ALGEBRAIC ACTIONS OF DISCRETE GROUPS: THE p-ADIC METHOD

THE p-ADIC METHOD

SERGE CANTAT AND JUNYI XIE

ABSTRACT. We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based onp-adic analysis, the second makes use of isoperimetric inequalities and Lang- Weil estimates. For instance, we show that ifSL_n(Z)acts faithfully on a com- plex quasi-projective varietyX by birational transformations, then dim(X)≥ n−1 andXis rational if dim(X) =n−1.

CONTENTS

1. Introduction 1

2. Tate analytic diffeomorphisms of the p-adic polydisk 5 3. Good models, p-adic integers, and invariant polydisks 14 4. Regular actions ofSL_n(Z)on quasi-projective varieties 19

5. Actions ofSL_n(Z)in dimensionn−1 23

6. Mapping class groups and nilpotent groups 29

7. Periodic orbits and invariant polydisks 33

8. Birational actions of lattices on quasi-projective varieties 45

9. Appendix 48

References 50

1. INTRODUCTION

1.1. Automorphisms and birational transformations. Let X be a quasi-pro- jective variety of dimension d, defined over the field of complex numbers. Let Aut(X)denote its group of (regular) automorphisms andBir(X)its group of bi- rational transformations. A good example is provided by the affine space A^dC of dimensiond≥2: Its group of automorphisms is “infinite dimensional” and con- tains elements with a rich dynamical behavior (see [36, 3]); its group of birational transformations is the Cremona group Cr_d(C), and is known to be much larger thanAut(A^d_C).

Date: 2014/2017 (last version, May 2017).

1

We present two new arguments that can be combined to study finitely generated groups acting by automorphims or birational transformations: They lead to new constraints on groups of birational transformations,in any dimension.

The first argument is based on p-adic analysis and may be viewed as an ex- tension of two classical strategies from a linear to a non-linear context. The first strategy appeared in the proof of the theorem of Skolem, Mahler, and Lech, which says that the zeros of a linear recurrence sequence occur along a finite union of arithmetic progressions. This method plays now a central role in arithmetic dy- namics (see [7, 6, 55]). The second strategy has been developed by Bass, Milnor, and Serre when they obtained rigidity results for finite dimensional linear rep- resentations of SL_n(Z) as a corollary of the congruence subgroup property (see [1, 62]). Here, we combine these strategies for nonlinear actions of finitely gener- ated groups of birational transformations.

Our second argument makes use of isoperimetric inequalities from geometric group theory and of the Lang-Weil estimates from diophantine geometry. We now list the main results that follow from the combination of those arguments.

1.2. Actions ofSL_n(Z). Consider the groupSL_n(Z)ofn×nmatrices with inte- ger entries and determinant 1. LetΓbe a finite index subgroup ofSL_n(Z); it acts by linear projective transformations on the projective spacePⁿ⁻¹_C , and the kernel of the homomorphismΓ→PGL_n(C)contains at most 2 elements. The following result shows thatΓdoes not act faithfully on any smaller variety.

Theorem A.LetΓbe a finite index subgroup ofSL_n(Z). Let X be an irreducible, complex, quasi-projective variety. If Γ embeds into Aut(X), then dim_C(X) ≥ n−1. Ifdim_C(X) =n−1there is an isomorphismτ: X→Pⁿ⁻¹_C which conjugates the action ofΓon X to a linear projective action onPⁿ⁻¹_C .

Letkandk⁰be fields of characteristic 0. Every field of characteristic 0 which is generated by finitely many elements embeds intoC. Since finite index subgroups ofSL_n(Z)are finitely generated, Theorem A implies: (1)The groupSL_n(Z)em- beds intoAut(A^dk)if and only ifd≥n; (2) ifAut(A^dk)embeds intoAut(A^d

0

k⁰)(as abstract groups) then d ≤d⁰. Previous proofs of Assertion (2) assumedk to be equal toC(see [29, 45]).

1.3. Lattices in simple Lie groups. Theorem A may be extended in two direc- tions, replacing SL_n(Z)by more general lattices, and looking at actions by bira- tional transformations instead of automorphisms. Let Sbe an absolutely almost simple linear algebraic group which is defined overQ; fix an embedding ofSin GL_n (over Q). The Q-rank of S is the maximal dimension of a Zariski-closed

subgroup of Sthat is diagonalizable over Q; theR-rank of Sis the maximal di- mension of a Zariski-closed subgroup that is diagonalizable overR. The subgroup S(Z)is a lattice inS(R); it is co-compact if and only if theQ-rank ofSvanishes (see [8]).

Theorem B.Let X be an irreducible complex projective variety. LetS⊂GL_n be a linear algebraic group, over the field of rational numbersQ. Assume that Sis absolutely almost simple, and that the latticeS(Z)is not co-compact inS(R). If a finite index subgroup ofS(Z)embeds intoBir(X), thendim_C(X)≥rank_R(S).If dim(X) =rank_R(S)≥2, thenS_RisR-isogeneous toSL_dim(X)+1,R.

As a corollary,the Cremona groupCr_d(k)does not embed intoCr_d⁰(k⁰)ifkand k⁰have characteristic0andd>d⁰.

Remark 1.1. Ifrank_R(S)≥2, every latticeΓofS(R)is almost simple: Its normal subgroups are finite and central, or co-finite (see § 4.2 and 8.1). Thus, the assump- tion “Γembeds intoBir(X)” can be replaced by “there is a homomorphism from ΓtoBir(X)with infinite image”. Using the Margulis arithmeticity theorem, one can replace Sby any simple real Lie group H with rank_R(H)≥2, and S(Z) by any non-uniform lattice ofH in the statement of Theorem B.

Remark 1.2. The statement of Theorem B concerns non-uniform lattices because the proof makes use of the congruence subgroup property, and the congruence kernel is known to be finite for all those lattices. There are also uniform lattices for which this property is known and the same proof applies (for instance for all lattices inQ-anisotropic spin groups for quadratic forms inm≥5 variables with real rank≥2, see [43, 64]).

Remark 1.3. The main theorems of [11, 19] extend Theorem B to all types of lattices (including co-compact lattices) in simple real Lie groups but assume that the action is by regular automorphisms on a compact kähler manifold. WhenX is compact,Aut(X)is a complex Lie group: It may have infinitely many connected components, but its dimension is finite. The techniques of [11, 19] do not apply to arbitrary quasi-projective varieties (for instance toX =A^d_C) and to groups of bi- rational transformations. See [17, 13, 28] for groups of birational transformations of surfaces.

Example 1.4. There is a lattice in SO_1,9(R) which acts faithfully on a rational surface by regular automorphisms: This is due to Coble (see [16], § 3.4). A similar phenomenon holds for general Enriques surfaces (see [23]). Thus, lattices in simple Lie groups of large dimension may act faithfully on small dimensional varieties.

1.4. Finite fields and Hrushovski’s theorem. LetΓbe a finite index subgroup ofS(Z), whereSis as in Theorem B. To prove Theorems A and B, we first change the field of definition, replacing C by the field of p-adic numbersQ_p for some large prime number p; indeed, since the ring generated by the coefficients of the formulae defining the variety X and the generators of the group Γ⊂ Bir(X) is finitely generated, we may embed this ring inZ_pfor some large prime p.

Then, we prove that there exists a finite extensionKofQ_pand ap-adic polydisk inX(K)which is invariant under the action of a finite index subgroupΓ⁰ofΓ, and on whichΓ⁰acts by Tate analytic diffeomorphisms. Those polydisks correspond to periodic orbits for the action of Γ on X(F), where F is the residue field of the non-archimedian field K. Thus, an important step toward Theorem B is the existence of pairs (m,Γ⁰), where m is in X(F), Γ⁰ is a finite index subgroup of Γ, and all elements ofΓ⁰ are well-defined at m and fix m(no element of Γ⁰ has an indeterminacy at m). For cyclic groups of transformations, this follows from a theorem of Hrushovski (see [41]). Here, we combine the Lang-Weil estimates with isoperimetric inequalities from geometric group theory: The existence of the pair(m,Γ⁰) is obtained for groups with Kazhdan Property (T) in Theorems 7.10 and 7.13; the argument applies also to other types of groups (see Section 7.2.5).

Once such invariant polydisks are constructed, several corollaries easily follow (see § 7.4.2). For instance, we get:

Theorem C.If a discrete group Λ with Kazhdan Property (T) acts faithfully by birational transformations on a complex projective variety X , the groupΛis resid- ually finite and contains a torsion-free, finite index subgroup.

1.5. The p-adic method. When an invariant p-adic polydisk is constructed, a theorem of Bell and Poonen provides a tool to extend the action of every element γ in our group into a Tate analytic action of the additive group Z_p. WhenΓ has finite index inS(Z), as in Theorem B, andrank_R(S)≥2, this may be combined with the congruence subgroup property: We prove that the action of the lattice extends to an action of a finite index subgroup of the p-adic groupS(Z_p)by Tate analytic transformations (Theorem 2.11). Thus, starting with a countable group of birational transformations, we end up with an analytic action of ap-adic Lie group to which Lie theory may be applied. This is how Theorems A and B are proven;

our strategy applies also to actions of other discrete groups, such as the mapping class group of a closed surface of genusg, or the group of outer automorphisms of a free group (see Section 6 and [2]). Let us state a sample result.

Given a groupΓ, letma(Γ)be the smallest dimension of a complex irreducible variety on which some finite index subgroup of Γ acts faithfully by automor- phisms. Let Mod(g) be the mapping class group of a closed orientable surface

of genusg. It is known that ma(Mod(g))≤6g−6 for allg≥2 (see § 6.1), and that ma(Mod(1)) =1 (because a finite index subgroup of GL₂(Z) embeds into PGL₂(C)).

Theorem D.IfMod(g)acts faithfully on a complex variety X by automorphisms, thendim(X)≥2g−1. Thus,2g−1≤ma(Mod(g))≤6g−6for every g≥2.

1.6. Margulis super-rigidity and Zimmer program. LetΓbe a lattice in a sim- ple real Lie groupS, withrank_R(S)≥2. According to the Margulis super-rigidity theorem, unbounded linear representations of the discrete groupΓ“come from”

linear algebraic representations of the group Sitself. As a byproduct, the small- est dimension of a faithful linear representation ofΓcoincides with the smallest dimension of a faithful linear representation ofS(see [51]).

The Zimmer program asks for an extension of this type of rigidity results to non-linear actions ofΓ, for instance to actions ofΓby diffeomorphisms on com- pact manifolds (see [67, 68], and the recent survey [34]). Theorems A and B are instances of Zimmer program in the context of algebraic geometry.

WhenΓ=SL_n(Z)orSp_2n(Z), Bass, Milnor and Serre obtained a super-rigidity theorem from their solution of the congruence subgroup problem (see [1], § 16, and [62]). Our proofs of Theorems A and B may be considered as extensions of their argument to the context of non-linear actions by algebraic transformations.

1.7. Notation. To specify the field (or ring) of definitionKof an algebraic variety (or scheme)X, we use the notationX_K. IfK⁰is an extension ofK,X(K⁰)is the set ofK⁰-points ofX. The group of automorphisms (resp. birational transformations) ofX which are defined overK⁰is denotedAut(X_K⁰)(resp. Bir(X_K⁰)).

1.8. Acknowledgement. Thanks to Yves de Cornulier, Julie Déserti, Philippe Gille, Sébastien Gouezel, Vincent Guirardel, and Peter Sarnak for interesting dis- cussions related to this article. We thank the referees for numerous insightful re- marks, and in particular for their suggestions regarding Sections 2.4, 6 and 7. This work was supported by the ANR project BirPol and the foundation Del Duca from the French Academy of Sciences, and the Institute for Advanced Study, Princeton.

2. TATE ANALYTIC DIFFEOMORPHISMS OF THE p-ADIC POLYDISK

In this section, we introduce the group of Tate analytic diffeomorphisms of the unit polydisk

U

^=Zdp, describe its topology, and study its finite dimensional subgroups. The main result of this section is Theorem 2.11.

2.1. Tate analytic diffeomorphisms.

2.1.1. The Tate algebra (see [59], § 6). Let p be a prime number. Let K be a field of characteristic 0 which is complete with respect to an absolute value| · | satisfying|p|=1/p; such an absolute value is automatically ultrametric (see [44], Ex. 2 and 3 Chap. I.2). Good examples to keep in mind are the fields of p-adic numbers Q_p and its finite extensions. Let Rbe the valuation ring of K, i.e. the subset of K defined by R={x∈K; |x| ≤1}; in the vector space K^d, the unit polydisk is the subsetR^d.

Fix a positive integerd, and consider the ringR[x] =R[x₁, ...,x_d]of polynomial functions indvariables with coefficients inR. For f inR[x], define the normk f k to be the supremum of the absolute values of the coefficients of f:

k f k=sup

I

|a_I| (2.1)

where f =∑_{I=(i1,...,id)}a_Ix^I. By definition, theTate algebraRhxiis the completion ofR[x]with respect to the normk · k. The Tate algebra coincides with the set of formal power series f =∑Ia_Ix^I, I∈Z^d₊, converging (absolutely) on the closed unit polydiskR^d. Moreover, the absolute convergence is equivalent to|a_I| →0 as kIk→∞.

For f and gin Rhxiand cin R+, the notation f ∈ p^cRhximeans k f k≤ |p|^c and the notation

f ≡g(modp^c) (2.2)

meansk f−gk≤ |p|^c; we then extend such notations component-wise to(Rhxi)^m for all m≥1. For instance, with d =2, the polynomial mapping f(x) = (x₁+ p,x₂+px₁x₂)satisfies f ≡id(modp), where id(x) =xis the identity.

2.1.2. Tate diffeomorphisms. Denote by

U

the unit polydisk of dimensiond, that is

U

^{=Rd. For x and y in}

U

, the distance dist(x,y) is defined by dist(x,y) = max_i|x_i−y_i|, where thex_i andy_i are the coordinates of xandyin R^d. The non- archimedean triangle inequality implies that |h(y)| ≤1 for every h in Rhxi and y∈

U

. Consequently, every element gin Rhxi^d determines a Tate analytic map g:

U

^→

U

^.

If g= (g₁, . . . ,g_d) is an element of Rhxi^d, the norm kgk is defined as the maximum of the normskg_ik(see Equation (2.1)); one has

kgk≤1 and dist(g(x),g(y))≤kgkdist(x,y), (2.3) so thatgis 1-Lipschitz.

For indeterminatesx= (x₁, . . . ,x_d)andy= (y₁, . . . ,y_m), the compositionRhyi×

Rhxi^m→Rhxiis well defined, and hence coordinatewise we obtain Rhyiⁿ×Rhxi^m→Rhxiⁿ.

In particular, withm=n=d, we get a semigroupRhxi^d. The group of (Tate)ana- lytic diffeomorphismsof

U

is the group of invertible elements in this semigroup;

we denote it byDiff^an(

U

). Elements ofDiff^an(

U

⁾are transformations f:

U

^→

U

given by

f(x) = (f₁, . . . ,f_d)(x)

where each f_iis inRhxiand f has an inverse f⁻¹:

U

^→

U

that is also defined by power series in the Tate algebra. The distance between two Tate analytic diffeo- morphisms f andgis defined ask f−gk; by the following Lemma, this endows Diff^an(

U

⁾with the structure of a topological group.

Lemma 2.1. Let f , g, and h be elements of Rhxi^d. (1) kg◦f k≤kgk;

(2) if f is an element ofDiff^an(

U

^{)thenkg◦f k=kgk;}

(3) kg◦(id+h)−gk≤khk;

(4) k f⁻¹−idk=k f−idkif f is a Tate analytic diffeomorphism.

Proof. Lets∈Randc>0 satisfy|p|^c=|s|=kgk. Then(1/s)gis an element of Rhxi^d. It follows that(1/s)g◦f is an element ofRhxi^dtoo, and thatkg◦f k≤ |p|^c. This proves Assertion (1). The second assertion follows becauseg= (g◦f)◦f⁻¹. To prove Assertion (3), writeh= (h₁,h₂, . . . ,h_d)where eachh_isatisfieskh_ik≤k hk. Theng◦(id+h)takes the form

g◦(id+h) =g+A₁(h) +

∑

i≥2

A_i(h)

where eachA_iis a homogeneous polynomial in(x₁, . . . ,x_d)of degreeiwith coef- ficients inR. Assertion (3) follows. For Assertion (4), assume that f is an analytic diffeomorphism and apply Assertion (2): k f⁻¹−idk=kid−f k.

This lemma easily implies the following proposition (see [18] for details).

Proposition 2.2. For every real number c>0, the subgroup of all elements f ∈ Diff^an(

U

^{)with f ≡id(modpc)}is a normal subgroup ofDiff^an(

U

^).

Lemma 2.3. Let f be an element ofDiff^an(

U

^{). If f(x)≡id(mod pc), with c≥1,}

and p^N divides l, then the l-th iterate of f satisfies f^l(x)≡id(mod p^c+N). In particular, if f ≡id(mod p), then f^{p`</sup> ≡id(mod p<sup>`}).

Proof. Write f(x) =x+sr(x) wherer is in Rhxi^d ands∈R satisfies|s| ≤ |p|^c. Then

f◦f(x) = x+sr(x) +sr(x+sr(x))

= x+2sr(x) +s²u₂(x)

for some u₂ ∈Rhxi^d. After k iterations one gets f^k(x) =x+ksr(x) +s²u_k(x), withu_k∈Rhxi^d. Takingk= p, we obain

f^p(x) = x+psr(x) +s²u_p(x)

≡ x(modp^c+1)

becausec≥1. Then, f^p2(x)≡x(mod p^c+2)and f^pN(x)≡x(modp^c+N).

2.2. From cyclic groups to p-adic flows.

2.2.1. From cyclic groups to R-flows. The following theorem is due to Bell and to Poonen (see [55], as well as [7] Lemma 4.2, and [6] Theorem 3.3).

Theorem 2.4. Let f be an element of Rhxi^d with f ≡id(modp^c) for some real number c>1/(p−1). Then f is a Tate diffeomorphism of

U

^{=Rd and there}

exists a unique Tate analytic mapΦ:

U

^×R→

U

^{such that}

(1) Φ(x,n) = fⁿ(x)for all n∈Z;

(2) Φ(x,t+s) =Φ(Φ(x,s),t)for all t, s in R;

(3) Φ: t∈R7→Φ(·,t)is a continuous homomorphism from the abelian group (R,+)to the group of Tate diffeomorphismsDiff^an(

U

^);

(4) Φ(x,t)≡x(modp^{c−1/(p−1)})for all t ∈R.

We shall refer to this theorem as the “Bell-Poonen theorem”, or “Bell-Poonen extension theorem”. An analytic mapΦ:

U

^×R→

U

which defines an action of the group(R,+)will be called anR-flow, or simplya flow. See below, in § 2.2.2, how it is viewed as the flow of an analytic vector field. A flowΦwill be considered either as an analytic action Φ:

U

^×R→

U

of the abelian group (R,+), or as a morphismΦ: t∈R7→Φt=Φ(·,t)∈Diff^an(

U

); we use the same vocabulary (and the same letterΦ) for the two maps. The Bell-Poonen theorem implies that every element f ofRhxi^d with f ≡id(modp²)is included in an analyticR-flow.

Corollary 2.5. Let f be an element of Rhxi^d with f ≡id(modp^c) for some real number c>1/(p−1). Then f is a Tate diffeomorphism of

U

^=Rd, and if f has finite order inDiff^an(

U

^{), then f =id.}

To prove it, assume that f has orderk≥1 and apply the Bell-Poonen theorem.

For everyx∈

U

, the analytic curvet7→Φ(x,t)−xvanishes on the infinite setZk;

hence, it vanishes identically, f(x) =Φ(x,1) =xfor allx∈

U

^{, and f =id.}

Remark 2.6. Theorem 2.4 is not stated as such in [55]. Poonen constructs a Tate analytic map Φ:

U

^×R→

U

which satisfies Property (1) for n≥0; his proof implies also Properties (3) and (4). We now deduce Property (1) for n∈Z. We already know that the relation Φ(x,n+1) = f ◦Φ(x,n) holds for every integer

n≥0. Thus, for everyxin

U

, the two Tate analytic functionst7→Φ(x,t+1)and t7→ f◦Φ(x,t)coincide on Z₊, hence onR by the isolated zero principle. This impliesΦ(x,t+1) = f ◦Φ(x,t)in Rhx,ti^d. Taket =−1 to deduce that f is an analytic diffeomorphism of

U

^{and f−1=Φ(·,}−1). Then, by induction, one gets Φ(x,n) = fⁿ(x)for alln∈Z. Property (2) follows from (1) forsandt inZ, and then for all values ofsandt inRby the isolated zero principle.

2.2.2. Flows and Tate analytic vector fields. Consider the Lie algebra Θ(

U

^{) of}

vector fields X=∑^d_i=1u_i(x)∂_i where each u_i is an element of the Tate algebra Rhxi. The Lie bracket with a vector fieldY=∑iv_i(x)∂_iis given by

[X,Y] =

d

∑

j=1

w_j(x)∂_j, with w_j=

d

∑

i=1

u_i∂v_j

∂x_i −v_i∂u_j

∂x_i

.

Lemma 2.7. LetΦ:

U

^×R→

U

be an element of Rhx,ti^d that defines an analytic flow. ThenX=

∂Φ

∂t

|t=0is an analytic vector field. It is preserved byΦt: For all t∈R,(Φ_t)_∗X=X. Moreover,X(x₀) =0if and only ifΦ_t(x₀) =x₀for all t∈R.

The analyticity andΦ_t-invariance are easily obtained. Let us show thatX(x₀) = 0 if and only if x₀ is a fixed point of Φ_t for all t. Indeed, if X vanishes at x₀, thenXvanishes along the curveΦ(x₀,t),t ∈R, becauseXisΦ_t-invariant. Thus,

∂_tΦ(x₀,t) =0 for allt, and the result follows.

Corollary 2.8. If f is an element of Diff^an(

U

^{) with f ≡id(modpc) for some}

c>1/(p−1), then f is given by the flowΦf, at time t=1, of a unique analytic vector fieldX_f. The zeros ofX_f are the fixed points of f .

Two such diffeomorphisms f and g commute if and only if[X_f,X_g] =0.

Proof. The first assertion follows directly from Lemma 2.7 and the Bell-Poonen theorem (Theorem 2.4). Let us prove the second assertion. IfX_f commute toX_g, thenΦ_f andΦ_g commute too, meaning thatΦ_f(Φ_g(x,t),s) =Φ_g(Φ_f(x,s),t)for every pair(s,t)∈R×R. Taking(s,t) = (1,1)we obtain f◦g=g◦ f. If f andg commute, thenΦ_f(Φ_g(x,n),m) = f^m◦gⁿ=Φ_g(Φ_f(x,m),n)for every pair(m,n) of integers, and the principle of isolated zeros implies that the flowsΦ_f andΦ_g

commute; hence,[X_f,X_g] =0.

2.3. A pro-p structure. Recall that a pro-p group is a topological group G which is a compact Hausdorff space, with a basis of neighborhoods of the neutral element 1_Ggenerated by subgroups of index a (finite) power ofp. In such a group, the index of every open normal subgroup is a power of p. We refer to [30] for a good introduction to pro-pgroups.

In this subsection, we assume that K is a finite extension ofQ_p. The residue field, i.e. the quotient ofRby its maximal idealm_K={x∈K; |x|<1}, is a finite field of characteristicp. It hasqelements, withqa power of p, and the number of elements of the ringR/m^k_K is a power of pfor everyk. We also fix an elementπ that generates the idealm_K.

2.3.1. Action modulom^k_K. Recall that

U

denotes the polydiskR^d. Let f be an el- ement ofDiff^an(

U

). Its reduction modulom^k_Kis a polynomial transformation with coefficientsa_I in the finite ringR/m^k_K; it determines a permutation of the finite set (R/m^k_K)^d. Thus, for eachk≥1, reduction modulom^k_Kprovides a homomorphism θ_k: Diff^an(

U

^{)→Perm((R/mk}_K^)d)into the group of permutations of the finite set (R/m^k_K)^d.

Another way to look at the same action is as follows. Each element ofDiff^an(

U

⁾

acts isometrically on

U

with respect to the distance dist(x,y) (see § 2.1.2); in particular, for every radiusr, Diff^an(

U

⁾ acts by permutations on the set of balls of

U

^{of radius}r. Since the set of balls of radius|π|^−k is in bijection with the set (R/m^k_K)^d, the action of Diff^an(

U

⁾ on this set of balls may be identified with its action on(R/m^k_K)^d after reduction modulom^k_K.

As a consequence, an element f of Diff^an(

U

⁾ is the identity if and only if dist(f(x),x) ≤ |π|^k for all x and all k, if and only if its image in the group of permutations of(R/m^k_K)^dis trivial for allk.

2.3.2. A pro-p completion. Given a positive integerr, defineDiff^an(

U

⁾r as the subgroup of Diff^an(

U

⁾ whose elements are equal to the identity modulo p^r. Forr=1, we set

D=Diff^an(

U

⁾1={f ∈Diff^an(

U

^{); f ≡id(modp)}. (2.4)}

By definition, f is an element of D if it can be written f =id+ph whereh is inRhxi^d. Thus,Dacts trivially on (R/pR)^d (here, pR=m^{`</sup><sub>K</sub> with|π<sup>`}|=1/pfor some`).

Now we show that the imageθ_`m(D) in Perm((R/p^mR)^d)is a finite p-group.

Indeed, by Lemma 2.3, for any f ∈D, we have f^pm−1≡id(modp^m).Thus f^pm−1 acts trivially on(R/p^mR)^d. It follows that the order of every element inθ<sub>`m</sub>(D)is a power of p. Sinceθ`m(D)is a finite group, Sylow’s theorem implies thatθ`m(D) is a p-group.

We endow the finite groups Perm((R/p^mR)^d) with the discrete topology, and we denote by ˆDthe inverse limit of the p-groupsθ_{`m</sub>(D)⊂Perm((R/p<sup>m</sup>R)<sup>d</sup>); ˆD is a pro-p group: It is the closure of the image ofDin ∏<sub>m</sub>Perm((R/p<sup>m</sup>R)<sup>d</sup>) by the diagonal embedding(θ<sub>`m})_m≥1. We denote by

T

the topology of ˆD(resp. the

induced topology on D); the kernels of the homomorphismsθ_`m form a basis of neighborhoods of the identity for this topology.

Since the action on(R/p^mR)^d is the action on the set of balls of radius p^−m in

U

, the Tate topology is finer than the topology

T

: The identity map f 7→ f is a continuous homomorphism with respect to the Tate topology on the source, and the topology

T

on the target; we shall denote this continuous injective homomor- phism by

f ∈D7→ fˆ∈D.ˆ

Remark 2.9. Fix a prime p and consider the field K=Q_p, with valuation ring R=Z_p. Assume that the dimensiondis 2. The sequence of polynomial automor- phisms of the affine plane defined by h_n(x,y) = (x,y+p(x+x²+x³+· · ·+xⁿ)) determines a sequence of elements of D. No subsequence of (h_n) converges in the Tate topology, but in the compact group ˆD one can extract a converging sub- sequence. A better example is provided by the sequenceg_n(x,y) = (x,y+s_n(x)) withs_n(x) =xⁿ(x^pn−pn−1−1). This sequence converges towards the identity in ˆD becauses_nvanishes onZ/pⁿZ, but does not converge inDfor the Tate topology.

2.4. Extension theorem.

2.4.1. Analytic groups (see[47], § IV, or[30, 60]and[10]Chapter III). LetGbe a topological group. We say thatGis ap-adic analytic group if there is a structure of p-adic analytic manifold onGwhich is compatible with the topology ofGand the group structure: The group law(x,y)∈G×G7→xy⁻¹is p-adic analytic (see [30], Chapter 8). If such a structure exists, it is unique (see [30], Chapter 9). The dimension dim(G)is the dimension ofGas a p-adic manifold.

IfGis a compact, p-adic analytic group, thenG contains a finite index, open, normal subgroup G₀ which is a (uniform) pro-p group. Moreover, G₀ embeds continuously inGL_d(Z_p)for somed(see [30], Chapters 7 and 8).

Letg be an element of the pro-pgroup G. The homomorphism ϕ: m∈Z7→

g^m∈G extends automatically to the pro-pcompletion ofZ, i.e. to a continuous homomorphism of pro-pgroups

ϕ: Z_p→G.

For simplicity, we denote ϕ(t) by g^t fort in Z_p (see [30], Proposition 1.28, for embeddings ofZ_pinto pro-pgroups).

Lemma 2.10. Let G be a p-adic analytic pro-p group of dimension s=dim(G).

LetΓbe a dense subgroup of G. There exist an integer r≥s and elementsγ₁, ..., γ_r inΓsuch that the mapπ: (Z_p)^r→G,π(t₁, . . . ,t_r) = (γ₁)^t1· · ·(γ_r)^tr, satisfies the following properties:

(1) πis a surjective, p-adic analytic map;

(2) the restriction of π to {(t_i)|t_j =0 if j >s} is a local diffeomorphism onto its image;

(3) as l runs over the set of positive integers, the setsπ((p^lZ_p)^r)form a basis of neighborhoods of the neutral element in G.

Proof. Let g be the Lie algebra of G; as a finite dimensional Q_p-vector space, g coincides with the tangent space of G at the neutral element 1_G. There are finite index open subgroups H ofG for which the exponential map defines a p- adic analytic diffeomorphism from a neighborhood of the origin ing(Z_p)onto the groupH itself (see the notion of standard subgroups in [10, 30]). LetH be such a subgroup.

SinceΓ is dense inG its intersection withH is dense inH. Each α_i∈Γ∩H corresponds to a tangent vectorνi∈gsuch that exp(tν_i) =α^t_i fort∈Z_p. SinceΓ is dense inH, the subspace ofg generated by all theν_i is equal tog. Thus, one can find elements α1, ..., α_s ofΓ, with s=dim(G), such that the ν_i generateg.

Then, the mapπ: (Z_p)^d→H defined by

π(t₁, . . . ,t_s) =exp(t₁ν1)· · ·exp(t_sνs) =α^t₁¹· · ·α^t_s^s

is analytic and, by the p-adic inverse function theorem, it determines a local ana- lytic diffeomorphism from a neighborhood of 0 ing to a neighborhoodV of 1_G. The groupGcan then be covered by a finite number of translatesh_jV, j=1,. . ., s⁰. Since Γis dense, one can find elements βj inΓwithβ⁻¹_j h_j∈V. The lemma follows if one setsr=s+s⁰,γ_i=α_ifor 1≤i≤s, andγ_i=β_i−s,s≤i≤r.

2.4.2. Actions by Tate analytic diffeomorphisms. LetGbe a compact p-adic an- alytic group. Let Γ be a finitely generated subgroup of G. We say that G is a virtual pro-pcompletionofΓif there exists a finite index subgroupΓ0ofΓsuch that (1) the closure of Γ₀ in G is an open pro-p subgroup G₀ of G, and (2) G₀ coincides with the pro-pcompletion ofΓ₀. Note that, by compactness ofG, the groupG₀has finite index inG. A good example to keep in mind isΓ=SL_n(Z)in G=SL_n(Z_p)(see §4.2.3 below).

We now study homomorphisms fromΓ to the groupDiff^an(

U

). Thus, in this paragraph, the same prime number pplays two roles since it appears in the defi- nition of the pro-pstructure ofG, and of the Tate topology onDiff^an(

U

^).

Theorem 2.11. Let G be a compact, p-adic analytic group. LetΓ be a finitely generated subgroup of G. Assume that G is a virtual pro-p completion ofΓ.

Let Φ: Γ→Diff^an(

U

⁾1 be a homomorphism into the group of Tate analytic diffeomorphisms of

U

which are equal to the identity modulo p. Then, there exists

a finite index subgroupΓ0ofΓfor whichΦ_|Γ0extends to the closure G₀=Γ0⊂G as a continuous homomorphism

Φ: G₀→Diff^an(

U

⁾¹

such that the action G₀×

U

^→

U

^{given by(g,x)7→Φ(g)(x)}is analytic.

DenoteDiff^an(

U

⁾1 byD, as in Equation (2.4). Recall from §2.3.2 thatD em- beds continuously into the pro-pgroup ˆD. LetΓ₀ be a finite index subgroup ofΓ whose closureG₀inGis the pro-pcompletion ofΓ₀. We obtain:

(1) The homomorphism Γ₀→Dˆ extends uniquely into a continuous homo- morphism ˆΦfromG₀=Γ0to ˆD.

Then, the following property is automatically satisfied.

(2) Let f be an element of D. By the Bell-Poonen extension theorem (The- orem 2.4), the homomorphism t ∈Z7→ f^t extends to a continuous mor- phism Z_p→D via a Tate analytic flow. If ˆf denotes the image of f in D, thenˆ n7→(fˆ)ⁿ is a homomorphism from Z to the pro-p group ˆD; as such, it extends canonically to the pro-pcompletion Z_p, giving rise to a homomorphismt∈Z_p7→(fˆ)^t∈D. These two extensions are compatible:ˆ (df^t) = (fˆ)^t for alltinZ_p.

Thus, given any one-parameter subgroup Z of Γ, we already know how to ex- tendΦ: Z⊂Γ→Dinto Φ: Z_p⊂G₀→Din a way that is compatible with the extension ˆΦ: G₀→D.ˆ

For simplicity, we now denoteΓ₀andG₀byΓandG.

Lemma 2.12. Let(α_n)be a sequence of elements ofΓthat converges towards1_G in G. ThenΦ(α_n)converges towards the identity inDiff^an(

U

^).

Proof. Writeαn=π(t₁(n), . . . ,t_r(n)) = (γ₁)^t1(n)· · ·(γ_r)^tr(n), whereπand theγiare given by Lemma 2.10 and the t_i(n) are in Z_p. Sinceα_n converges towards 1_G, we may assume, by Lemma 2.10, that each (t_i(n)) converges towards 0 in Z_p as ngoes to +∞. By the Bell-Poonen theorem (Theorem 2.4), each f_i :=Φ(γ_i) gives rise to a flowt7→ f_i^t,t in Z_p; moreover, k f_i^t−idk≤ p^mif |t|<p^m (apply Lemma 2.3 and the last assertion in the Bell-Poonen theorem). Thus, the lemma follows from Lemma 2.1 and the equality

Φ(α_n) = f₁^t1(n)· · ·f_r^tr(n). (2.5) To prove this equality, one only needs to check it in the group ˆDbecauseDembeds into ˆD. But in ˆD, the equality holds trivially because the homomorphismΓ₀→Dˆ extends toG₀continuously (apply Properties (1) and (2) above).

Lemma 2.13. If(α_m)_m≥1is a sequence of elements ofΓthat converges towards an element α_∞ of G, then Φ(α_m) converges to an element of Diff^an(

U

^{) which}

depends only onα_∞.

Proof. Since (α_m) converges, α_m◦α⁻¹_m0 converges towards the neutral element 1_G as mand m⁰ go to +∞. Consequently, Lemma 2.12 shows that the sequence (Φ(α_m))is a Cauchy sequence inDiff^an(

U

), hence a convergent sequence.¹ The limit depends only on α_∞, not on the sequence (α_m) (if another sequence (α⁰_m) converges towardα_∞, consider the sequenceα1,α⁰₁,α2,α⁰₂, ...).

We can now prove Theorem 2.11. Lemmas 2.12 and 2.13 show thatΦextends, in a unique way, to a continuous homomorphism Φ: G→D. Moreover, this extension coincides with Bell-Poonen extensions Z_p→D along one parameter subgroups ofGgenerated by elements ofΓ. According to Lemma 2.10, one can findselementsγ₁, ...,γ_sofΓ, withs=dim(G), such that the map

(t₁, . . . ,t_s)7→π(t₁, . . . ,t_d) =γ^t₁¹· · ·γ^t_s^s

determines an analytic diffeomorphism from a neighborhood of 0 inZ^s_pto a neigh- borhood of the identity inG. By the Bell-Poonen theorem, the map

(t₁, . . . ,t_s,x)∈(Z_p)^s×

U

^7→Φ(γ1)^t1◦ · · · ◦Φ(γ_s)^ts(x)

is analytic. Thus, the action of G on

U

determined by Φis analytic. This con- cludes the proof of Theorem 2.11.

3. GOOD MODELS, p-ADIC INTEGERS,AND INVARIANT POLYDISKS

Start with an irreducible complex varietyX of dimensiond, a finitely generated groupΓ, and a homomorphismρ: Γ→Bir(X). First, we explain how to replace the field C, or any algebraically closed fieldk of characteristic 0, by the ring of p-adic integers Z_p, for some prime p, the variety X by a variety X_Zp which is defined overZ_p, and the homomorphismρby a homomorphism intoBir(X_Zp).

In a second step, we look for a polydisk

U

^'ZdpinX_Zp(Q_p)which is invariant under the action ofΓin order to apply the Bell-Poonen extension theorem (Theo- rem 2.4) on

U

. This is easy whenρ(Γ)is contained inAut(X), but much harder whenΓacts by birational transformations; Section 7 addresses this problem.

1WriteΦ(α_m◦α⁻¹_m0) =id+ε_m,m⁰ whereε_m,m⁰ is equivalent to the constant map 0 in Rhxi^d modulo|p|^k(m,m0), withk(m,m⁰)that goes to+∞asmandm⁰do. Then, apply Lemma 2.1.

3.1. From complex to p-adic coefficients: Good models. Let k be an alge- braically closed field of characteristic 0. LetX=X_k be a quasi-projective variety defined over k, for instance X =A^d, the affine space. Let Γ be a subgroup of Bir(Xk)with a finite, symmetric set of generatorsS={γ₁, . . . ,γ_s}.

3.1.1. From complex to p-adic coefficients. Fix an embedding ofX into a projec- tive spaceP^Nk and write

X=

Z

^(a)\

Z

^(b)

whereaandbare two homogeneous ideals ink[x₀, . . . ,x_N]and

Z

^(a)denotes the zero-set of the ideal a. Choose generators(F_i)_1≤i≤a and(G_j)_1≤j≤b for a andb respectively.

LetCbe a finitely generatedQ-algebra containing the setBof all coefficients of theF_i, theG_j, and the polynomial formulas defining the generatorsγ_k∈S; more precisely, eachγ_k is defined by ratios of regular functions on affine open subsets

V

l =X\W_l and one includes the coefficients of the formulas for these regular functions and for the defining equations of the Zariski closed subsetsW_l. One can viewX andΓas defined over Spec(C).

Lemma 3.1(see [48], §4 and 5, and [7], Lemma 3.1). Let L be a finitely generated extension of Qand B be a finite subset of L. The set of primes p for which there exists an embedding of L intoQ_pthat maps B intoZ_phas positive density among the set of all primes.

Bypositive density, we mean that there existε>0 andN₀>0 such that, among the firstNprimes, the proportion of primespthat satisfy the statement is bounded from below byεifN≥N₀.

Apply this lemma to the fraction fieldL=Frac(C)and the setBof coefficients.

This provides an odd prime p and an embedding ι: L→ Q_p with ι(B)⊂ Z_p. Applyingιto the coefficients of the formulas that defineX and the elements ofΓ, we obtain what will be called a “model of the pair(X,Γ)overZ_p”; in particular, Γ embeds into Bir(X_Zp), or in Aut(X_Zp) ifΓ is initially a subgroup ofAut(X_k).

The following paragraphs clarify this idea.

3.1.2. Good models. Let R be an integral domain. Let X_R andY_R be separated and reduced schemes of finite type defined over R. Assume, moreover, that the morphismX_R→Spec(R)is dominant on every irreducible component ofX_R. Let

U

^and

V

be two dense open subsets ofX_R. Two morphisms ofR-schemes f:

U

^→

Y_Randg:

V

^→YRare equivalent if they coïncide on some dense open subsetW of U∩V; rational maps f: X_R 99KY_R are equivalence classes for this relation ([40, 7.1.]). For any rational map f, there is a maximal open subset Dom(f)⊂X_R on

which f induces a morphism: If a morphism

V

^→YR is in the equivalence class of f, then

V

is contained in Dom(f). This open subset Dom(f)is the domain of definition of f ([40, 7.2.]); its complement is the indeterminacy locus Ind(f).

A rational map f is birational if there is a rational mapg:Y_R99KX_R such that g◦ f =Id and f ◦g= Id. The group of birational transformations f: X_R 99K X_R is denoted Bir(X_R); the group of regular automorphisms is denoted Aut(X_R).

Consider a birational map f: X99KY and denote bygthe inverse map f⁻¹. The domains of definition Dom(f) and Dom(g) are dense open subsets ofX andY respectively, for the Zariski topology. Then, setU_R,f = (f_|Dom(f))⁻¹(Dom(g)).

Since f is birational, U_R,f is open and dense in X. The restriction of f to this open subset is an open immersion ofU_R,f intoY; indeed, f(U_R,f)is the open set (g_|Dom(g))⁻¹(U_R,f)andgis a morphism on f(U_R,f)such thatg◦f =Id. Moreover, U_R,f is the largest open subset ofX on which f is locally, for the Zariski topology, an open immersion. In what follows, we denote byB_R,f the complement ofU_R,f in X_R: this nowhere dense Zariski closed subset is the set of bad points; on its complement, f is an open immersion.

Fory∈Spec(R), denote byX_y the reduced fiber ofX_R above y. IfX_y∩B_R,f is nowhere dense inX_y, then f induces a birational transformation f_y: X_y99KX_y; we have Ind(f_y)⊂X_y∩B_R,f, and this inclusion may be strict.

If η is the generic point of Spec(R), we can always restrict f to X_η. This map f 7→ f_η determines an isomorphism of groupsi:Bir(X_R)→Bir(X_η). More precisely,letKbe the fraction field of the integral domainR; letX_Rbe a separated and reduced scheme over R; assume that X_R is of finite type overR, and that the morphismX_R→Spec(R)is dominant on every irreducible component ofX_R; then, the mapi:Bir(X_R)→Bir(X_K)is bijective. Indeed,iis injective becauseX_R is of finite type over R and the map X_R→Spec(R) is dominant on every irreducible component of X_R. It is surjective, because f_η can be defined on a dense affine open subset U =Spec(R[x₁, . . . ,x_m]/I) of X_R by polynomial functions G_i with coefficients in K. There is an element d of R such that the functions dG_i have coefficients inR; then, f_η extends as a morphism on Spec(R[1/d,x₁, . . . ,x_m]/I)).

(see [18], § 9.1 for details)

Let k be an algebraically closed field of characteristic 0. Let X_k be an irre- ducible variety which is defined over k. Let Γbe a subgroup of Bir(X_k) (resp.

Aut(Xk)). LetR be a subring of k. We say that the pair (X,Γ) is defined over R if there is a separated, reduced, irreducible scheme X_R over R for which the structure morphismX_R →Spec(R)is dominant, and an embeddingΓ→Bir(X_R) (resp. inAut(X_R)) such that bothX_k andΓare obtained fromX_R by base change:

X_k=X_R×_Spec(R)Spec(k)and similarly for all elements f ∈Γ.

Let p be a prime number. A model for the pair (X,Γ) over the ring Z_p is given by the following data. First, a ringR⊂kon which X and Γ are defined, and an embedding ι: R→Z_p. Then, an irreducible scheme X_Zp over Z_pand an embeddingρ: Γ→Bir(XZ_p)(resp. inAut(XZ_p)) such that

(i) X_Zp'X_R×_Spec(R)Spec(Z_p)is the base change ofX_Randρ(f)is the base change of f ∈Bir(X_R)for every f inΓ.

Agood modelfor the pair(X,Γ)over the ringZ_pis a model such that

(ii) the special fiberX_Fp ofX_Zp →Spec(Z_p) is absolutely reduced and irre- ducible and its dimension is

dimF_p(X_Fp) =dim_Qp X_Zp×_Spec(R)Spec(Q_p) (iii) ∀f ∈Γ, the special fiberX_Fp is not contained inB_Zp,ρ(f).

IfKis a finite extension ofQ_pand

O

Kis its valuation ring, one can also introduce the notion of good models over

O

K. The following is proven in the Appendix.

Proposition 3.2. Let X be an irreducible complex projective variety, andΓbe a finitely generated subgroup ofBir(X)(resp. ofAut(X)). Then, there exist infinitely many primes p≥3such that the pair(X,Γ)has a good model overZ_p.

3.2. From birational transformations to local analytic diffeomorphisms.

3.2.1. Automorphisms and invariant polydisks. Now, for simplicity, assume that X is the affine space A^d. Let p be an odd prime number, and let Γ be a sub- group ofAut(A^d_Zp); all elements ofΓare polynomial automorphisms of the affine space defined by formulas with coefficients inZ_p. Reduction modulo pprovides a homomorphism fromΓto the groupAut(A^d_Fp): Every automorphism f ∈Γde- termines an automorphism f of the affine space with coefficients inF_p. One can also reduce modulo p², p³, ...

IfR₀ is a finite ring, thenA^d(R₀) andGL_d(R₀)are both finite. Therefore, the automorphisms f ∈Γwith f(m) =m(modp²)andd f_m=Id(modp)for all points min A^d(Z_p) form a finite index subgroupΓ0 of Γ. Every element ofΓ0 can be written

f(x) =p²A₀+ (Id+pB₁)(x) +

∑

k≥2

A_k(x)

whereA₀is a point with coordinates inZ_p, B₁is ad×d matrix with coefficients inZ_p, and ∑k≥2A_k(x) is a finite sum of higher degree homogeneous terms with coefficients inZ_p. Rescaling, one gets

p⁻¹f(px) = pA₀+ (Id+pB₁)(x) +

∑

k≥2

p^k−1A_k(x).

Thus, the Bell-Poonen extension theorem (Theorem 2.4) applies to p⁻¹f(px)be- cause p≥3.

A similar argument applies to automorphism groupsΓof any quasi-projective varietyX of dimensiond. One first replacesQ_p by a finite extensionK to assure the existence of at least one pointminX(R/m_K)(withRthe valuation ring ofK).

Then, the stabilizer ofmmodulom_K is a finite index subgroup, becauseX(R/m_K) is a finite set; this group fixes a polydisk in X(K)and the Bell-Poonen theorem can be applied to a smaller, finite index subgroup. This provides the following statement, the proof of which is given in [6] (Propositions 4.4 and 2.2), when the groupΓis cyclic. Propositions 3.2 and 3.4 are inspired by [6] and also imply this statement.

Proposition 3.3(see [6]). Let X_Zp be a quasi-projective variety defined over Z_p and let Γ be a subgroup of Aut(X_Zp). There exists a finite extension K of Q_p, a finite index subgroupΓ₀ ofΓ, and an analytic diffeomorphism ϕfrom the unit polydisk

U

^=Rd⊂Kd to an open subset

V

^{of X(K)such that}

V

^isΓ0-invariant and the action ofΓ0on

V

is conjugate, viaϕ, to a subgroup ofDiff^an(

U

⁾1.

Combining this result with Proposition 3.2, we get: If a finitely generated group Γadmits a faithful action by automorphisms on some irreducible d-dimensional complex variety, there is a finite index subgroupΓ₀ inΓ, a prime p, and a finite extension K of Q_p such that Γ₀ admits a faithful action by Tate analytic diffeo- morphisms on a polydisk

U

^⊂Kd. This will be used as a first step in the proofs of Theorems A and D.

3.2.2. Birational transformations and invariant polydisks. Let us now deal with invariant polydisks for groups of birational transformations. Let X_Zp be a pro- jective variety defined overZ_p and let Γbe a subgroup ofBir(X_Zp)with a finite symmetric set of generators S. Let X_Fp be the special fiber ofX_Zp. We assume that the special fiber is not contained inB_Zp,sfor anys∈S; this implies thatX_Fp is not contained inB_Zp,gfor everyg∈Γ. By restriction, we obtain a homomorphism Γ→Bir(X_Fp). These assumptions are satisfied by good models.

Let K be a finite extension of Q_p, O_K be the valuation ring of K, and F the residue field ofO_K; by definition,F=O_K/m_K wherem_K is the maximal ideal of O_K. Denote by| · |_pthe p-adic norm onK, normalized by|p|_p=1/p. Set

X_OK =X_Zp×_Spec(Zp)Spec(O_K).

The generic fiber X_Zp×_Spec(Zp)Spec(K) is denoted by X_K, and the special fiber isX_F=X_Zp×_Spec(Zp)Spec(F). Denote byr: X_K(K)→X_F(F) =X(F)the reduc- tion map. Letx be a smooth point inX(F)and

V

be the open subset of X_K(K) consisting of pointszsatisfyingr(z) =x.

Proposition 3.4(see also [6], Proposition 2.2). There exists an analytic diffeomor- phismϕfrom the unit polydisk

U

^{= (O}K)^d to the open subset

V

^{of X}K(K)such that, for every f ∈Bir(X_OK)with x∈/B_OK,f and f(x) =x, the set

V

is f -invariant and the action of f on

V

is conjugate, viaϕ, to a Tate analytic diffeomorphism on

U

^{. Thus, ifΓ⊂Bir(X}Zp)satisfies

(i) x is not contained in any of the sets B_OK,f (for f ∈Γ), (ii) f(x) =x (for every f inΓ),

then

V

^isΓ-invariant andϕconjugates the action ofΓon

V

to a group of analytic diffeormorphisms of the polydisk

U

^.

Thus, once a good model has been constructed, the existence of an invariant polydisk on which the action is analytic is equivalent to the existence of a smooth fixed pointx∈X_F(F) in the complement of the bad lociB_OK,f, f inΓ. Periodic orbits correspond to polydisks which are invariant by finite index subgroups. This will be used to prove Theorems B and C.

We shall prove this proposition in the Appendix. Note that Proposition 3.4, Proposition 3.2, and the rescaling argument of Section 3.2.1 provide a proof of Proposition 3.3.

4. REGULAR ACTIONS OFSL_n(Z)ON QUASI-PROJECTIVE VARIETIES

In this section, we prove the first assertion of Theorem A together with one of its corollaries. Thus, our goal is the following statement.

Theorem 4.1. Let n≥2be an integer. LetΓbe a finite index subgroup ofSL_n(Z).

IfΓembeds into the group of automorphisms of a complex quasi-projective variety X , thendim(X)≥n−1; if X is a complex affine space, thendim(X)≥n.

4.1. Dimension1. When dim_C(X) =1, the group of automorphisms ofX is iso- morphic toPGL₂(C)ifXis the projective line and is virtually solvable otherwise.

On the other hand, every finite index subgroup ofSL_n(Z)contains a non-abelian free group ifn≥2 (see [25], Chapter 1). Theorems A and 4.1 follow from these remarks when n=2 or dim(X) =1. In what follows, we assume dim_C(X)≥2 andn≥3.

4.2. Congruence subgroups ofSL_n(Z); see[1, 62].

4.2.1. Normal subgroups. LetΓbe a finite index subgroup ofSL_n(Z). Forn≥3, the group Γis a lattice in the higher rank almost simple Lie groupSL_n(R). For such a lattice, every normal subgroup is either finite and central, or co-finite. In particular, the derived subgroup[Γ,Γ]has finite index inΓ.

4.2.2. Strong approximation. For anyn≥2 andm≥1, denote byΓ_mandΓ^∗_mthe following subgroups ofSL_n(Z):

Γm = {B∈SL_n(Z)|B≡Id(modm)},

Γ^∗_m = {B∈SL_n(Z)| ∃a∈Z, B≡aId(modm)}.

By definition,Γ_mis the principal congruence subgroup of levelm.

Let p be a prime number. The closure of Γm in SL_n(Z_p) is the finite index, open subgroup of matrices which are equal to Id modulom; thus, ifm=p^urwith r∧p =1, the closure of Γ_m in SL_n(Q_p) coincides with the open subgroup of matricesM∈SL_n(Z_p)which are equal to Id modulo p^u.

The strong approximation theorem states that the image ofSL_n(Z)is dense in the productΠpSL_n(Z_p)(product over all prime numbers). IfΓhas finite index in SL_n(Z), its closure inΠ_pSL_n(Z_p) is a finite index subgroup; it contains almost allSL_n(Z_p).

4.2.3. Congruence subgroup property. A deep property that we shall use is the congruence subgroup property, which holds forn≥3. It asserts that every finite index subgroupΓofSL_n(Z)contains a principal congruence subgroupΓm; ifΓis normal, there exists a unique integermwithΓ_m⊂Γ⊂Γ^∗_m. We shall come back to this property in Section 8.1 for more general algebraic groups (the congruence subgroup property is not known in full generality for co-compact lattices).

Another way to state the congruence subgroup and strong approximation prop- erties is to say that the profinite completion of SL_n(Z)coincides with the prod- uct Π_pSL_n(Z_p). If Γ has finite index in SL_n(Z), its profinite completion is a product Π_qG_q ⊂ Π_qSL_n(Z_q) where each G_q has finite index in SL_n(Z_q), and G_q=SL_n(Z_q)for almost all primesq.

Remark 4.2. Fix n≥ 3. For every prime number q, the group SL_n(Z_q) is a perfect group, because it is generated by the elementary matricese_{i j}(r), r∈Z_q, and every elementary matrix is a commutator. Thus, every homomorphism from SL_n(Z_q) to a p-group is trivial, because every p-group is nilpotent. Thus, the pro-pcompletion ofSL_n(Z_q)is trivial.

Before stating the following lemma, recall that the concept of virtual pro-p completion is introduced in Section 2.4.2.