 2 ) ⊂ X 1 ∀ γ ∈ Γ 1 {id}, γ ′ (X 1 ) ⊂ X 2 ∀ γ ′ ∈ Γ 2 {id}.
Then Γ is isomorphic to the free product

Γ 1 * Γ 2 .
such that ω(φ • ), α(φ • ) = ω(ψ • ), α(ψ • ) . Proof. Assume by contradiction that ω(φ • ), α(φ • ) = ω(ψ • ), α(ψ • ) .
Then the Ping-Pong argument implies that there exist two integers n and m such that ψ n and φ m generate a subgroup of G isomorphic to the free group F 2 (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]). But φ, ψ is a solvable group: contradiction.

Let G be an infinite solvable, non virtually abelian, subgroup of Bir(P 2 C ). Assume that G contains a hyperbolic map φ. Let α(φ • ) and ω(φ • ) be the two fixed points of φ • on ∂H P 2

C

, and Ax(φ • ) be the geodesic passing through these two points. As G is solvable there exists a subgroup of G of index ≤ 2 that preserves α(φ • ), ω(φ • ), and Ax(φ • ) (see [START_REF] Cantat | Sur les groupes de transformations birationnelles des surfaces[END_REF]Theorem 6.4]); let us still denote by G this subgroup. Note that there is no twist in G since a parabolic isometry has a unique fixed point on ∂H P 2

C

. One has a morphism κ : G → R >0 such that ψ • (ℓ) = κ(ψ)ℓ for any ℓ in Z P 2

C

lying on Ax(φ • ). The kernel of κ is an infinite subgroup that contains only elliptic maps. Indeed the set of elliptic elements of G coincides with ker κ; and [G, G] ⊂ ker κ so if ker κ is finite, G is abelian up to finite index which is by assumption impossible.

Gap property. If ψ is an hyperbolic birational map of G, then κ(ψ) = L(ψ • ) = log λ(ψ). Recall that λ(φ) is an algebraic integer with all Galois conjugates in the unit disk, that is a Salem number, or a Pisot number. The smallest known number is the Lehmer number λ L ≃ 1, 176 which is a root of X 10 + X 9 -X 7 -X 6 -X 5 -X 4 -X 3 + X + 1. Blanc and Cantat prove in [2, Corollary 2.7] that there is a gap in the dynamical spectrum

Λ = λ(φ) | φ ∈ Bir(P 2 C )}: there is no dynamical degree in ]1, λ L [.
The gap property implies that in fact κ is a morphism from G to a subgroup of R >0 isomorphic to Z. ). Let G toric be its image.

Elliptic subgroups of the

One can now apply [START_REF] Delzant | Kähler groups, real hyperbolic spaces and the Cremona group[END_REF]Theorem 4] that says that if there exists a short exact sequence 1 -→ A -→ N -→ B -→ 1 where N ⊂ Bir(P 2

C ) contains at least one hyperbolic element, and A ⊂ Bir(P 2 C ) is an infinite and elliptic 1 group, then N is up to conjugacy a subgroup of G toric . Hence up to birational conjugacy G ⊂ G toric . Recall now that if ψ is a hyperbolic map of the form (x a y b , x c y d ), then to preserve α(ψ • ) and ω(ψ

still denoted α(φ • ) that represents α(φ • ). Take ϕ ∈ G such that ϕ(α(φ • )) = α(φ • ). Then ψ = ϕφϕ -1 is parabolic and fixes the unique element α(ψ • ) of LZ P 2 C proportional to ϕ(α(φ • )). Take ε > 0 such that U(α(φ • ), ε) ∩ U(α(ψ • ), ε) = / 0 where U(α,ε) = ℓ ∈ LZ P 2 C | α • ℓ < ε . Since ψ • is parabolic, then for n large enough ψ n • U(α(φ • ), ε) is included in a U(α(ψ • ), ε). For m sufficiently large φ m • ψ n • U(α(φ • ), ε) ⊂ U(α(φ • ), ε/2) U(α(φ • ), ε) ; hence φ m • ψ n
• is hyperbolic. You can by this way build two hyperbolic maps whose sets of fixed points are distinct: this gives a contradiction with Lemma 3.1. So for any ϕ ∈ G one has : α(φ • ) = α(ϕ • ); one can thus state the following result. ) that contains a twist φ. Then

• if φ is a Jonquières twist, then G preserves a rational fibration, that is up to birational conjugacy G is a subgroup of PGL(2, C(y)) ⋊ PGL(2, C), • if φ is a Halphen twist, then G preserves an elliptic fibration. Note that if G is a subgroup of Bir(P 2 C ) that preserves an elliptic fibration, then G is countable ( [START_REF] Cantat | Dynamique des automorphismes des surfaces complexes compactes[END_REF]). Let us explain briefly why. A smooth rational projective surface S is a Halphen surface if there exists an integer m > 0 such that the linear system C . This fibration is relatively minimal in the sense that there is no (-1)-curve contained in a fiber. The following properties are equivalent:

| -mK S | is of dimension 1,
• S is a Halphen surface of index m,

• there exists an irreducible pencil of curves of degree 3m with 9 base points of multiplicity m in P 2

C such that S is the blow-up of the 9 base points and |-mK S | is the proper transform of this pencil (the base points set may contain infinitely near points). C ) preserves a unique fibration that is rational then one can assume that up to birational conjugacy this fibration is given, in the affine chart z = 1, by y = cst. If φ preserves y = cst fiberwise, then

• φ is contained in a maximal abelian subgroup denoted Ab(φ) that preserves y = cst fiberwise ( [START_REF] Déserti | Sur les automorphismes du groupe de Cremona[END_REF]), • the centralizer of φ is a finite extension of Ab(φ) (see [START_REF] Cerveau | Centralisateurs dans le groupe de Jonquières[END_REF]Theorem B]). This allows us to establish that if φ preserves a fibration not fiberwise, then the centralizer of φ is virtually solvable. For instance if φ = x + a(y), y + 1 (resp. b(y)x, βy or x + a(y), βy with β ∈ C * of infinite order) preserves a unique fibration, then the centralizer of φ is solvable and metabelian ([9, Propositions 5.1 and 5.2]). The total degree tdeg φ of φ ∈ Bir(M) is defined as the volume of Γ φ with respect to the fixed metric on M × M:

tdeg φ = Γ φ π * 1 κ + π * 2 κ n = M Ind φ κ + φ * κ n .
Let d ≥ 1 be a natural integer, and set

Bir d (M) = φ ∈ Bir(M) | tdeg φ ≤ d .
A subgroup G of Bir(M) has bounded degree if it is contained in Bir d (M) for some d ∈ N * . Any subgroup G of Bir(M) that has bounded degree can be regularized, that is up to birational conjugacy all indeterminacy points of all elements of G disappear simultaneously: Theorem 3.7 ([27]). Let M be a complex projective variety, and let G be a subgroup of Bir(M). If G has bounded degree, there exists a smooth, complex, projective variety M ′ , and a birational map ψ : M ′ M such that ψ -1 Gψ is a subgroup of Aut(M ′ ).

The proof of this result can be found in [START_REF] Huckleberry | Actions of groups of birationally extendible automorphisms[END_REF][START_REF] Zaitsev | Regularization of birational group operations in the sense of Weil[END_REF]; an heuristic idea appears in [START_REF] Cantat | Morphisms between Cremona groups, and characterization of rational varieties[END_REF]. ). Let us recall that if G is a group, the descending central series of G is defined by

C 0 G = G C n+1 G = [G,C n G] ∀ n ≥ 0.
We say that G is nilpotent if there exists j ≥ 0 such that C j G = {id}. If j is the minimum non-negative number with such a property we say that G is of nilpotent class j. Nilpotent subgroups of the Cremona group have been described: Theorem 4.1 ([14]). Let G be a nilpotent subgroup of Bir(P 2 C ). Then • either G is up to finite index metabelian,

• or G is a torsion group.

We find an alternative proof of [START_REF] Déserti | Sur les sous-groupes nilpotents du groupe de Cremona[END_REF]Lemma 4.2] for G infinite: Lemma 4.2. Let G be an infinite, nilpotent, non virtually abelian subgroup of Bir(P 2 C ). Then G does not contain a hyperbolic map.

Proof. The group G is also solvable. Assume by contradiction that G contains a hyperbolic map; then according to Theorem A up to birational conjugacy and finite index there exists ϒ ⊂ C * × C * infinite such that G is generated by φ = (x p y q , x r y s ) and

(αx, βy) | (α, β) ∈ ϒ .

The group

C 1 G contains [φ, (αx, βy)] | (α, β) ∈ ϒ = (α p-1 β q x, α r β s-1 y) | (α, β) ∈ ϒ that is infinite since ϒ is infinite. Suppose that C i G contains the infinite set (α ℓ i β n i x, α k i β m i y) | (α, β) ∈ ϒ (ℓ i , n i , k i
and m i are some functions in p, q, r and s); then C i+1 G contains

(α (p-1)ℓ i +qm i β (p-1)k i +qn i x, α rℓ i +qm i β rk i +(s-1)n i y) | (α, β) ∈ ϒ that is still infinite.
So any nilpotent and infinite subgroup of Bir(P 2 C ) falls in case 1., 2., 3. of Theorem A. If it falls in case 2. or 3. then G is virtually metabelian ([14, Proof of Theorem 1.1.]). Finally if G falls in case 1., we can prove as in [START_REF] Déserti | Sur les sous-groupes nilpotents du groupe de Cremona[END_REF] that either G is a torsion group, or G is virtually metabelian.

Soluble length of a nilpotent subgroup of Bir(P 2

C ). Let us recall the following statement due to Epstein and Thurston ( [START_REF] Epstein | Transformation groups and natural bundles[END_REF]): let M be a connected complex manifold. Let h be a nilpotent Lie subalgebra of the complex vector space of rational vector fields on M. Then h (n) = {0} if n ≥ dim M; hence the solvable length of h is bounded by the dimension of M. We have a similar statement in the context of birational maps; indeed a direct consequence of Theorem 4.1 is the following property: let G ⊂ Bir(P 2 C ) be a nilpotent subgroup of Bir(P 2 C ) that is not a torsion group, then the soluble length of G is bounded by the dimension of P 2 C . 4.3. Favre's question. In [START_REF] Favre | Le groupe de Cremona et ses sous-groupes de type fini[END_REF] Favre asked few questions; among them there is the following: does any solvable, finitely generated subgroup G of Bir(P 2 C ) contain a subgroup H of finite index such that [H, H] is nilpotent ? We will prove that the answer is no if [G, G] is not a torsion group.

Take G a solvable and finitely generated subgroup of the Cremona group; besides suppose that [G, G] is not a torsion group. Assume that the answer of Favre's question is yes. Up to finite index one can assume that [G, G] is nilpotent. According to Theorem 4.1 the group G (1) = [G, G] is up to finite index metabelian; in other words up to finite index G (2) = [G (1) , G (1) ] is abelian and so G (3) = [G (2) , G (2) ] = id , i.e. the soluble length of G is bounded by 3 up to finite index. Consider the subgroup (x + y 2 , y), (x(1 + y), y), x, y 1 + y , (x, 2y) of Bir(P 2 C ). It is solvable of length 4 (see [START_REF] Martelo | Derived length of solvable groups of local diffeomorphisms[END_REF]): contradiction. In [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF] we prove that there is no embedding of BS(m; n) into Bir(P 2 C ) as soon as |n|, |m|, and 1 are distinct; it corresponds exactly to the case BS(m; n) is not solvable. Indeed BS(m; n) is solvable if and only if |m| = 1 or |n| = 1 (see [START_REF] Souche | Quasi-isométrie et quasi-plans dans l'étude des groupes discrets[END_REF]Proposition A.6]). Proposition 4.3. Let ρ be an embedding of BS(1; n) = r, s | rsr -1 = s n , with n = 1, into the Cremona group. Then

• the image of ρ doesn't contain a hyperbolic map,

• and ρ(s) = x, y + 1 ρ(r) = ν(x), n(y + a(x))

with ν ∈ PGL(2, C) and a ∈ C(x).

Proof. According to [3, Proposition 6.2, Lemma 6.3] one gets that ρ(s) = (x, y + 1) and ρ(r) = ν(x), n(y + a(x)) for some ν ∈ PGL(2, C) and a ∈ C(x). Furthermore ρ(s) can neither be conjugate to an automorphism of the form (αx, βy)

(see [START_REF] Blanc | Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the Cremona groups[END_REF]), nor to a hyperbolic birational map of the form (γx p y q , δx r y s ) with p q r s ∈ GL(2, Z) of spectral radius > 1. As a consequence Proposition 3.2 implies that ρ(BS(1; n)) does not contain a hyperbolic birational map.

2 C

 2 Cremona group with a large normalizer. Consider in P the complement of the union of the three lines {x = 0}, {y = 0} and {z = 0}. Denote by U this open set isomorphic to C * × C * . One has an action of C * × C * on U by translation. Furthermore GL(2, Z) acts on U by monomial maps p q r s → (x, y) → (x p y q , x r y s ) One thus has an injective morphism from (C * × C * ) ⋊ GL(2, Z) into Bir(P 2 C

Remark 3 . 4 .

 34 Both cases are mutually exclusive.

  has no fixed component, and has no base point. The smallest positive integer for which S satisfies such a property is the index of S. If S is a Halphen surface of index m, then K 2 S = 0 and, by the genus formula, the linear system | -mK S | defines a genus 1 fibration S → P 1

Corollary 3 . 5 .Example 3 . 6 .

 3536 As a corollary of the classification of relatively minimal elliptic surfaces the relative minimal model of a rational elliptic surface is a Halphen surface of index m ([22, Chapter 2, §10]). Up to conjugacy G is a subgroup of Aut(S ) where S denotes a Halphen surface of index m. The action of G on NS(S ) is almost faithful, and G is a discrete (it preserves the integral structure of NS(S )) and virtually abelian (it preserves the intersection form and the class of the elliptic fibration) subgroup of Aut(S ). So one has: If G is an uncountable, solvable, non abelian subgroup of Bir(P 2 C ), then G doesn't contain a Halphen twist. Let us come back to the example given in §1. If φ ∈ Bir(P 2

3. 3 .

 3 Solvable groups with no hyperbolic map, and no twist. Let M be a smooth, irreducible, complex, projective variety of dimension n. Fix a Kähler form κ on M. If ℓ is a positive integer, denote by π i : M ℓ → M the projection onto the i-th factor. The manifold M ℓ is then endowed with the Kähler form ℓ ∑ i=1 π * i κ which induces a Kähler metric. To any φ ∈ Bir(M) one can associate its graphΓ φ ⊂ M × M defined as the Zariski closure of (z, φ(z)) ∈ M × M | z ∈ M Indφ .By construction Γ φ is an irreducible subvariety of M × M of dimension n. Both projections π 1 , π 2 : M × M → M restrict to a birational morphism π 1 , π 2 : Γ φ → M.

1 .

 1 Nilpotent subgroups of Bir(P 2 C

4. 4 .

 4 Baumslag-Solitar groups. For any integers m, n such that mn = 0, the Baumslag-Solitar group BS(m; n) is defined by the following presentationBS(m; n) = r, s | rs m r -1 = s n

  2 C ). If • (deg φ n ) n∈N is bounded, we say that φ is an elliptic map, • (deg φ n ) n∈N grows linearly, we say that φ is a Jonquières twist, • (deg φ n ) n∈N grows quadratically, we say that φ is a Halphen twist, • (deg φ n ) n∈N grows exponentially, we say that φ is a hyperbolic map.

	Theorem 2.1

  the infimum is a minimum, then -either it is equal to 0 and φ • has a fixed point in H S , φ • is thus elliptic, -or it is positive and φ • is hyperbolic. Hence the set of points p ∈ H S such that dist(p, φ • (p)) is equal to L(φ • ) is a geodesic line Ax(φ • ) ⊂ H S . Its boundary points are represented by isotropic vectors ω(φ • ) and α

  The Ping-Pong argument allows us to prove the following: Lemma 3.1. A solvable, non abelian subgroup of Bir(P 2 C ) cannot contain two hyperbolic maps φ and ψ

  • ) is equivalent to preserve the Let G be an infinite solvable, non virtually abelian, subgroup of Bir(P 2 C ). If G contains a hyperbolic birational map, then G is, up to conjugacy and finite index, a subgroup of the group generated by (x p y q , x r y s ), (αx, βy) | α, β ∈ C *

	eigenvectors of the matrix	a b c d	. We can now thus state:
	Proposition 3.2. where p q r s	denotes an element of GL(2, Z) with spectral radius > 1.

3.2. Solvable groups with a twist. Consider a solvable, non abelian subgroup G of Bir(P 2 C ). Let us assume that G contains a twist φ; the map φ preserves a unique fibration F that is rational or elliptic. Let us prove that any element of G preserves F . Denote by α(φ • ) ∈ ∂H P 2 C the fixed point of φ • . Take one element in LZ P 2 C

  Let G be a solvable, non abelian subgroup of Bir(P 2 C

	Proposition 3.3.
	1. A subgroup of Bir(P 2 C ) is elliptic if it fixes a point in H P 2 C	.

Author partially supported by ANR Grant "BirPol" ANR-11-JS01-004-01. 1