aa r X i v : . [ m a t h . AG ] J un DISTORTION IN CREMONA GROUPS
SERGE CANTAT AND YVES DE CORNULIERA
BSTRACT . We study distortion of elements in 2-dimensional Cremona groupsover algebraically closed fields of characteristic zero. Namely, we obtain thefollowing trichotomy: non-elliptic elements (i.e., those whose powers have un-bounded degree) are undistorted; elliptic elements have a doubly exponential, orexponential distortion according to whether they are virtually unipotent.
1. I
NTRODUCTION
Let k be an algebraically closed field. The goal of this paper is to study thedistorsion in the Cremona group Bir ( P k ) . We characterize distorted elements, andstudy their distorsion function. The three main tools are: (1) an upper bound onthe distorsion which is obtained via height estimates, using basic number theory(this holds in arbitrary dimension); (2) a result of Blanc and Déserti concerningbase points of birational transformations of the plane; (3) a non-distorsion result forparabolic elements in Bir ( P k ) , obtained via Noether inequalities and the study ofthe action of Bir ( P k ) on the Picard-Manin space, an infinite dimensional hyperbolicspace. This third step sheds new light on the geometry of the action of Bir ( P k ) onthis hyperbolic space.1.1. Distorsion. If f and g are two real valued functions on R + , we write f (cid:22) g ifthere exist three positive constants C , C ′ , C ′′ such that f ( x ) ≤ Cg ( C ′ x ) + C ′′ for all x ∈ R + . We write f ≃ g when f (cid:22) g (cid:22) f .Let G be a group. If S and T are two subsets of G containing the neutral element 1,we write S (cid:22) T if S ⊂ T k for some integer k ≥
0, and S ≃ T if S (cid:22) T (cid:22) S . Let c be anelement of G . Let S be a finite symmetric subset of G containing 1; if the subgroup G S generated by S contains c , we define the distortion function δ c , S ( n ) = sup { m ∈ N : c m ∈ S n } Date : June 5, 2017.2010
Mathematics Subject Classification.
Primary 14E07, Secondary 14J50, 20F65.
By definition, δ c , S ( n ) = ∞ if and only if c has finite order. Clearly, if S ⊂ T then δ c , S ≤ δ c , T . Also, δ c , S k ( n ) = δ c , S ( kn ) . In particular, if S ⊂ T k , then δ c , S ≤ δ c , T ( kn ) .If S (cid:22) T , it follows that δ c , S (cid:22) δ c , T , and if S ≃ T then δ c , S ≃ δ c , T .If S and T both generate G then S ≃ T and δ c , S ≃ δ c , T . Thus, when G is finitelygenerated, the ≃ -equivalence class of the distorsion function only depends on ( G , c ) ,not on the finite generating subset; it is is called the distortion function of c in G ,and is denoted δ Gc , or simply δ c . The element c is called undistorted if δ c ( n ) (cid:22) n ,and distorted otherwise. Example 1.1.
Fix a pair of integers k , ℓ ≥
2. In the Baumslag-Solitar group B k = h t , x | txt − = x k i , we have δ B k x ( n ) ≃ exp ( n ) . In the "double" Baumslag-Solitargroup, one finds double exponential distorsion (see [22] and § 8).It is natural to consider distortion in groups which are not finitely generated. Wesay that an element c ∈ G is undistorted if δ Hc ( n ) ≃ n for every finitely generatedsubgroup H of G containing c . Changing H may change the distorsion function δ Hc ; for instance, if c is not a torsion element, it is undistorted in H = c Z but maybe distorted in larger groups. Also, there are examples of pairs ( G , c ) such that c becomes more and more distorted, in larger and larger subgroups of G (see § 8).Thus, we have a good notion of distorsion, but the distorsion is not measured by anequivalence class of a function " δ Gc ".We shall say that the distorsion type (or class) of c in G is at least f if there isfinitely generated subgroup H containing c with f (cid:22) δ Hc , and at most g if δ Hc (cid:22) g for all finitely generated subgroup H containing c . If the distorsion type is at least f and at most f simultaneously, we shall say that f is the distorsion type of c . Forinstance, c may be exponentially, or doubly exponentially distorted in G . Example 1.2.
Let k be a field. Let c be an element of the general linear group GL d ( K ) ; we have one of the following (see [25, 24] and § 3) • c is not virtually unipotent, i.e. at least one of its eigenvalues in an algebraicclosure is not a root of unity, and then c is undistorted; • c is virtually unipotent of infinite order, and then δ c ( n ) ≃ exp ( n ) (this ispossible only if k has characteristic zero); • c has finite order.The dimension d does not intervene in this description. In contrast, the unipotentelementary matrix e ( ) = Id + δ , is undistorted in SL ( Z ) but has exponentialdistortion in SL d ( Z ) for d ≥ ISTORTION IN CREMONA GROUPS 3
Distorsion in Cremona groups.
Distortion in groups of homeomorphisms isan active subject (see [2, 9, 23, 27, 28]). For instance, in the group of homeomor-phisms of the sphere S d , every element is distorted. Our goal in this paper is tostudy distorsion in groups of birational transformations.If M is a projective variety over a field k , we denote by Bir ( M k ) its group ofbirational transformations over k . When M is the projective space P m k , this group isthe Cremona group in m variables Cr m ( k ) = Bir ( P m k ) = Bir ( A m k ) . The problem isto describe the elements of Bir ( M k ) which are distorted in Bir ( M k ) , and to estimatetheir distorsion functions.1.2.1. Degree sequences.
Let H be a hyperplane section of M , for some fixed em-bedding M ⊂ P N k . The degree of a birational transformation f : M M withrespect to the polarization H is the intersection product deg H ( f ) = H m − · f ∗ ( H ) ,where m = dim ( M ) . When M is P m k and H is a hyperplane, then deg H ( f ) is thedegree of the homogeneous polynomial functions f i , without common factor ofpositive degree, such that f = [ f : · · · : f m ] in homogeneous coordinates.The degree function is almost submultiplicative (see [17, 29, 32]): there is aconstant C M , H such that for all f and g in Bir ( M k ) deg H ( f ◦ g ) ≤ C M , H deg H ( f ) deg H ( g ) . (1.1)Thus, we can define the dynamical degree λ ( f ) by λ ( f ) = lim n → + ∞ ( deg H ( f n ) / n ) . By definition, λ ( f ) ≥
1, and the following well known lemma implies that λ ( f ) = f is distorted (see Section 2). Lemma 1.3.
Let G be a group with a finite symmetric set S of generators. Let | w | denote the word length of w ∈ G with respect to the set of generators S. Then, (1) | · | is sub-additive: | vw | ≤ | v | + | w | ; (2) the stable length sl ( c ) : = lim n → ∞ n | c n | is a well-defined element of R + ; (3) c is distorted if and only if sl ( c ) = . Distortion in dimension . Assume, for simplicity, that the field k is alge-braically closed. Typical elements of Cr d ( k ) have dynamical degree >
1. At theopposite, we have the notion of algebraic elements . A birational transformation f : M → M is algebraic , or bounded , if deg H ( f n ) is a bounded sequence of in-tegers; by a theorem of Weil (see [34]), f is bounded if and only if there exists aprojective variety M ′ , a birational map ϕ : M ′ M , and an integer m >
0, such that ϕ − ◦ f m ◦ ϕ is an element of Aut ( M ′ ) (the connected component of the identity inthe group of automorphisms Aut ( M ′ ) ). In the case of surfaces, bounded elementsare also called elliptic ; we shall explain this terminology in Section 4. ISTORTION IN CREMONA GROUPS 4
Theorem 1.4.
Let k be a field. If an element f ∈ Cr ( k ) is distorted, then f iselliptic. If k is algebraically closed and of characteristic , and f ∈ Cr ( k ) iselliptic, then either f has finite order, or its distortion is exponential, or its distortionis doubly exponential and in that case f is conjugate to a unipotent automorphismof P k . The first assertion extends to
Bir ( X ) for all projective surfaces (see Theorems and ),but the second does not. For instance, if X is a complex abelian surface and Aut ( X ) has only finitely many connected components, every translation of infinite orderis undistorted and elliptic. Consider, in Cr ( k ) , the element ( x , y ) s ( x , xy ) ; it isnot elliptic and by the above theorem, it is not distorted in Cr ( k ) . On the otherhand, the natural embedding Cr ( k ) ⊂ Cr ( k ) maps it to ( x , y , z ) ( x , xy , z ) , whichis exponentially distorted in Bir ( A k ) , while its degree growth remains linear. ThusTheorem 1.4 is specific to the projective plane. Question 1.5. (see Section 3)(A) In Theorem 1.4, can we remove the restriction concerning the characteristicor the algebraic closedness of the field k ?(B) Can we find more than double exponential distortion in the Cremona groupCr m ( C ) , for some m ≥ Hyperbolic spaces, horoballs, and distortion.
Our proof of Theorem 1.4makes use of the action of Cr ( k ) on an infinite dimensional hyperbolic space H ∞ ,already at the heart of several articles (see [13]). There are elements f of Cr ( k ) acting as parabolic isometries on H ∞ , with a unique fixed point ξ f at the boundaryof the hyperbolic space. We shall show that the orbit of a sufficiently small horoballcentered at ξ f under the action of Cr ( k ) is made of a family of pairwise disjointhoroballs. We refer to Theorem C in Section 6 for that result. Theorem B, proved inSection 4, is a general result for groups acting by isometries on hyperbolic spacesthat provides a control of the distorsion of parabolic elements.1.4. Remark and Acknowledgement.
One step towards Theorem 1.4 is to provethat the so-called Halphen twists of Cr ( k ) (a certain type of parabolic elements) arenot distorted. Blanc and Furter obtained simultaneously another proof of that result;instead of looking at the geometry of horoballs, as in our Theorem 4.1, they provea very nice result on the length of elements of Cr ( k ) in terms of the generatorsprovided by Noether-Castelnuovo theorem (the generating sets being PGL ( k ) andtransformations preserving a pencil of lines). Our proof applies directly to Halphentwists on non-rational surfaces. ISTORTION IN CREMONA GROUPS 5
We thank Jérémy Blanc and Jean-Philippe Furter, as well as Vincent Guirardel,Anne Lonjou, and Christian Urech for interesting discussions on this topic.2. D
EGREES AND UPPER BOUNDS ON THE DISTORTION
The following proposition shows that the degree growth may be used to controlthe distortion of a birational transformation.
Proposition 2.1.
Let ( M , H ) be a polarized projective variety, and f be a birationaltransformation of M. (1) If deg H ( f n ) grows exponentially, then f is undistorted. (2) If deg ( f n ) (cid:23) n α for some α > , the distortion of f is at most exponential.Proof. According to Equation (1.1), the degree function is almost submultiplicative;replace it by deg ′ H ( f ) : = deg H ( f ) / C M , H to get a submultiplicative function.If S is a finite symmetric subset of Bir ( M ) , and D is the maximum of deg ′ H ( g ) for g in S , then D n is an upper bound for deg ′ H on the ball S n . Hence if deg ′ ( f m ) ≥ Cq m for some constants C > q >
1, and if f m ∈ S n we have Cq m ≤ D n . Takinglogarithm, we get m log ( q ) + C ≤ n log ( D ) , and then m ≤ log ( q ) − ( nlog ( D ) − C ) .Thus δ f , S ( n ) ≤ ( nlog ( D ) − C ) log ( q ) (cid:22) n and the first assertion is proved. Now, assume that deg ′ H ( f m ) ≥ cm α for somepositive constants c and α . Then cm α ≤ D n , so m ≤ c − / α D n / α . Thus δ f , S ( n ) ≤ c − / α D n / α (cid:22) exp ( n ) and the second assertion follows. (cid:3) Remark 2.2.
More generally, consider an increasing function α such that α ( m ) ≤ log deg ′ H ( f m ) for all m ≥
1. Let β be a decreasing inverse of α , i.e. a function β : R + → R + such that β ( α ( m )) = m for all m . We have α ( m ) ≤ log ( deg ′ H ( f m )) ≤ n log ( D ) if f m is in S n , hence δ f , S ( n ) ≤ β ( n log ( D )) . However, we do not know any exampleof birational transformation with intermediate (neither exponential nor polynomi-ally bounded) degree growth. See [33] for a lower bound on the degree growthwhen f ∈ Aut ( A m k ) . 3. H EIGHTS AND DISTORTION
In this section we study the distortion of automorphisms of P m k in the groups Aut ( P m k ) and Cr m ( k ) = Bir ( P m k ) . ISTORTION IN CREMONA GROUPS 6
Distortion and monomial transformations.
Let k be an algebraically closedfield of characteristic zero. Here, we show that all elements of PGL m + ( k ) aredistorted in Cr m ( k ) , and we compute their distortion rate.3.1.1. Monomial transformations and distortion of semi-simple automorphisms.
The group GL m ( Z ) acts by automorphisms on the m -dimensional multiplicativegroup G m m : if A = [ a i , j ] is in GL m ( Z ) , then A ( x , . . . , x m ) = ( y , . . . , y m ) with y j = ∏ i x a i , j i . (3.1)The group G m m ( k ) acts also on itself by translations. Altogether, we get an embed-ding of GL m ( Z ) ⋉ G m m ( k ) in Bir ( P m k ) .If s is a fixed element of k × , we denote by ϕ s : Z m → G m m the homomorphismdefined by ϕ s ( n , . . . , n d ) = ( s n , . . . , s n d ) . This homomorphism is injective if andonly if s is not a root of unity. Its image ϕ s ( Z m ) is normalized by the monomialgroup GL d ( Z ) ; in this way, every element s ∈ k × of infinite order determines anembedding of GL m ( Z ) ⋉ Z m into Bir ( P m k ) , the image of which is GL m ( Z ) ⋉ ϕ s ( Z m ) .The following lemma is classical (see [25, 24] for instance). Lemma 3.1.
For every m ≥ , the abelian subgroup Z m is exponentially distortedin GL m ( Z ) ⋉ Z m . More precisely, | g n | ≃ log ( n ) for every non-trivial element g inthe (multiplicative) abelian group Z m . For u ∈ k × , the subgroup ϕ u ( Z m ) of G m m ( k ) acts by translations on G m m ( k ) . Thisdetermines a subgroup V u of Cr m ( k ) acting by diagonal transformations ( x , . . . , x m ) ( u n x , . . . , u n m x m ) . By the previous lemma, the distorsion of every element in V u isat least exponential in Cr m ( k ) (when u is a root of unity, the distorsion is infinite).Now let u be an arbitrary diagonal transformation: u ( x ) = ( u x , . . . , u m x m ) , where ( u i ) ∈ G m m ( k ) . Consider the transformations g i = ( x , . . . , x i − , u i x i , x i + , . . . , x m ) .Then the g i pairwise commute and u = g . . . g m . Since g i ∈ V u i , it is at least expo-nentially distorted in GL m ( Z ) ⋉ G m m ( k ) . Thus, u is at least exponentially distortedin GL m ( Z ) ⋉ G m m ( k ) . We have proved: Lemma 3.2.
Let k be a field and m ≥ be an integer. In Bir ( P m k ) , every linear,diagonal transformation is at least exponentially distorted. Distortion of unipotent automorphisms.
Lemma 3.3.
If U is a unipotent element of SL m + ( k ) , then U is at least expo-nentially distorted in SL m + ( k ) , and it is at least doubly exponentially distorted in Bir ( P m k ) for m ≥ . ISTORTION IN CREMONA GROUPS 7
Consequently, the image of U has finite order in every linear representation of(large enough subgroups of) the Cremona group. Note that (in characteristic zero)this already indicates that Cr ( k ) ⊂ Cr ( k ) is distorted in the sense that the transla-tion x x +
1, which has exponential distortion in Cr ( k ) ≃ PGL ( k ) , has doubleexponential distortion in Cr ( k ) . Proof.
Unipotent elements of SL m + ( k ) have finite order if the characteristic of thefield is positive; hence, we assume that char ( k ) =
0. Consider the element U = (cid:18) (cid:19) (3.2)of SL ( k ) . Let A ∈ SL ( k ) be the diagonal matrix with coefficients 2 and 1 / A n UA − n = U n and U is exponentially distorted in the subgroup of SL ( k ) generated by U and A . Similarly, consider a unipotent matrix U i , j = Id + E i , j , where E i , j is the ( m + ) × ( m + ) matrix with only one non-zero coefficient, namely e i , j =
1; then U i , j is exponentially distorted in SL m + ( k ) : there is a diagonal matrix A such that | U ni , j | ≃ log ( n ) in the group h U i , j , A i , for all n ≥
1. This implies thatunipotent matrices are exponentially distorted in SL m + ( k ) .As a second step, consider a 3 × U = , U n = n n ( n − ) /
20 1 n . (3.3)We want to prove that U is doubly exponentially distorted in Cr ( k ) . Take iterates U K n for some integer K >
1. Then, conjugating by A n , and multiplying by B , with A = K
00 0 1 , B = −
10 0 1 , C = K , (3.4)we get a new matrix BA − n U K n A n = [ v i , j ( n )] which is upper triangular; its coeffi-cients are equal to 1 on the diagonal, v , = K n , v , = K n ( K n − ) / v , = C n changes v , into v ′ , = v , into v ′ , = ( K n − ) /
2. Mul-tiplying by the unipotent matrix D = Id − E , + / E , changes v ′ , into 0 and v ′ , into K n . One more conjugacy by C n gives a matrix E with constant coefficients.Thus U K n is a word of finite length (independent of n ) in A n , C n , and a fixed, fi-nite number of unipotent matrices ( B , D , E ). Since A and C are diagonal matrices,they satisfy | A n | ∼ log ( n ) and | C n | ∼ log ( n ) in some finitely generated subgroup ofCr ( k ) . Thus, U is doubly exponentially distorted.This argument and a recursion starting at m = (cid:3) ISTORTION IN CREMONA GROUPS 8
Distortion of linear projective transformations.
Every A ∈ PGL m + ( k ) is theproduct of a semi-simple element S A with a unipotent element U A such that S A and U A commute. When k is algebraically closed, S A is diagonalizable. By Lemmas 3.2and 3.3, A is at least exponentially distorted (resp. doubly exponentially distorted if S A has finite order).3.2. Heights and upper bounds.Theorem 3.4.
Let k be an algebraically closed field of characteristic zero. Let Abe an element of Aut ( P m k ) given by a matrix in SL m + ( k ) of infinite order. Then,its distortion in the Cremona group Bir ( P m k ) is doubly exponential if the matrix isvirtually unipotent, and simply exponential otherwise. To prove this result, we use basic properties of heights of polynomial functions.We start with a proof of this theorem when k = Q is an algebraic closure of the fieldof rational number; the general case is obtained by a specialization argument.3.2.1. Heights of polynomial functions.
Let K be a finite extension of Q , and let M K be the set of places of K ; to each place, we associate a unique absolute value | · | v on K , normalized as follows (see [6], §1.4). First, for each prime number p , the p -adic absolute value on Q satisfies | p | p = / p , and | · | ∞ is the standard absolutevalue. Then, if v ∈ M K is a place that divides p , with p prime or ∞ , then | x | v = | Norm K / Q ( x ) | / [ K : Q ] p (3.5)for every x ∈ K . With such a choice, the product formula reads ∑ v ∈ M K log | x | v = x ∈ K \ { } .Let m be a natural integer. If f ( x ) = ∑ I a I x I is a polynomial function in thevariables x = ( x , . . . , x m ) , with a I ∈ K for each multi-indice I = ( i , . . . , i m ) , we set | f | v = max I | a I | v (3.7)for every place v ∈ M K . If f =
0, we define its height h ( f ) by h ( f ) = ∑ v ∈ M K log | f | v . (3.8)If ˆ f = ( f , . . . , f m ) is an endomorphism of A m + K , the height h ( ˆ f ) is the maximumof the heights h ( f i ) , and | ˆ f | v is the maximum of the | f i | v . (Note that the affinecoordinates system x is implicitely fixed.) ISTORTION IN CREMONA GROUPS 9
Remark 3.5.
Let f and g be non-zero elements of K [ x , . . . , x m ] .(1).– The product formula implies that h ( a f ) = h ( f ) , ∀ a ∈ K \ { } .(2).– From this, we see that h ( f ) ≥ f ∈ K [ x , . . . , x m ] \ { } . Indeed, onecan multiply f by the inverse of a coefficient a I = | f | v ≥ v ∈ M K .(3).– The Gauss Lemma says that | f g | v = | f | v | g | v when v is not archimedean.This multiplicativity property fails for places at infinity.(4).– If L is an extension of K , then the height of f ∈ K [ x , . . . , x m ] is the same asits height as an element of L [ x , . . . , x m ] (see [6], Lemma 1.3.7). Thus, the height iswell defined on Q [ x , . . . , x m ] . Theorem 3.6 (see [6], 1.6.13) . Let f , . . . , f s be non-zero elements of Q [ x , . . . , x m ] ,and let f be their product f · · · f s . Let ∆ ( f ) be the sum of the partial degrees of fwith respect to each of the variables x i . Then − ∆ ( f ) log ( ) + s ∑ i = h ( f i ) ≤ h ( f ) ≤ ∆ ( f ) log ( ) + s ∑ i = h ( f i ) . If deg ( f ) denotes the degree of f , then ∆ ( f ) ≤ ( m + ) deg ( f ) . For s = h ( f ) ≤ h ( f ) − h ( f ) + ( m + ) log ( ) deg ( f ) . (3.9)3.2.2. Heights of birational transformations.
Consider a birational transformation f : P m Q P m Q , and write it in homogeneous coordinates f [ x : . . . : x m ] = [ f : . . . : f m ] (3.10)where the f i ∈ Q [ x , . . . , x m ] are homogeneous polynomial functions of the samedegree d with no common factor of positive degree. Then, d is the degree of f (see§ 1.2.1), and the f i are uniquely determined modulo multiplication by a commonconstant a ∈ Q \ { } . Thus, Remark 3.5(1) shows that the real number h ( f ) = max i h ( f i ) (3.11)is well defined. This number h ( f ) is, by definition, the height of the birationaltransformation f . It coincides with the height of the lift of f as the endomorphismˆ f = ( f , . . . , f m + ) of A m + k (see § 3.2.1).3.2.3. Growth of heights under composition.
Let S = { f , . . . , f s } be a finite sym-metric set of birational transformations of P m Q ; the symmetry means that f ∈ S if and only if f − ∈ S . Consider the homomorphism from the free group F s = h a , . . . , a s | /0 i to Bir ( P m Q ) defined by mapping each generator a j to f j . Then, to ISTORTION IN CREMONA GROUPS 10 every reduced word w ℓ ( a , . . . , a s ) of length ℓ in the generators a j corresponds anelement w l ( S ) = w l ( f , . . . , f s ) (3.12)of the Cremona group Bir ( P m Q ) .For each f i ∈ S , we fix a system of homogeneous polynomials f ij ∈ Q [ x : . . . : x m ] defining f , as in § 3.2.2: f i = [ f i : . . . : f im ] and the f ij have degree d i = deg ( f i ) .Moreover, we choose the f ij so that for every i at least one of the coefficients of the f ij is equal to 1. Once the f ij have been fixed, we have a canonical lift of each f i toa homogeneous endomorphism ˆ f i of A m + Q , given byˆ f i ( x , . . . , x m ) = ( f i , . . . , f im ) . (3.13)Thus, every reduced word w ℓ of length ℓ in F s determines also an endomorphismˆ w ℓ ( S ) = w ℓ ( ˆ f , . . . , ˆ f s ) of the affine space.Let d S be the maximum of { , d , . . . , d s } , so that d S ≥
2. Then, the degree of theendomorphism ˆ w ℓ ( S ) is at most d ℓ S .Let K be the finite extension of Q which is generated by all the coefficients a ij , I of the polynomial functions f ij = ∑ a ij , I x I . We shall say that a place v ∈ M K is active if | a ij , I | v > v ∈ M K , we set M ( v ) = max | a ij , I | v = max | ˆ f i | v , (3.14)the maximum of the absolute values of the coefficients; our normalization impliesthat M ( v ) ≥ M ( v ) = v is not active. Lemma 3.7.
Let v be a non-archimedean place. If w ℓ ∈ F s is a reduced word oflength ℓ , then log | ˆ w ℓ ( S ) | v ≤ log ( M ( v )) d ℓ S . Thus, if v is not active, then log | ˆ w ℓ ( S ) | v = .Proof. Set d = d S . Write ˆ w ℓ ( S ) as a composition ˆ g ℓ ◦ · · · ◦ ˆ g , where each ˆ g k is oneof the ˆ f i (here we use that S is symmetric). By definition, | ˆ g | v ≤ M ( v ) . Then,assume that | ˆ g k − ◦ · · · ◦ ˆ g | v ≤ M ( v ) + d + ··· + d k − for some integer 2 ≤ k ≤ ℓ . Writeˆ g k − ◦ · · · ◦ ˆ g = ( u , . . . , u m ) for some homogeneous polynomials u j . The Gausslemma (see Remark 3.5) says that | u i · · · u i m m | v = | u | i m v · · · | u m | i v ≤ ( M ( v ) + d + ··· + d k − ) d (3.15) ISTORTION IN CREMONA GROUPS 11 for every multi-index I = ( i , . . . , i m ) of length ∑ i j ≤ d . The endomorphism ˆ g k hasdegree ≤ d , and the absolute values of its coefficients are bounded by M ( v ) , hence | ˆ g k ◦ · · · ◦ ˆ g | v ≤ M ( v ) + d + ··· + d k − . (3.16)By recursion, this upper bound holds up to k = ℓ . For k = ℓ we obtain the estimatelog | ˆ w ℓ ( S ) | v ≤ log ( M ( v )) d ℓ because 1 + d + · · · + d ℓ − ≤ d ℓ . (cid:3) Lemma 3.8.
Let v be an archimedean place. If w ℓ ∈ F s is a reduced word of length ℓ , then log | ˆ w ℓ ( S ) | v ≤ ( M ( v )) d ℓ S + log ( md mS ) d ℓ S . Proof.
Consider a monomial x I = x i · · · x i m m of degree ≤ d . Let u , . . . , u m be homo-geneous polynomials of degree ≤ D with D ≥ | c | v ≤ C . Note that the space of homogeneous polynomials of degree D in m vari-ables has dimension (cid:0) D + mm (cid:1) .Then | u i · · · u i m m | v ≤ (cid:18) D + mm (cid:19) d C d ≤ ( D + m ) md C d ≤ ( mD m ) d C d (3.17)Indeed, every coefficient in the product u i · · · u i m m is obtained as a sum of at most (cid:0) D + mm (cid:1) d terms, each of which is a product of at most d coefficients of the u j .Then, to estimate the absolute values of the coefficients of ˆ w ℓ ( S ) , we proceed byrecursion as in the proof of Lemma 3.7. Set B = md mS . For a composition ˆ g k ◦ · · · ◦ ˆ g of length k we obtain | ˆ g k ◦ · · · ◦ ˆ g | v ≤ B ( k − ) d k − S M ( v ) d k − S . (3.18)The conclusion follows from ℓ d ℓ S ≤ d ℓ S . (cid:3) Putting these lemmas together, we get h ( ˆ w ℓ ( S )) ≤ ∑ v active ( M ( v )) d ℓ S + ∑ v | ∞ log ( md mS ) d ℓ S (3.19)This inequality concerns the height of the endomorphism ˆ w ℓ ( S ) ; to obtain the bira-tional transformation w ℓ ( S ) , we might need to divide by a common factor q ( x , . . . , x m ) .Since the degree of ˆ w ℓ ( S ) is no more than d ℓ S , Theorem 3.6 provides the upper bound h ( w ℓ ( S )) ≤ ( m + ) log ( ) + ∑ v active log ( M ( v )) + ∑ v | ∞ log ( md mS ) ! d ℓ S . (3.20)This proves the following proposition. ISTORTION IN CREMONA GROUPS 12
Proposition 3.9.
Let F s = h a , . . . , a s | /0 i be a free group of rank s ≥ . For everyhomomorphism ρ : F s → Bir ( P m Q ) , there exist two constants C m ( ρ ) and d ( ρ ) ≥ such that h ( ρ ( w )) ≤ C m ( ρ ) d ( ρ ) | w | for every w ∈ F s , where | w | is the length of w asa reduced word in the generators a i . Proof of Theorem 3.4.
We may now prove Theorem 3.4. When k = Q , thisresult is a direct corollary of Proposition 3.9 and Section 3.1.3; we start with thiscase and then treat the general case via a specialization argument.3.3.1. Number fields.
Let A be an element of SL m + ( Q ) of infinite order. Afterconjugation, we may assume A to be upper triangular. First, suppose that A isvirtually unipotent (all its eigenvalues are roots of unity). Then h ( A n ) grows like τ log ( n ) as n goes to + ∞ . Thus, if A n is a word of length ℓ ( n ) in some fixed, finitelygenerated subgroup of Bir ( P m Q ) , Proposition 3.9 shows that τ log ( n ) ≤ Cd ℓ ( n ) (3.21)for some positive constants C and d >
1. Thus, A is at most doubly exponentiallydistorted; from Lemma 3.3, it is exactly doubly exponentially distorted. Now, sup-pose that an eigenvalue α of A is not a root of unity. Kronecker’s lemma provides aplace v ∈ M Q ( α ) for which | α | v > h ( A n ) grows like τ n for some positive constant τ as n goes to + ∞ (see Remark 3.5(2)), and A is at mostexponentially distorted in Bir ( P m Q ) . From Section 3.1.3, we obtain Theorem 3.4when k = Q .3.3.2. Fields of characteristic zero.
Let k be an algebraically closed field of char-acteristic zero and let A be an element of SL m + ( k ) . Let S = { f , . . . , f m } be afinite symmetric subset of Bir ( P m k ) such that the group generated by S contains A .For each n , denote by ℓ ( n ) the length of A n as a reduced word in the f i .Write each f i in homogeneous coordinates f i = [ f i : . . . : f im ] , as in Section 3.2.3;and denote by C the set of coefficients of the matrix A and of the polynomial func-tions f ij = ∑ a ij , I x I . This is a finite subset of k , generating a finite extension K of Q .This finite extension is an algebraic extension of a purely trancendental extension Q ( t , . . . , t r ) , where r is the transcendental degree of K over Q . Then, the elementsof C are algebraic functions with coefficients in Q (such as ( t t − ) / + t ); thering of functions generated by C (over Q ) may be viewed as the ring of functions ofsome algebraic variety V C (defined over Q ).If u is a point of V C ( Q ) and c ∈ C is one of the coefficients, we may evaluate c at u to obtain an algebraic number c ( u ) . Similarly, we may evaluate, or specialize, ISTORTION IN CREMONA GROUPS 13 A and the f i at u . This gives an element A u in SL m + ( Q ) (the determinant is 1),and rational transformations f iu of P m Q . For some values of u , f iu may be degenerate,identically equal to [ . . . : 0 ] ; but for u in a dense, Zariski open subset of V C , the f iu are birational transformations of degree deg ( f iu ) = deg ( f i ) . Pick such a point u ∈ V C ( Q ) . If A n is a word of length ℓ ( n ) in the f i , then A nu is a word of the samelength in the f iu . From the previous section we deduce that A is at most doublyexponentially distorted. Moreover, if one of the eigenvalues α ∈ k of A is not a rootof unity, we may add α to the set C and then choose the point u such that α ( u ) isnot a root of unity either. Then, A u and thus A is at most exponentially distorted.This concludes the proof of Theorem 3.4.4. N ON - DISTORTION
In this section, we prove Theorem 4.1, which provides an upper bound for the dis-tortion of parabolic isometries in certain groups of isometries of hyperbolic spaces.4.1.
Hyperbolic spaces and parabolic isometries.
Hyperbolic spaces.
Let H be a real Hilbert space of dimension m + m canbe infinite). Fix a unit vector e of H and a Hilbert basis ( e i ) i ∈ I of the orthogonalcomplement of e . Define a new scalar product on H by h u | u ′ i = a a ′ − ∑ i ∈ I a i a ′ i (4.1)for every pair u = a e + ∑ i a i e i , u ′ = a ′ e + ∑ i a ′ i e i of vectors. Define H m to be theconnected component of the hyperboloid { u ∈ H | h u | u i = } that contains e , andlet dist be the distance on H m defined by (see [3])cosh ( dist ( u , u ′ )) = h u | u ′ i . (4.2)The metric space ( H m , dist ) is a model of the hyperbolic space of dimension m (see[3]). The projection of H m into the projective space P ( H ) is one-to-one onto itsimage. In what follows, H m is identified with its image in P ( H ) and its boundary isdenoted by ∂ H m ; hence, boundary points correspond to isotropic lines in the space H for the scalar product h·|·i .4.1.2. Hyperbolic plane.
A useful model for H is the Poincaré model: H is iden-tified to the upper half-plane { z ∈ C ; Im ( z ) > } , with its Riemanniann metric givenby ds = ( x + y ) / y . Its group of orientation preserving isometries coincides with ISTORTION IN CREMONA GROUPS 14
PSL ( R ) , acting by linear fractional transformations. The distance between twopoints z and z satisfiessinh (cid:18) dist H ( z , z ) (cid:19) = | z − z | ( Im ( z ) Im ( z )) / . (4.3)4.1.3. Isometries.
Denote by O , m ( R ) the group of linear transformations of H preserving the scalar product h·|·i . The group of isometries Iso ( H m ) coincides withthe index 2 subgroup O + , m ( R ) of O ( H ) that preserves the chosen sheet H m of thehyperboloid { u ∈ H | h u | u i = } . This group acts transitively on H m , and on its unittangent bundle.If h ∈ O + , m ( R ) is an isometry of H m and v ∈ H is an eigenvector of h with eigen-value λ , then either | λ | = v is isotropic. Moreover, since H m is homeomorphicto a ball, h has at least one eigenvector v in H m ∪ ∂ H m . Thus, there are three typesof isometries [8]:(1) An isometry h is elliptic if and only if it fixes a point u in H m . Since h·|·i is negative definite on the orthogonal complement u ⊥ , the linear transformation h fixes pointwise the line R u and acts by rotation on u ⊥ with respect to h·|·i .(2) An isometry h is parabolic if it is not elliptic and fixes a vector v in the isotropiccone. The line R v is uniquely determined by the parabolic isometry h . If z is a pointof H m , there is an increasing sequence of integers m i such that h m i ( z ) convergestowards the boundary point ξ determined by v .(3) An isometry h is loxodromic if and only if h has an eigenvector v + h with eigen-value λ >
1. Such an eigenvector is unique up to scalar multiplication, and there isanother, unique, isotropic eigenline R v − h corresponding to an eigenvalue <
1; thiseigenvalue is equal to 1 / λ . On the orthogonal complement of R v + h ⊕ R v − h , h actsas a rotation with respect to h·|·i . The boundary points determined by v + h and v − h are the two fixed points of h in H ∞ ∪ ∂ H ∞ : the first one is an attracting fixed point,the second is repelling. Moreover, h ∈ Iso ( H ∞ ) is loxodromic if and only if its translation length L ( h ) = inf { dist ( x , h ( x )) | x ∈ H ∞ } (4.4)is positive. In that case, λ = exp ( L ( h )) is the largest eigenvalue of h and dist ( x , h n ( x )) grows like nL ( h ) as n goes to + ∞ for every point x in H m .When h is elliptic or parabolic, the translation length vanishes (there is a point u in H m with L ( h ) = dist ( u , h ( u )) if h is elliptic, but no such point exists if h isparabolic). ISTORTION IN CREMONA GROUPS 15
Horoballs.
Let ξ be a boundary point of H m , and let ε be a positive realnumber. The horoball H ξ ( ε ) in H ∞ is the subset H ξ ( ε ) = { v ∈ H m ; 0 < h v | ξ i < ε } . It is a limit of balls with centers converging to the boundary point ξ . An isometry h fixing the boundary point ξ maps H ξ ( ε ) to H ξ ( e L ( h ) ε ) .4.2. Distortion estimate.
Our goal is to prove the following theorem.
Theorem 4.1.
Let G be a subgroup of
Iso ( H m ) . Let f be a parabolic element of G,and let ξ ∈ ∂ H m be the fixed point of f . Suppose that the following two propertiesare satisfied. (i) There are positive constants C and C ′ > and a point x ∈ H m such that dist ( f n ( x ) , x ) ≥ C log n − C ′ for all large enough values of n. (ii) There exists a horoball B centered at ξ such that for every g ∈ G eithergB = B or gB ∩ B = /0 .Then f is at most n / C -distorted in G. In particular C ≤ and if C = then f isundistorted in G. When looking at the Cremona group Cr ( k ) , we shall see examples of isometriesin H ∞ with dist ( f n ( x ) , x ) ∼ C log n for C = C = Complements of horoballs.
Let X be a metric space. Let W be a subset of X .Let dist W c (or d X when W = /0 ) be the induced intrinsic distance on the complementof W ; namely, for x and y in W c , we have dist W c ( x , y ) = sup ε > d W c , ε ( x , y ) with d W c , ε ( x , y ) = inf ( n − ∑ i = d ( x i , x i + ) : n ≥ , x = x , x n = y , sup i d ( x i , x i + ) ≤ ε ) for points x i which are all in X \ W . It is a distance as soon as it does not take the ∞ value. In the cases we shall consider, X will be the hyperbolic space H m , W will bea union of horoballs, and W c will be path connected. In that case, dist W c ( x , y ) is theinfimum of the length of paths connecting x to y within W c .Similarly, if Y is a subset of X , we denote by dist Y the induced intrinsic distanceon Y (hence, dist Y = dist ( X \ Y ) c ). Lemma 4.2.
Let B ⊂ H m be an open horoball, with boundary ∂ B. Let x and y bepoints on the horosphere ∂ B. Then dist B c ( x , y ) = dist ∂ B ( x , y ) = ( dist ( x , y ) / ) . ISTORTION IN CREMONA GROUPS 16
Proof.
The statement being trivial when x = y , we assume x = y in what follows.Let ξ be the center at infinity of B . Then x , y , and the boundary point ξ are containedin a unique geodesic plane P . Since the projection of H m onto P is a 1-Lipschitzmap, we have dist ∂ B ( x , y ) = dist ∂ B ∩ P ( x , y ) and dist B c ( x , y ) = dist B c ∩ P ( x , y ) . Hence,we can replace H m by the 2-dimensional hyperbolic space P ≃ H . To conclude,we use the Poincaré half-plane model of H . There is an isometry P → H mappingthe horosphere ∂ B ∩ P to the line i + R , the points x and y to i + t and i + t ′ , and ξ to ∞ . If γ ( s ) = x ( s ) + iy ( s ) , s ∈ [ a , b ] , is a path in P ∩ B c that connects x to y , its lengthsatisfies length ( γ ) = Z ba ( x ′ ( s ) + y ′ ( s ) ) / y ( s ) ds ≤ Z ba | x ′ ( s ) | ds because y ≤ B c ∩ P . Thus, the geodesic segment from x to y for dist B c (resp. dist ∂ B ) is the euclidean segment γ ( s ) = i + s , with s ∈ [ t , t ′ ] , and dist B c ( x , y ) = dist ∂ B ( x , y ) = | t − t ′ | . We conclude with Formula (4.3), that gives sinh ( dist H ( x , y ) / ) = | t − t ′ | / (cid:3) Lemma 4.3.
Let ( B i ) be a family of open horoballs in H m with pairwise disjointclosures and let Q = S B i . Then (1) dist Q c is a distance on X r Q; (2) for every index i and every pair of points ( x , y ) on the boundary of ∂ B i , wehave dist Q c ( x , y ) = dist ∂ B i ( x , y ) .Proof. Let ( x , y ) be a pair of points in Q c . Consider the unique geodesic segment of H m that joins x to y . Denote by [ u j , u ′ j ] the intersection of this segment with B j . Let C be a positive constant such that 2 sinh ( s / ) ≤ Cs for all s ∈ [ , dist ( x , y )] (such aconstant depends on dist ( x , y ) ). From Lemma 4.2 we obtain dist Q c ( x , y ) ≤ dist ( x , y ) + ∑ j dist ∂ B j ( u j , u ′ j ) (4.5) ≤ dist ( x , y ) + ∑ j C dist ( u j , u ′ j ) (4.6) ≤ ( + C ) dist ( x , y ) (4.7)Thus, dist Q c ( x , y ) is finite: this proves (1).For (2), note that Q c ⊂ B ci and Lemma 4.2 imply dist Q c ( x , y ) ≥ dist B ci ( x , y ) = dist ∂ B i ( x , y ) , and that dist ∂ B i ( x , y ) ≥ dist Q c ( x , y ) because ∂ B i ⊂ Q c . (cid:3) ISTORTION IN CREMONA GROUPS 17
Proof of Theorem 4.1.
Changing B in a smaller horoball, we can supposethat B is open and that g ¯ B ∩ ¯ B is empty for all g ∈ G with gB = B . Let Q be theunion of the horoballs gB for g ∈ G . Let x be a point on the horosphere ∂ B . Let D > ( D ) = C ′ + dist ( x , x ) . From the first hypothesis, we know that dist ( f n ( x ) , x ) ≥ C log n − ( D ) for all sufficiently large values of n . By Lemmas 4.2 and 4.3, we get dist Q c ( x , f n ( x )) = ( dist ( x , f n ( x )) / ) ≥ ( C log ( n ) / − log ( D )) ≥ D − n C / − Dn − C / for large enough n . Let us now estimate the distortion of f in G . Let S be a finitesymmetric subset of G and let D S be the maximum of the distances dist Q c ( g ( x ) , x ) for g in S . Suppose that f n = g ◦ g ◦ · · · ◦ g ℓ is a composition of ℓ elements g i ∈ S .The group generated by the g i acts by isometries on Q c for the distance dist Q c . Thus, dist Q c ( f n ( x ) , x ) = dist Q c ( g ◦ · · · ◦ g ℓ ( x ) , x ) ≤ dist Q c ( g ◦ · · · ◦ g ℓ ( x ) , g ◦ · · · ◦ g ℓ − ( x ))+ dist Q c ( g ◦ · · · ◦ g ℓ − ( x ) , x ) ≤ dist Q c ( g ℓ ( x ) , x ) + dist Q c ( g ◦ · · · ◦ g ℓ − ( x ) , x ) ≤ ℓ ∑ j = dist Q c ( g j ( x ) , x ) ≤ ℓ D S . This shows that D − n C / − Dn − C / ≤ D S × ℓ for large values of n (and ℓ ), and theconclusion follows.5. T HE P ICARD -M ANIN SPACE AND HYPERBOLIC GEOMETRY
In this section, we recall the construction of the Picard-Manin space of a projec-tive surface X (see [13, 26] for details).5.1. Picard Manin spaces.
Let X be a smooth, irreducible, projective surface. Wedenote its Néron-Severi group by Num ( X ) ; when k = C , Num ( X ) can be identifiedto H , ( X ; R ) ∩ H ( X ; Z ) . The intersection form ( C , D ) C · D (5.1)is a non-degenerate quadratic form on Num ( X ) of signature ( , ρ ( X ) − ) . ThePicard-Manin space Z ( X ) is the limit lim π : X ′ → X Num ( X ′ ) obtained by looking at ISTORTION IN CREMONA GROUPS 18 all birational morphisms π : X ′ → X , where X ′ is smooth and projective. By con-struction, Num ( X ) embeds naturally as a proper subspace of Z ( X ) , and the inter-section form is negative definite on the infinite dimensional space Num ( X ) ⊥ . Example 5.1.
The group Pic ( P k ) is generated by the class e of a line. Blow-upone point q of the plane, to get a morphism π : X → P k . Then, Pic ( X ) is a freeabelian group of rank 2, generated by the class e of the exceptional divisor E q , andby the pull-back of e under π (still denoted e in what follows). After n blow-upsone obtains Pic ( X n ) = Num ( X n ) = Ze ⊕ Ze ⊕ . . . ⊕ Ze n (5.2)where e (resp. e i ) is the class of the total transform of a line (resp. of the excep-tional divisor E q i ) by the composite morphism X n → P k (resp. X n → X i ). The directsum decomposition (5.2) is orthogonal with respect to the intersection form: e · e = , e i · e i = − ∀ ≤ i ≤ n , and e i · e j = ∀ ≤ i = j ≤ n . (5.3)Taking limits, Z ( P k ) splits as a direct sum Z ( P k ) = Ze ⊕ L q Ze q where q runsover all possible points of the so-called bubble space B ( P k ) of P k (see [26, 18, 4]).5.2. The hyperbolic space H ∞ ( X ) . Denote by Z ( X , R ) and Num ( X , R ) the ten-sor products Z ( X ) ⊗ Z R and Num ( X ) ⊗ Z R . Elements of Z ( X , R ) are finite sums u X + ∑ i a i e i where u X is an element of Num ( X , R ) , each e i is the class of an excep-tional divisor, and the coefficients a i are real numbers. Allowing infinite sums with ∑ i a i < + ∞ , one gets a new space Z ( X ) , on which the intersection form extendscontinuously [12, 7]. Fix an ample class e in Num ( X ) ⊂ Z ( X ) . The subset ofelements u in Z ( X ) such that u · u = H ∞ ( X ) = { u ∈ Z ( X ) | u · u = u · e > } (5.4)is the sheet of that hyperboloid containing ample classes of Num ( X , R ) . With thedistance dist ( · , · ) defined by cosh dist ( u , u ′ ) = u · u ′ , (5.5) H ∞ ( X ) is isometric to the hyperbolic space H ∞ described in Section 4.1.We denote by Iso ( Z ( X )) the group of isometries of Z ( X ) with respect to the inter-section form, and by Iso ( H ∞ ( X )) the subgroup that preserves H ∞ ( X ) . As explainedin [12, 13, 26], the group Bir ( X ) acts by isometries on H ∞ . The homomorphism f ∈ Bir ( X ) f • ∈ Iso ( H ∞ ( X )) (5.6)is injective. ISTORTION IN CREMONA GROUPS 19
Types and degree growth.
Since
Bir ( X ) acts faithfully on H ∞ ( X ) , there arethree types of birational transformations: Elliptic , parabolic , and loxodromic , ac-cording to the type of the associated isometry of H ∞ ( X ) . We now describe howeach type can be characterized in algebro-geometric terms.5.3.1. Degrees, distances, translation lengths and loxodromic elements.
Let h ∈ Num ( X , R ) be an ample class with self-intersection 1. The degree of f with respectto the polarization h is deg h ( f ) = f • ( h ) · h = cosh ( dist ( h , f • h )) . Consider for in-stance an element f of Bir ( P k ) , with the polarization h = e given from the classof a line; then the image of a general line by f is a curve of degree deg h ( f ) whichgoes through the base points q i of f − with certain multiplicities a i , and f • e = deg h ( f ) e − ∑ i a i e i (5.7)where e i is the class corresponding to the exceptional divisor that one gets whenblowing up the point q i .If the translation length L ( f • ) is positive, we know that the distance dist ( f n • ( x ) , x ) grows like nL ( f • ) for every x ∈ H ∞ ( X ) (see Section 4.1). We get: the logarithmlog ( λ ( f )) ofthedynamicaldegreeof f isthetranslationlength L ( f • ) oftheisome-try f • . In particular, f is loxodromic if and only if λ ( f ) > Classification.
Elliptic and parabolic transformations are also classified interms of degree growth. Say that a sequence of real numbers ( d n ) n ≥ grows linearly(resp. quadratically) if n / c ≤ d n ≤ cn (resp. n / c ≤ d n ≤ cn ) for some c > Theorem 5.2 (Gizatullin, Cantat, Diller and Favre, see [20, 10, 11, 16]) . Let Xbe a projective surface, defined over an algebraically closed field k , and h be apolarization of X . Let f be a birational transformation of X . (1) f is elliptic if and only if the sequence deg h ( f n ) is bounded. In this case,there exists a birational map φ : Y X and an integer k ≥ such that φ − ◦ f ◦ φ is an automorphism of Y and φ − ◦ f k ◦ φ is in the connectedcomponent of the identity of the group Aut ( Y ) . (2) f is parabolic if and only if the sequence deg h ( f n ) grows linearly or quadrat-ically with n. If f is parabolic, there exists a birational map ψ : Y Xand a fibration π : Y → B onto a curve B such that ψ − ◦ f ◦ ψ permutes thefibers of π . The fibration is rational if the growth is linear, and elliptic (orquasi-elliptic if char ( k ) ∈ { , } ) if the growth is quadratic. (3) f is loxodromic if and only if deg h ( f n ) grows exponentially fast with n:There is a constant b h ( f ) > such that deg h ( f n ) = b h ( f ) λ ( f ) n + O ( ) . ISTORTION IN CREMONA GROUPS 20
Elliptic elements of Cr ( k ) . Every elliptic, infinite order element of
Bir ( P k ) isconjugate to an automorphism f ∈ PGL ( k ) when k is algebraically closed (see [5]).Thus, Theorem 3.4 stipulates that elliptic elements of infinite order are • exactly doubly exponentially distorted if they are conjugate to a virtuallyunipotent element of PGL ( k ) ; • exactly exponentially distorted otherwise.5.5. Loxodromic elements of Cr ( k ) . Loxodromic elements have an exponentialdegree growth; by Proposition 2.1, they are not distorted. This result applies to allloxodromic elements f ∈ Bir ( X ) , for all projective surfaces.5.6. Parabolic elements of Cr ( k ) . According to Theorem 5.2, there are two typesof parabolic elements, depending on the growth of the sequence deg ( f n ) : Jon-quières and Halphen twists. Here, we collect extra informations on these trans-formations, and study their distortion properties in Sections 6 and 7.5.6.1. Jonquières twists.
Let f be an element of Cr ( k ) for which the sequencedeg ( f n ) grows linearly with n . Then, f is called a Jonquières twist . Examples aregiven by the transformations f ( X , Y ) = ( X , Q ( X ) Y ) with Q ∈ k ( X ) of degree ≥ Normal form .– There is a birational map ϕ : P k × P k P k that conjugates f to anelement g of Bir ( P k × P k ) which preserves the projection π : P k × P k → P k onto thefirst factor. More precisely, there is an automorphism A of P k such that π ◦ g = A ◦ π .If x and y are affine coordinates on each of the factors, then g ( x , y ) = ( A ( x ) , B ( x )( y )) (5.8)where ( A , B ) is an element of the semi-direct product PGL ( k ) ⋉ PGL ( k ( x )) . Al-ternatively, f is conjugate to an element g ′ of Cr ( k ) that preserves the pencil oflines through the point [ ] . Action on H ∞ ( P k ) .– Assume now that g ′ preserves the pencil of lines through thepoint q : = [ ] . Let e ∈ Z ( P k ; R ) be the class of the exceptional divisor E that one gets by blowing-up q . Then g ′• preserves the isotropic vector e − e (corresponding to the class of the linear system of lines through q ), and the uniquefixed point of g ′• on ∂ H ∞ ( P k ) is determined by e − e . Let d denote the degreedeg e ( g ′ ) . Let q i denote the base points of ( g ′ ) − (including infinitely near basepoints) and e ( q i ) be the corresponding classes of exceptional divisors. From [4, 1], ISTORTION IN CREMONA GROUPS 21 one knows that there are 2 d − q ), and that g ′• e = d e − ( d − ) e ( q ) − d − ∑ i = e ( q i ) (5.9) g ′• e ( q ) = ( d − ) e − ( d − ) e ( q ) − d − ∑ i = e ( q i ) . (5.10) Degree growth .– The sequence n deg e ( f n ) converges toward a number α ( f ) . Theset { α ( h f h − ) ; h ∈ Cr ( k ) } admits a minimum; this minimum is of the form µ ( f ) for some integer µ ( f ) >
0, and there is an integer a ≥ α ( f ) = µ ( f ) a .Blanc and Déserti prove also that a = f preserves a pencil oflines in P k (thus, the conjugate g ′ of f satisfies α ( g ′ ) = µ ( f ) ). Moreover, when f preserves such a pencil, one knows from [4], Lemma 5.7, that deg e ( f n ) is a subad-ditive sequence. Thus, n deg e ( f n ) ≥ µ /
2, and µ / n deg e ( f n ) . InSection 7, we shall describe how Blanc and Déserti interpret µ ( f ) as an asymptoticnumber of base points.5.6.2. Halphen twists.
Let f be an element of Cr ( k ) for which the sequence deg ( f n ) grows quadratically with n . Then, f is called a Halphen twist . The following prop-erties follow from [4, 14, 15].
Normal form.–
There is a rational surface X , together with a birational map ϕ : X P k and a genus 1 fibration π : X → P k such that g = ϕ − ◦ f ◦ ϕ is a regular auto-morphism of X that preserves the fibration π . More precisely, there is an element A in Aut ( P k ) of finite order such that π ◦ g = A ◦ π . Changing g into g k where k is theorder of g , we may assume that the action on the base of π is trivial; then, g acts bytranslations along the fibers of π .There is a classification of genus 1 pencils of the plane up to birational conjugacy,which dates back to Halphen (see [19, 21]): a Halphen pencil of index l is a pencilof curves of degree 2 l with 9 base-points of multiplicity l . Every Halphen twist f preserves such a pencil; on X , the pencil corresponds to the genus 1 fibration whichis g -invariant. Action on H ∞ ( P k ) and degree growth.– Let c be the class of the fibers of π inNum ( X ) (resp. in Z ( X ) = Z ( P k ) ). This class is g -invariant (resp. f • -invariant)and isotropic. Thus c ∈ Z ( P k ) determines the unique fixed point of the parabolicisometry f • on ∂ H ∞ ( P k ) .After conjugacy, we may assume that the genus 1 fibration π comes from aHalphen pencil of the plane of index l with nine base points q , . . . , q . This linear ISTORTION IN CREMONA GROUPS 22 system corresponds to the class c such that1 l c = e − ∑ j = e ( q j ) . (5.11)Thus, after conjugacy, we may assume that the Halphen twist g fixes such a class.Under this hypothesis, Lemma 5.10 of [4] provides the following inequality q deg e ( g n + m ) ≤ q deg e ( g n ) + q deg e ( g m ) (5.12)for all integers n , m ≥
0. In particular, the number τ ( g ) = inf n > n q deg e ( g n ) = lim n → + ∞ n q deg e ( g n ) (5.13)is a well defined positive real number, and deg e ( g n ) ≥ τ ( g ) n for all n ≥
1. Blancand Déserti prove that the minimum κ ( g ) = min τ ( hgh − ) for h ∈ Cr ( k ) is a posi-tive rational number and that lim n → + ∞ n deg e ( g n ) = κ ( g ) a for some integer a ≥ ARABOLIC ELEMENTS OF Cr ( k ) AND THEIR INVARIANT HOROBALLS
For simplicity, the hyperbolic space H ∞ ( P k ) will be denoted by H ∞ . In this sec-tion, we prove Theorem 6.1, which states that sufficiently small horoballs invariantby Jonquières or Halphen twists are pairwise disjoint. Combined with Theorem 4.1,this result implies that Halphen twists are not distorted.6.1. Small horoballs associated to Halphen and Jonquières twists.
Fixed points of Jonquières and Halphen twists.
Let f be an element of Cr ( k ) acting as a parabolic isometry on the hyperbolic space H ∞ . Then, f fixes a uniquepoint ξ on the boundary ∂ H ∞ . Up to conjugacy, there are two possibilities: • f is a Jonquières twist, and f preserves the pencil of lines through a point q of P k . Then, setting e = e ( q ) , the boundary point ξ is represented bythe ray R + w , where w J = e − e . (6.1) • f is a Halphen twist. Then, up to conjugacy, ξ is R + w with w H = e − e − e − e − e − e − e − e − e − e , (6.2)where the e i are the classes given by the blow-up of the base-points of aHalphen pencil. ISTORTION IN CREMONA GROUPS 23
Disjonction of horoballs. If w is an element of the Picard-Manin space with w = w · e >
0, the ray R + w determines a boundary point of H ∞ . Let ε be apositive real number. The horoball H w ( ε ) is defined in Section 4.1.4; its elementsare characterized by the following three constraints: v = , v · e > , < v · w < ε . (6.3)When f is a Jonquières or Halphen twist then, after conjugacy, f • preserves thehoroballs centered H w J ( ε ) or H w H ( ε ) . Define ε J = ( √ − ) / ≃ . ε H : = s √ + − √ ≃ . Theorem 6.1.
Let w J be the class e − e ∈ H ∞ ( P k ) determined by the pencil oflines through a point q . If < ε < ε J , the horoballs h ( H w J ( ε )) , for h ∈ Cr ( k ) , arepairwise disjoint; more precisely, given h in Cr ( k ) , either h ( H w J ( ε )) = H w J ( ε ) or h ( H w J ( ε )) ∩ H w J ( ε ) = /0 . Let w H be the class e − e − e − e − e − e − e − e − e − e determined bya Halphen pencil. If < ε ≤ ε H , the horoballs h ( H w H ( ε )) , h ∈ Cr ( k ) , are pairwisedisjoint; more precisely, given h in Cr ( k ) , either h ( H w H ( ε )) = H w H ( ε ) or h ( H w H ( ε )) ∩ H w H ( ε ) = /0 . Proof of the first assertion. w instead of w J . Let h be an element of Cr ( k ) . If h • fixes the line R + w , then it fixes w and its dynamical degree is equal to 1; thus, h fixes the horoballs H w ( ε ) . We may therefore assume that h • does not fix w . Write h • ( w ) = h • ( e − e ) = m e − ∑ i r i e i (6.5)for some multiplicities r i in Z + . Since w =
0, we get m = ∑ i r i . (6.6)For later purpose, we shall write r = m − s for some integer s ≥
0. Then, s + ∑ j ≥ r j = ms . (6.7) Remark 6.2.
We have h • ( w ) = w if and only if m = r =
1, if and only if s =
0. Indeed, if s =
0, then the last equation implies that all r j vanish for j ≥ h • ( w ) = mw for some m ≥ h is parabolic, and m must be equal to thedynamical degree of h , so that m = ISTORTION IN CREMONA GROUPS 24 h • ( H w ( ε )) intersects H w ( ε ) . Then, there exists a point u in theintersection. Write u = α e − ∑ i α i e i . (6.8)By definition of H w ( ε ) , we have 0 < w · u < ε and 0 < h • ( w ) · u < ε , i.e.0 < α − α < ε and 0 < m α − ∑ i r i α i < ε (6.9)We shall write α = α − τ with 0 < τ < ε . Since u · e > α > u = ∑ i α i = α − , (6.10)and therefore τ + ∑ j ≥ α j = α τ − . (6.11)6.2.3. In a first step, we prove a lower estimate for α . By Equation (6.9), m α < ε + ∑ i α i r i . (6.12)Apply Cauchy-Schwartz inequality and use Equations (6.5) and (6.10) to obtain m α < ε + ( ∑ i α i ) / ( ∑ i r i ) / = ε + ( α − ) / ( m ) / . (6.13)This gives m α ( − ( − / α ) / ) < ε . (6.14)Then, remark that ( − t ) / ≤ − t /
2, to deduce 1 − ( − / α ) / ≥ α , and injectthis relation in the previous inequality to get m ε < α . (6.15)6.2.4. Isolate r α in Equation (6.9), i.e. write m α − r α − ∑ j ≥ α j r j < ε , toobtain s α + m τ < ε + s τ + ∑ j ≥ α j r j . (6.16)Then, remark that m τ ≥
0, and apply Cauchy-Schwartz estimate to the vectors ( s , ( r j ) j ≥ ) and ( τ , ( α j ) j ≥ ) ; from Equations (6.11) and (6.7) we get s α < ε + ( α τ − ) / ( ms ) / (6.17) < ε + ( α ε ) / ( ms ) / (6.18) ISTORTION IN CREMONA GROUPS 25 because 0 < τ < ε . This gives (cid:16) s m α (cid:17) / < ε ( ms α ) / + ( ε ) / and the inequality α > m / ( ε ) gives (cid:16) s ε (cid:17) / < ε / ( m s / ) / + ( ε ) / . In particular, ( / ε ) / < / ε / + ( ε ) / and 1 / < ε + ε . This is a contradic-tion because ε < ε J .6.3. Proof of the second assertion.
The proof follows the same lines.6.3.1. For simplicity, we write w instead of w H . Let h be an element of Cr ( k ) . If h • the line R w , it fixes also the class w , and its dynamical degree is equal to 1; thus, h • fixes the horoballs H w ( ε ) . Thus, we may assume that h • does not fix w . Write h • ( w ) = me − ∑ i r i e i (6.19)for some r i in Z + . Since w =
0, we get m = ∑ i r i . (6.20)For later purpose, we shall write r i = ( m / ) − s i for each index 1 ≤ i ≤
9. Then ∑ i = s i + ∑ j ≥ r j = ( / ) mS . (6.21)with S : = ∑ i = s i . (6.22) Remark 6.3.
We have h • ( w ) = w if and only if m = r i = ≤ i ≤ S =
0. Indeed, if S =
0, then the last inequality implies that allmultiplicities r j vanish for j ≥
10, and all s i vanish for 1 ≤ i ≤
9. Thus, h • ( w ) = mw , m must be equal to the dynamical degree of h , and m = h • ( H w ( ε )) intersects H w ( ε ) . Then, there exists a point u in theintersection. Write u = α e − ∑ i α i e i . By definition, we have 0 < w · u < ε and0 < h • ( w ) · u < ε , i.e.0 < α − ∑ i = α i < ε and 0 < m α − ∑ i r i α i < ε (6.23) ISTORTION IN CREMONA GROUPS 26
We shall write α i = ( / ) α − τ i for 1 ≤ i ≤
9, and T = ∑ i = τ i . Then,0 < T < ε . (6.24)Since u · e > α >
0, and since u = ∑ i α i = α − . (6.25)Thus, ∑ i = τ i + ∑ j ≥ α j = ( / ) α T − . (6.26)6.3.3. The following lower estimate is obtained as in the case w = w J : m ε < α . (6.27)6.3.4. Now, isolate the terms r i α i , for i between 1 and 9, in Equation (6.23): m α − ∑ i = r i α i − ∑ j ≥ α j r j < ε (6.28)We obtain ( m − ( / ) ∑ i = r i ) α + ∑ i r i τ i < ε + ∑ j ≥ α j r j (6.29)i.e. ( / ) S α + ( / ) mT < ε + ∑ i = s i τ i + ∑ j ≥ α j r j (6.30)Apply again, the fact that mT ≥ ( / ) S α − ε < (( / ) α T − ) / (( / ) mS ) / < ( / )( α ε ) / ( mS ) / (6.31)because 0 < T < ε . This gives (cid:18) Sm α (cid:19) / < ε ( mS α ) / + ( / )( ε ) / (6.32)and the inequality α > m / ( ε ) implies (cid:18) S ε (cid:19) / < ε / ( m S / ) / + ( / )( ε ) / (6.33)In particular, ( / ε ) / < / ε / + ( / )( ε ) / , in contradiction with ε < ε H . ISTORTION IN CREMONA GROUPS 27
Consequence: Halphen twists are not distorted.
Let h ∈ Cr ( k ) be a Halphentwist. After conjugacy, we may assume that h • preserves the class w H associated tosome Halphen pencil. We know from Section 5.6.2 that the degree growth of h isquadratic, with deg e ( h n ) ≥ ( τ ( h ) n ) . (6.34)Since deg e ( h n ) is equal to cosh ( dist ( h • e , e )) , we obtain the lower boundlog dist ( h • e , e )) ≥ ( n ) − ( τ ( h )) . (6.35)Set H m = H ∞ ( P k ) , f = g • , G = Cr ( k ) , B = H w H ( ε H / ) , and C =
2. By Theo-rem 6.1 if g is an element of G then g ( B ) = B or g ( B ) ∩ B = /0 . Thus, we may applyTheorem 4.1 to f = h • and we get the desired result: h is undistorted in Cr ( k ) .6.5. Non-rational surfaces.
The previous paragraph makes use of the explicit de-scription of Halphen pencils in P k . Here, we consider a smooth projective surface X , over the algebraically closed field k , and assume that • X is not rational; • f is a birational transformation of X with deg ( f n ) ≃ n (we shall say that f is a Halphen twist of X ).Then, from Theorem 5.2, we know that f preserves a unique pencil of genus 1. Lemma 6.4.
The Kodaira dimension of X is equal to or . The surface X has aunique minimal model X , and Bir ( X ) = Aut ( X ) .Proof. A Halphen twist has infinite order, thus
Bir ( X ) is infinite, and the Kodairadimension of X is <
2. If it is equal to − ∞ , then X is a ruled surface, and since X isnot rational, the ruling is unique and Bir ( X ) -invariant. Thus, f must preserve twopencils. These two rational fibrations determine two f • -invariant isotropic classesin Z ( X ) , in contradiction with the fact that f • is parabolic. This proves the firstassertion. The second one is a well known consequence of the first. (cid:3) We can therefore conjugate f to an automorphism f of X , and assume that Bir ( X ) = Aut ( X ) . Thus, the distortion of f in Bir ( X ) is now equivalent to the dis-tortion of f in Aut ( X ) . Instead of looking at the infinite dimensional vector space Z ( X ) , we can look at the action of Aut ( X ) on the Néron-Severi group Num ( X ) .Identify Num ( X ) to Z r , where r is the Picard number of X , and denote by q the intersection form on Num ( X ) . Then, the image of Aut ( X ) in GL ( Num ( X )) is a subgroup of the orthogonal group O + ( q ; Z ) preserving the hyperbolic space H r ⊂ Num ( X ; R ) defined by q . The quotient V = H r / O + ( q ; Z ) is a hyperbolicorbifold, and the fixed point ξ of f in ∂ H r gives a cusp of V . A sufficiently small ISTORTION IN CREMONA GROUPS 28 horoball B centered at ξ determines a neighborhoods of this cusp (see [30]). Thus,if g is an element of O + ( q ; Z ) , then g ( B ) = B or g ( B ) ∩ B = /0 , as in Theorem 6.1.From Theorem 4.1, we deduce that f is undistorted. We have proved: Theorem 6.5.
Let k be an algebraically closed field. Let X be a smooth projectivesurface, defined over k . If f ∈ Bir ( X ) is a Halphen twist (i.e. deg ( f n ) ≃ n ), then fis not distorted in Bir ( X ) .
7. J
ONQUIÈRES TWISTS ARE UNDISTORTED
The argument presented in Section 6.4 to show that Halphen twists are undis-torted is not sufficient for de Jonquières twists; it only gives a quadratic upper boundon the distortion function. As we shall see, the following result follows from [5].
Theorem 7.1.
Let k be an algebraically closed field, and let X k be a projectivesurface. Let f be an element of Bir ( X ) . If f is a Jonquières twist (i.e. if the sequence deg ( f n ) grows linearly) then f is not distorted in Bir ( X ) . In the Cremona group.
We first describe the proof when X is the projectiveplane. Denote by bp : Cr ( k ) → Z + the function number of base-points : bp ( f ) is the number of base-points of the homaloidal net of f , i.e. of the linear systemof curves obtained by pulling-back the system of lines in P . Indeterminacy pointsare examples of base-points, but the base-point set may also include infinitely nearpoints. The number of base-points is also the number of blow-ups needed to con-struct a minimal resolution of the indeterminacies of f . If f • denotes the action of f on the Picard-Manin space, and e is the class of a line, then ( f − ) • e = de − ∑ i m i e ( p i ) (7.1)where d is the degree of f and m i is the multiplicity of the homaloidal system f ∗ O ( ) at the base-point p i ; thus, bp ( f ) is just the number of classes for which themultiplicity m i is positive. The number of base-points is non-negative, is subaddi-tive, and is symmetric (see [5]): bp ( f ◦ g ) ≤ bp ( f ) + bp ( g ) and bp ( f ) = bp ( f − ) .As a consequence, the limit α ( f ) = lim n → + ∞ n bp ( f n ) (7.2)exists and is non-negative. It is symmetric, i.e. α ( f − ) = α ( f ) , invariant underconjugacy, and it vanishes if f is distorted, because if f is distorted its stable lengthvanishes (Lemma 1.3) and this implies α ( f ) = ISTORTION IN CREMONA GROUPS 29
Blanc and Déserti prove that α ( f ) is a non-negative integer, and that it vanishesif and only if f is conjugate to an automorphism by a birational map π : X P . In particular, bp ( f ) > f is a Jonquières twist, α ( f ) coïncides with the integer µ ( f ) which was defined inSection 5.6.1. Theorem 7.1 follows from those results.7.2. In Bir ( X ) . The definition of bp ( f ) extends to birational transformations ofarbitrary smooth surfaces; again, its stable version α ( f ) is invariant under con-jugacy, vanishes when f is distorted, and may be interpreted as the number ofterms in the decomposition f n e = u X + ∑ i a i e i in the Picard-Manin space Z ( X ) = Num ( X ) ⊕ i Ze i (see Section 5), where e is any ample class in Num ( X ) . (The proofsof Blanc and Déserti extend directly to this general situation.)If f ∈ Bir ( X ) is a parabolic element with deg ( f n ) ≃ n , and if X is not a rationalsurface, one can do a birational conjugacy to assume that X is the product C × P k of a curve of genus g ( C ) ≥ Bir ( X ) preserves theprojection π : X → C , acting by automorphisms on the base.The Néron-Severi group of X has rank 2, and is generated by the class v of avertical line { x } × P k and by the class h of a horizontal section C × { y } . Thecanonical class k X is 2 ( g − ) v − h , where g is the genus of the curve C . Blowing-up X , the canonical class of the surfaces X ′ → X determines a limit˜ k = ( g − ) v − h + ∑ i e i (7.3)where the e i are the classes of all exceptional divisors, as in Section 5. This limit isnot an element of the Picard-Manin space Z ( X ) , but it determines a linear form onthe Z -module Z ( X ) ; this form is invariant under the action of Bir ( X ) on Z ( X ) .As an ample class, take e = √ − ( v + h ) . This is an element of H ∞ ( X ) . If f is an element of Bir ( X ) , it preserves the class v of the fibers of π : X → C ; hence √ f • ( e ) = h + d v − ∑ a i e i for some multiplicities a i ∈ Z + . Applied to √ f • ( e ) ,the invariance of the canonical class leads to the following constraint:2 ( d − ) = ∑ i a i . (7.4)And the invariance of the intersection form gives2 ( d − ) = ∑ i a i . (7.5) ISTORTION IN CREMONA GROUPS 30
Thus, a i = ( d − ) non-zero terms in the sum ∑ i a i e i .We get √ f • ( e ) = h + d v − ( d − ) ∑ i = e i (7.6)When f is a Jonquières twist, then deg ( f n ) ≃ n , and the number bp ( f n ) of terms inthe sum also grows linearly, like 2 deg ( f n ) . Thus, α ( f ) >
0, extending the result ofBlanc and Déserti to all surfaces. This concludes the proof of Theorem 7.1.8. A
PPENDIX : TWO EXAMPLES
Baumslag-Solitar groups.
Fix a pair of integers k , ℓ ≥
2. In the Baumslag-Solitar group B k = h t , x | txt − = x k i , we have δ x ( n ) ≃ exp ( n ) (see [22], § 3.K1). Inthe "double" Baumslag-Solitar group B k ,ℓ = h t , x , y | txt − = x k , xyx − = y ℓ i , we have t n xt − n = x k n ∈ S n + and x k n yx − k n = y ℓ kn ∈ S n + ; hence, δ y , S ( n + ) ≥ ℓ k n and the distortion of y in B k ,ℓ is at least doubly exponential. In fact, we can check δ y ( n ) ≃ exp exp ( n ) in B k ,ℓ as follows. Consider the homeomorphisms of the reallines R which are defined by Y ( s ) = s + X ( s ) = ℓ s , and T ( s ) = sign ( s ) | s | k ; therelations satisfied by t , x and y in B k ,ℓ are also satisfied by T , X and Y in Homeo ( R ) :this gives a homomorphism from B k ,ℓ to Homeo ( R ) . If f is any of the three home-omorphisms T , X and Y or their inverses, it satisfies | f ( s ) | ≤ max ( ℓ, | s | k ) . Thus,a recursion shows that every word w of length n in the generators is a homeomor-phism satisfying | w ( ) | ≤ ( ℓ ) k n . Since Y m ( ) = m , this shows that the distortion of y is at most doubly exponential.8.2. Locally nilpotent groups.
Consider the group M of upper triangular (infinite)matrices whose entries are indexed by the ordered set Q of rational numbers, thecoefficients are rational numbers, and the diagonal coefficients are all equal to 1:(1) M is perfect (it coincides with it derived subgroup), and torsion free;(2) M is locally nilpotent (every finitely generated subgroup is nilpotent);(3) for every integer d ≥
1, the elementary matrix U = Id + E , is in the d -thderived subgroup of a finitely generated, nilpotent subgroup N d of M .The first two assertions are described in [31] § 6.2; the last one follows from thefollowing two simple remarks: the elementary matrix Id + E d , d + is in the center ofthe group of upper triangular matrices of SL d + ( Q ) ; the translation α α − d is anorder preserving permutation of Q , and this actions determines an automorphism ofthe group M that maps Id + E d , d + to U . Property (3) implies that the distortion of ISTORTION IN CREMONA GROUPS 31 U in N d is n d . This implies that the distortion of U in M is at least n d for all d ; butits distortion is polynomial in every finitely generated subgroup of M .R EFERENCES [1] Maria Alberich-Carramiñana.
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NIVERSITÉ DE R ENNES
1, F
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AND U NIV L YON , U
NIV C LAUDE B ERNARD L YON
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NSTITUT C AMILLE J ORDAN ,43
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