RIGIDITY AND ARITHMETICITY Marc BURGER

1. Introduction. Statements of the results Lattices. Mostow’s rigidity theorem. Borel-Harish Chandra’s theorem. Margulis’ arithmeticity theorem. Margulis’ arithmeticity criterion. Margulis’ superrigidity theorem.

2. Invariant measures on homogeneous spaces Existence and uniqueness of Haar measures. Integration formulas. Group actions on functions and measures on a locally compact space. Existence and uniqueness of semi-invariant measures on homogeneous spaces. Applications to lattices. Morphisms of homogeneous spaces.

3. Examples of arithmetic lattices

3.A The inclusion SLn(Z) < SLn(R).

Minkowski’s theorem. The group SLn(Z) is a lattice. Mahler’s compactness criterion.

3.B Quaternion algebras and arithmetic lattices of SL2(R) Quaternion algebras. Cocompactness and arithmeticity of maximal orders. 3.C. Restriction of scalars and arithmetic lattices Almost k-simple groups. Lie and algebraic groups. Restriction of scalars.

4. Amenable groups Amenable groups via Fr´echet spaces. Abelian implies amenable. Extensions and amenability. Solvable implies amenable. Compact implies amenable. Minimal parabolic subgroups are amenable. Existence of invariant probability measures on a compact space.

5. Furstenberg maps 1 Existence of Γ-equivariant maps G/P → M (PVk). Connection with Teichm¨ullertheory.

6. The Zariski Support map

6.A The Support map is Borel and PGL(Vk)-equivariant.

6.B The Zariski Closure map is Borel and PGL(Vk)-equivariant. 1 6.C The Zariski Support map M (PVk) → Vark(PV ) is Borel and PGL(Vk)-equivariant.

7. Ergodic actions and Borel structures 7.A Ergodic actions Equivalence of positive Radon measures. Invariant classes. Ergodic actions. Actions of some semisimple elements in simple groups. Unitary representations. Ergodicity of a lattice action on G/P × G/P .

1 7.B Borel structures Borel spaces and σ-separated Borel spaces. Invariant measurable maps with ergodic source and σ-separated targets. Existence of a conull orbit. Locally closed orbits and σ-separated Borel structures. Topological condition for local closedness of orbits.

8. The boundary map. The ﬁrst case of dichotomy

8.A The boundary map Φ = SuppZ ◦ ϕ.

8.B Dynamics of PGL(Vk) on PVk. Furstenberg’s lemma. 8.C Stabilizers of measures. If Φ is essentially constant, then π(Γ) relatively compact.

8.D The case of SL2(R).

9. The boundary map. The second case of dichotomy

If Φ is not essentially constant, then there is a Γ-equivariant map G/P → Hk/Lk.

10. Commensurator Superrigidity

Construction of a Λ-equivariant map for Γ < Λ < CommG(Γ).

11. Higher Rank Superrigidity Essentially rational maps. Continuous extensions of homomorphisms of abstract groups.

12. Arithmeticity Arithmetic presentation in the higher rank case.

2 1. INTRODUCTION. STATEMENT OF THE MAIN RESULTS

The aim of these lectures is to expose the proof of a fundamental result due to Margulis, concerning the structure of lattices in semisimple Lie groups: the arithmeticity theorem. This chapter is devoted to the descrption of related results and of the main objects involved. Many of the notions used here will be introduced in more detail in subsequent chapters.

DEFINITION.— Let G be a locally compact group. A subgroup Γ of G is called a lattice if (i) it is discrete; equivalently, any compact subset of G meets Γ in ﬁnitely many points; (ii) the homogeneous space G/Γ admits a ﬁnite nonzero G-invariant measure.

REMARKS.— 1) In these lectures, a measure on a locally compact space will always refer to a regular Borel measure which is ﬁnite on compact sets. 2) As a consequence of a discussion of the existence of invariant measures on homogeneous spaces, we will see that if Γ is a lattice in G, the G-invariant measure on G/Γ is unique up to a scaling factor.

DEFINITION.— A closed subgroup H < G is called cocompact if G/H is compact. n n n EXAMPLES.— 1) G = R , Γ = Z . More generally, a1,... an being a basis of R , the Z- n n module Λ := Za1 + ... + Zan is a lattice in R . Observe that R /Λ is compact. It is a fact that all lattices of Rn are obtained this way. In particular, let Λ and Λ0 be lattices in Rn, and A :Λ → Λ0 be any group isomorphism. Then A extends to a continuous automorphism of Rn 0 n (an invertible linear map) Aext which obviously satisﬁes Aext(Λ) = Λ . Therefore, Z is up to continuous automorphisms of Rn, the unique lattice in Rn. 1 x z 2) G = { 0 1 y : x, y, z ∈ R} is a 2-step nilpotent Lie group – called the Heisenberg 0 0 1 1 x z group – and the subgroup Γ = { 0 1 y : x, y, z ∈Z} is a lattice in G. Observe that G/Γ 0 0 1 is compact. 3) G = SLn(R), Γ = SLn(Z). Here is a description of the homogeneous space SLn(R)/SLn(Z). n Let R denote the set of all lattices in R . The group GLn(R) acts transitively on R – see n example 1) – and the stabilizer of the point Z ∈R is GLn(Z). Hence we obtain a bijection ∼ GLn(R)/GLn(Z) → R g 7→ gZn.

1 n From this we deduce that if R := {Λ ∈ R : Vol(R /Λ) = 1}, then SLn(R) acts transitively on R1 and we obtain a bijection ∼ 1 SLn(R)/SLn(Z) → R g 7→ gZn.

We will see in section 3.A that SLn(Z) is a lattice in SLn(R). Given a connected simple group G, the main problem is to get a classiﬁcation of all lattices in G. Concerning the degree of generality adopted here, we will always think of G as being a classical group; in particular, it will always be a real matrix group. Concerning the classiﬁcation problem, a basic question is: does the group structure of Γ determine its position in G up to

3 automorphisms of G ? In other words, are two isomorphic lattices in G conjugate by an automorphism of the Lie group G ? The answer is given by Mostow’s rigidity theorem.

0 THEOREM (Mostow).— Let G, G be connected simple Lie groups with trivial center and Γ < G and Γ0 < G0 be lattices. Assume rk(G) ≥ 2. Then any group isomorphism θ :Γ −→∼ Γ0 extends to an isomorphism of Lie groups ∼ 0 θext : G −→ G .

REMARK. —This theorem does not hold for PSL2(R). The notion of rank is essential there.

EXAMPLES.— 1) The group SLn(R) is of rank n−1, it is the dimension of the group of diagonal matrices with positive entries and determinant 1, which is a maximal R-split torus. 2) G = SO(F ), the special orthogonal group of a quadratic form F : Rn → R of index p. In a suitable basis e1, ... en, F has matrix

J F0 , J

n−2p where J := [δi+j,p+1] is a p × p matrix and F0 : R → R is positive deﬁnite. As maximal R-split torus, one may choose

λ1 ... λp S = { In−2p : λi 6= 0} −1 λp ... −1 λ1 whose centralizer is SO(F0). Here rk SO(F ) = p. Although this theorem shows that the lattice Γ determines the ambient Lie group, it does not provide a method to construct lattices. The fundamental result of Margulis is that in Lie groups G of the type occuring in Mostow’s theorem, all lattices are obtained by an hh arithmetic ii construction. The Borel-Harish Chandra’s theorem justiﬁes the main step of this construction.

THEOREM (Borel-Harish Chandra).— Let G < GLn be an algebraic group deﬁned over Q, which is semisimple as a Lie group. Then G(Z) := G(Q) ∩ GLn(Z) is a lattice in GR. 2 2 2 EXAMPLE.— Set F = x1 + x2 − px3, for p a prime. Deﬁne also 1 1 t GF = SO(F ) = {g ∈GL3 : g 1 g = 1 , detg = 1}. −p −p

This is a simple algebraic group deﬁned over Q, and GF (R) is conjugate inside GL3(R) to SO2,1(R). Fixing such an isomorphism

4 ∼ j : GF (R) −→ SO2,1(R), we obtain a lattice ΓF := j GF (Z) ∈ SO2,1(R). Varying p, we obtain many non conjugate lattices in SO2,1(R). This example suggests that arithmetic lattices in a simple Lie group G should be lattices obtained via algebraic groups deﬁned over Q, whose real points are isomorphic to G as a Lie group. There is however a slight generalization of this construction.

LEMMA.— Let G, H be locally compact groups, Γ a lattice in H and p : H → G a surjective continuous homomorphism with compact kernel. Then p(Γ) is a lattice in G. Proof.— The map p is proper, therefore p(Γ) is discrete in G. The map p induces a continuous map ϕ : H/Γ → G/p(Γ) which equivariant, i.e., ϕ(hx) = p(h)ϕ(x), for all h ∈ H and x ∈ H/Γ. If α denotes an H-invariant measure on H/Γ, the direct image ϕ∗α is then a G-invariant ﬁnite measure on G/p(Γ). REMARK.— In the previous lemma, the compactness condition on the kernel is essential. 1 0 Indeed, take H = R2, Γ = Z ⊕ Z with α∈R \ Q. Set also G = R and p : R2 → R 1 α second projection. Then p(Γ) is dense in G. 0 0 DEFINITION.— Let G be a group and Γ and Γ be subgroups of G. Then Γ and Γ are called commensurable if Γ ∩ Γ0 is of ﬁnite index in Γ and Γ0. 0 REMARK.— Let Γ be a lattice in G locally compact, and Γ be commensurable with Γ. Then Γ0 is a lattice in G. This follows from the following diagram

G/(Γ ∩ Γ0) → G/Γ ↓ G/Γ0 And now (at last) the notion of an arithmetic lattice.

DEFINITION.— Let G be a simple Lie group. A lattice Γ∈G is called arithmetic if there exist an algebraic semisimple group H deﬁned over Q and a continuous surjective homomorphism p : H(R)◦ → G with compact kernel such that pH(Z) ∩ H(R)◦ and Γ are commensurable. The pair (p, H) will be referred to as an arithmetic presentation of Γ.

THEOREM (Margulis).— Let G be a connected simple Lie group with trivial center and rank ≥ 2. Every lattice in G is arithmetic. The proof of this theorem is based on the classiﬁcation of irreducible linear representations of Γ, of which Mostow’s rigidity theorem is a special case. While all lattices in higher rank are arithmetic, it has been shown for a long time that this is not true in PSL2(R); Gromov and Piatetski-Shapiro have shown that there exist non-arithmetic lattices in all groups SOn,1(R), n ≥ 2. So the question is, given a lattice, ﬁnd a necessary and suﬃcient condition for Γ to be arithmetic. Margulis found a criterion which is based on the commensurator of Γ∈G.

5 −1 DEFINITION.— The set CommG(Γ) := {g ∈G : Γ and gΓg are commensurable} is a subgroup containing Γ and called the commensurator subgroup of Γ∈G.

EXAMPLE.— Set G = SL2(R) and Γ = SL2(Z). Then CommG(Γ) > SL2(Q). Indeed, let g ∈SL2(Q) and m be the smallest common multiple of all denominators of the entries of g and g−1. Let 0 2 Γ := {γ ∈SL2(Z): γ ≡ idmodm }. 0 2 0 Clearly Γ is a subgroup of ﬁnite index in SL2(Z). Furthermore, if γ = id + m B is in Γ , where B has integral entries, then −1 −1 gγg = id + (mg)B(mg) ∈ SL2(Z). 0 −1 −1 Therefore gΓ g < SL2(Z) and gSL2(Z)g ∩ SL2(Z) is of ﬁnite index in SL2(Z) since it contains gΓ0g−1. The criterion of Margulis is the following.

THEOREM (Margulis).— Let G be a connected simple Lie group with trivial center. A lattice Γ∈G is arithmetic if and only if CommG(Γ)/Γ is inﬁnite.

REMARKS.— 1) This applies in arbitrary rank. 2) If Γ is not arithmetic, then CommG(Γ)/Γ is ﬁnite and therefore CommG(Γ) is a lattice in G. In fact, it is the unique maximal lattice of G containing Γ. The arithmeticity results will be a direct consequence of Margulis’ superrigidity theorem. Most of these lectures will be devoted to the proof of this theorem.

THEOREM (Margulis).— Let G be a connected almost R-simple algebraic group, ◦ Γ < G := GR be a lattice, and Λ be a subgroup such that

Γ < Λ < CommGΓ. Let k be a local ﬁeld of characteristic zero, H an adjoint connected almost k-simple group, and

π :Λ → Hk a homomorphism such that π(Γ) is Zariski dense in H and unbounded in Hk. Assume either that Λ is dense in G or that G is of rank ≥ 2. Then k is R or C and π extends to a k-homomorphism of algebraic groups:

πext : G → H.

2. INVARIANT MEASURES ON HOMOGENEOUS SPACES

Let G be a locally compact group. One has the following basic result due to Haar.

THEOREM 2.1.— On G there exists, up to a positive scaling factor, a unique left invariant positive Radon measure.

REMARKS.— 1) On a Lie group G, such a Haar measure is easily constructed via a nonzero left invariant diﬀerential n-form (for n = dimG).

6 2) We will frequently use Riesz’s representation theorem which identiﬁes continuous linear functionals on K(X), the space of continuous functions with compact support, with Radon measures on X (X locally compact). Let dg be a Haar measure on G. The following two facts are immediate consequences of the uniqueness statement in the above theorem. 1) There is a continuous homomorphism × ∆G : G → R+ such that Z Z −1 f(gh )dg = ∆G(h) f(g)dg G G h∈G, f ∈K(G). The homomorphism ∆G doesn’t depend on the choice of dg.

The group G is called unimodular if ∆G is identically equal to 1, that is any left invariant measure is also right invariant. 2) One has: Z Z −1 −1 f(g )∆G(g )dg = f(g)dg, G G for all f ∈K(G). n EXAMPLES. —1) Lebesgue measure on R . × 2) A connected semisimple Lie group has no nontrivial homomorphism to R+ and is therefore unimodular. The same argument shows that so is a compact group. 3) Let N be the group of n×n upper triangular matrices with 1’s on the diagonal. N = {[xij]: Q xii = 1 and xij = 0} for i > j. Then i

0 0 0 −1 0 Y ai use a n an = a a(a n a)n. One may check that ∆ + = . B a i

NOTATIONS. —Let G be a group and X a set on which G acts on the left. Let Y be a set and F the set of maps X → Y . Then G acts on the left on F via −1 g∗f(x) = f(g x). In particular, if X is locally compact and G acts by homeomorphisms on X, then G acts on K(X), and on the space M(X) of Radon measures:

7 −1 g∗f(x) = f(g x), f ∈K(X), −1 g∗µ(f) = µ(g∗ f), µ∈M(X). Let H be a closed subgroup of G.

DEFINITION.— A positive Radon measure µ on G/H is called semi-invariant if g∗µ and µ are proportional for all g ∈G. The proportionality factor deﬁnes then a continuous homomorphism ∗ χ : G → R× called the modulus of µ. ∗ THEOREM 2.2.— Let χ : G → R× be a continuous homomorphism. There exists a semi- ∆G |H invariant measure of modulus χ on G/H if and only if χ|H = . This measure is unique ∆H up to a positive scaling factor and after normalizing dg one has Z Z Z dµ(x) f(xh)dh = f(g)χ(g)−1dg, f ∈K(G). G/H H G Reference. See for example [R], preliminaries. COROLLARY 2.3.— If G and H are both unimodular, then there exists ( up to scaling ) a unique G-invariant measure on G/H. This happens in particular when G is connected simple and H is discrete. + As an application of corollary 2.3, we compute the Haar measure on SLn(R). Let B be as in example 4) and K = SOn(R). One has the Iwasawa decomposition + SLn(R) = B · K (this follows from Gram-Schmidt orthonormalization). We have therefore a B+-equivariant + identiﬁcation SLn(R)/K → B . Since both SLn(R) and K are unimodular, there is a SLn(R)- invariant measure dµ on SLn(R)/K. Via the above identiﬁcation, it must correspond to a left Haar measure db+ on B+. It follows then from theorem 2.2 and corollary 2.3 that Z Z db+ dkf(b+k) B+ K is a Haar measure on SLn(R). Finally we have another immediate consequence of theorem 2.2.

COROLLARY 2.4.— Let H1 < H2 be closed subgroups of G and assume that H1, H2 and G are unimodular. There is a choice of invariant measures on G/H1, G/H2 and H2/H1 such that Z Z Z f(y)dy = dx f(xz)dz. G/H1 G/H2 H2/H1

3. EXAMPLES OF ARITHMETIC LATTICES

3.A SLn(Z) < SLn(R) First we need the following basic result due to Minkowski. n n PROPOSITION 3.1 (Minkowski’s lemma).— Let Λ ⊂ R be a lattice and V ⊂ R be a closed n bounded balanced convex set such that: Vol(V ) ≥ 2n · Vol(R /Λ). Then: V ∩ (Λ \{0}) 6= ∅.

8 Proof.— Let π : Rn → R/Λ be the canonical projection. 1 Claim: π is not injective on 2 V . Assume the contrary. Then: 1 1 n (?) Vol π( 2 V ) = Vol 2 V ≥ Vol(R /Λ). 1 n Observe that π( 2 V ) ( R /Λ. Otherwise, one would have G 1 (??) Rn = (λ + V ). 2 λ∈Λ 1 This union is disjoint because π is injective on 2 V . Now a pathwise connected topological space cannot be a countable union of proper closed disjoint subsets. So (??) is impossible, 1 n 1 n hence π( 2 V ) ( R /Λ. But then Vol π( 2 V ) < Vol(R /Λ), which contradicts (?). Hence, π 1 is not injective on 2 V . 1 1 1 Therefore there exist x ∈ 2 V and λ ∈ Λ, λ 6= 0 with x + λ ∈ 2 V . Since 2 V is convex and 1 1 1 balanced, one has − 2 x + 2 (x + λ)∈ 2 V , and therefore λ∈V .

THEOREM 3.2 —SLn(Z) is a lattice in SLn(R).

Proof.— By induction on n. For n = 1, this is clear. Now assume n ≥ 2 and that SLn−1(Z) n is a lattice in SLn−1(R). Consider the transitive action of SLn(R) on R \{0}. Then

1 1 ∗ 0 0 ∗ · · · ∗ N := Stab = { ∈ SL (R)} ··· ··· · · · ∗ · · · n 0 0 ∗ · · · ∗

n−1 n−1 and NZ = SLn(Z) ∩ N, so that N = SLn−1(R) n R and NZ = SLn−1(Z) n Z .

By induction hypothesis, SLn−1(Z) is a lattice in SLn−1(R). Form this follows that NZ is a lattice in N. Consider the diagram ∼ n SLn(R)/NZ → SLn(R)/N = R \{0} ↓ SLn−1(R)/SLn−1(Z) n Let f be an integrable function on R \{0}. Since Vol(N/NZ) < ∞, the function

SLn(R)/NZ → R g 7→ f(ge) is integrable and – see corollary 2.4: Z Z f(ge)dg = Vol(N/NZ) f(x)dx. n SLn(R)/NZ R \{0} For the same reason, we have

Z X Z f(gγe) = f(ge)dg. SLn(R)/SLn(Z) SLn(R)/N γ∈SLn(Z)/NZ Z

9 X Hence, setting F (g) := f(gγe), we have

γ∈SLn(Z)/NZ Z Z (?) F (g)dg = Vol(N/NZ) f(x)dx. n SLn(R)/SLn(Z) R n Observe that the orbit under SLn(Z) of the ﬁrst canonical vector of Z is the set of vectors 1 whose coordinates have gcd=1. Recall that SLn(R)/SLn(Z) may be identiﬁed with R , the set of lattices Λ < Rn with Vol(Rn/Λ) = 1. Hence, we consider F as a function R1 → R. Then it follows that X F (Λ) = f(λ) λ∈Λ\{0},primitive Now, if f is the characteristic function of [−1; 1]n, it follows from proposition 3.1 that F (Λ) ≥ 1 for any Λ∈R1. Hence, using (?), we obtain Z n Vol SLn(R)/SLn(Z) ≤ F (g)dg = 2 · Vol(N/NZ) < ∞ SLn(R)/SLn(Z)

Now we turn to the question of the characterisation of relatively compact subsets in SLn(R)/SLn(Z). Indeed, as we will see below, although SLn(R)/SLn(Z) has ﬁnite invariant volume, this space is not compact. We will take R1 := {Λ ⊂ Rn : Λ lattice Vol(Rn/Λ) = 1}

(see proof above) as a model of SLn(R)/SLn(Z). One has the following intrinsic description of the topology on R1. Let B ⊂ Rn be any open ball centered at the origine and deﬁne for Λ, Λ0 ∈R1 0 0 0 dB(Λ, Λ ) := max {d(λ, Λ ∩ B), d(λ , Λ ∩ B)} λ∈Λ∩B λ0∈Λ0∩B n 0 where d is the Euclidean distance of R . Then a sequence (Λn)n∈Z converges to Λ if and only if 0 lim dB(Λn, Λ ) = 0 n→∞ for every ball B ⊂ Rn. 1 THEOREM 3.3 (Mahler’s compactness criterion) —A subset M ⊂ R is relatively compact if and only if there exists U ⊂ Rn neighborhood of {0} such that Λ ∩ U = {0} ∀Λ∈M. Proof.— (⇒) Assume that M is compact. Let B = B(0,R) be the closed ball of radius R > 0 and center 0. The map R1 → N Λ 7→ #(Λ ∩ B) 1 1 is upper semicontinuous. In fact, for every Λ∈R , there is a neighborhood VΛ ⊂ R such that #(Λ0 ∩ B) ≤ #(Λ ∩ B)

10 0 for all Λ ∈ VΛ (geometrically clear). From the assumption that M is compact follows that ﬁnitely many such neighborhoods VΛ cover M. Hence Λ 7→ #(Λ ∩ B) is bounded on M. From this follows easily that there exists Br = B(0, r) such that

Λ ∩ Br = {0} ∀Λ∈M. (⇐) Assume that there exists r > 0 such that Λ ∩ B(0, r) = {0} ∀Λ∈M. n n To every lattice Λ ⊂ R , we associate a closed convex subset CΛ ⊂ R deﬁned by n CΛ := {x∈R : d(x, 0) ≤ d(x, λ) ∀λ∈Λ}.

Then Cλ is the intersection of the family of half-spaces n Hλ := {x∈R : d(x, 0) ≤ d(x, λ)}, λ∈Λ. n Claim 1.— Cλ is a fundamental domain for the action of Λ on R . n n n (1) Any point x ∈ R is equivalent mod Λ to a point in Cλ; equivalently, π : R → R /Λ restricted to CΛ is surjective. Indeed, since Λ is discrete, the set {d(x, µ): µ∈Λ} has a minimum, say d(x, λ) for some λ∈Λ. The distance d being translation invariant, we obtain x − λ∈Cλ. n n (2) Two Λ-equivalent points in Cλ lie on the boundary ∂Cλ; equivalently, π : R → R /Λ restricted to the interior of CΛ is injective.

Indeed, assume x∈Cλ and x + λ ∈Cλ for some λ 6= 0. Then: d(x, 0) ≤ d(x, −λ) = d(x + λ, 0) ≤ d(x + λ, λ) = d(x, 0), [ hence d(x, 0) = d(x, −λ), or x∈Hλ. Since ∂Cλ = (Cλ ∩ ∂Hλ), this proves (2). λ∈Λ Claim 2.— There is a constant c(r) > 0, only depending on r, such that for all Λ∈M we have Cλ ⊂ B 0, c(r) . n n Since Cλ is a fundamental domain for Λ ⊂ R , we have Vol(Cλ) = Vol(R /Λ) = 1. Observe also that if Λ ∩ B(0, r) = {0}, then B(0, r/2) ⊂ CΛ. Therefore, for every x ∈ CΛ, the convex hull of x and B(0, r/2) is of volume ≤ 1. Hence k x k≤ C(r) for some constant c(r). This proves claim 2. [ In particular, Cλ = Hλ where SΛ is the ﬁnite set SΛ := Λ ∩ B 0, 2c(r) . Since for all

λ∈SΛ Λ∈M the points of Λ are distance r/2-apart, the cardinality of SΛ is bounded by the constant VolB(0, 2c(r) + r/2) κ(r) := , VolB(0, r/2) which depends only on r > 0.

Claim 3.— SΛ generates Λ. 0 Let Λ be the subgroup generated by SΛ, and CΛ0 the corresponding fundamental domain. For x∈CΛ0 we have

11 d(x, 0) = min d(x, λ0) ≤ min d(x, λ) λ0∈Λ0 λ∈Λ and hence x∈CΛ. Therefore CΛ0 ⊂ CΛ and

n 0 n Vol(R /Λ ) = Vol(CΛ0 ) = Vol(R /Λ). On the other hand, Λ0 is a subgroup of Λ and therefore

Vol(Rn/Λ0) = #(Λ/Λ0) · Vol(Rn/Λ).

From these, we conclude that #(Λ/Λ0) = 1, and hence Λ = Λ0. This prove claim 3.

Now we show that M is relatively compact by proving that any sequence (Λn)n∈N of M has 1 a convergent subsequence in R . Since #SΛn is bounded by κ(r), SΛn ⊂ B 0, 2c(r) , we may assume up to passing to a subsequence, that

lim SΛ = S n→∞ n where S ⊂ B0, 2c(r) is a ﬁnite set. This convergence is to be understood in the following sense.

(1) #SΛn = #S for all integer n.

(2) Any ε-neighborhood of S contains SΛn for n big enough. Now let Λ be the Z-module generated by S. We claim that Λ is a lattice in Rn and

lim Λn = Λ. n→∞ (a) For n big enough, we have a bijection ∼ bn : S −→ Sn X such that for every λ∈S, d(λ, bn.λ) → 0 as n → ∞. Now for λ = mµµ∈Λ, set µ∈S P λµ := µ∈S mµbn(µ) ∈ Λn.

Then lim λn = λ. In particular, if k λ k< r we have k λn k< r for n big and hence λn = 0, n→∞ since Λn ∩ B(0, r) = {0}. So λ = 0. This shows that Λ is discrete. (b) The rank of Λ is n. Let V be the vector space generated by S. Let x ∈ Rn be any vector orthogonal to V . Then we have

d(x, 0) < d(x, λ), ∀λ∈S.

Hence for n big enough we have d(x, 0) < d(x, µ) for all µ ∈ SΛn . In particular x ∈ CΛn ⊂ B0, 2c(r). So any vector x orthogonal to V is of length ≤ 2c(r). This implies clearly that V = Rn.

The proof that lim Λn = Λ is left to the reader as an exercise. n→∞

12 3.B Quaternion algebras and arithmetic lattices in SL2(R)

In this section we will construct cocompact arithmetic lattices in SL2(R) using quaternions algebras over Q. Let K be a ﬁeld with Q ⊂ K ⊂ C and a, b∈Z nonzero integers. Let

Ha,b(K) := K1 ⊕ Ki ⊕ Kj ⊕ Kk. be the 4-dimensional K-vector space with basis {1; i; j; k}. There is an associative K-algebra structure on Ha,b(K) whose product is deﬁned on the basis by: k = ij = −ji, i2 = a and j2 = b.

Ha,b(K) is called a quaternion algebra over K.

On Ha,b(K) we have an involution x = x1 + x2i + x3j + x4k 7→ x := x1 − x2i − x3j + x4k satisfying x.y = y.x, and a norm

N : Ha,b(K) → K 2 2 2 2 sending x to xx = x1 − x2a − x3b + x4ab and which is multiplicative: N(xy) = N(x)N(y). × × Let Ha,b(K) denote the group of invertible elements of Ha,b(K). Observe that x∈Ha,b(K) −1 if and only if N(x) 6= 0, in which case x = x/N(x). In particular, Ha,b(K) is a division algebra if and only if N −1(0) = {0}.

EXAMPLES. —1) If K = R and a = b = −1, one gets the Hamilton quaternion algebra. 2) K = Q, a = b = 1. Ha,b(Q) is not a division algebra since N(1 + i + j + k) = 0. 3) K = Q, b is a prime number, and a is not a square modulo b. Then Ha,b(Q) is a division algebra. Indeed, assume that N(x) = 0 for some nonzero x ∈ Ha,b(Q). We may assume 2 2 2 2 also that xi are integers, with gcd=1. Reducing x1 − x2a − x3b + x4ab modulo b, one gets 2 2 x1 = x2a ∈ Z/bZ. Hence x1 = x2 = 0 ∈ Z/bZ since b is not a square modulo b. Therefore x1 = y1b and x2 = y2b for yi ∈Z and 2 2 2 2 2 2 y1b − y2ab − x3b + x4ab = 0 hence 2 2 2 2 y1b − y2ab − x3 + x4a = 0. 2 2 Reducing modulo b, we get x4a = x3 ∈Z/bZ, and so x3 = x4 = 0∈Z/bZ, contradicting gcd=1.

NOTATIONS. —If A is any subring of C with unity, Ha,b(A) will denote the ring with unity A1 ⊕ Ai ⊕ Aj ⊕ Ak. It is invariant under x 7→ x, so 1 Ha,b(A) := {x∈Ha,b(A): N(x) = 1}. is a group. For the rest of this paragraph, we assume that a and b are positive integers. One veriﬁes that the map

h : Ha,b(R) → M2,2(R) √ √ √ x + x a x a + x ab x 7→ √1 2 √ 3 √4 x3 a − x4 ab x1 − x2 a is an isomorphism of R-algebras with N(x) = det h(x).

13 Therefore h induces the following Lie groups isomorphisms × h : Ha,b(R) → GL2(R); 1 h : Ha,b(R) → SL2(R). 1 Now Ha,b(Z), being the intersection of the discrete set Ha,b(Z) and the closed subgroup 1 1 1 Ha,b(R), is a discrete subgroup of Ha,b(R). Set Γa,b := h Ha,b(Z) .

THEOREM 3.4 —If Ha,b(Q) is a division algebra, the homogeneous space SL2(R)/Γa,b is com- pact.

REMARKS.— 1) Since SL2(R) and Γa,b are unimodular, there is an SL2(R)-invariant measure on SL2(R)/Γa,b – see corollary 2.3. When SL2(R)/Γa,b is compact, this measure is ﬁnite and Γa,b is a lattice in SL2(R). 2) We have shown previously that SL2(Z) is a lattice in SL2(R), but SL2(R)/SL2(Z) is not compact – see 3.1. 3) One can show that Γa,b is always a lattice in SL2(R) and that it is cocompact if and only if Ha,b(Q) is a division algebra. 4) We will show later on that Γa,b is indeed an arithmetic lattice in SL2(R). In order to prove theorem 3.4, we need some preliminary remarks and a lemma.

Every x∈Ha,b(R) determines two endomorphisms of the R-vector space Ha,b(R) via the left and right multiplications: Lx(y) := xy and Rx(y) := yx. In the basis {1; i; j; k}, the matrices of Lx and Rx are

x1 x2b x3b −x4ab x1 x2a x3b −x4ab x x x b −x b x x −x b x b 2 1 4 3 and 2 1 4 3 x3 −x4a x1 x2a x3 x4a x1 −x2a x4 −x3 x2 x1 x4 x3 x2 x1 2 and one veriﬁes that det(Lx) = det(Rx) = N(x) . Now for m∈Z \{0}, deﬁne m Ha,b(Z) := {x∈Ha,b(Z): N(x) = m}. 1 These sets are invariant under left and right multiplication by Ha,b(Z). 1 m LEMMA 3.5.— The group Ha,b(Z) has ﬁnitely many orbits in Ha,b(Z) when acting either by left or right multiplication. 1 m Proof.— Since Ha,b(Z) and Ha,b(Z) are invariant under x 7→ x, it suﬃces to show the lemma 1 for the action of Ha,b(Z) by left multiplication. m Now take x∈Ha,b(Z). Observe that Ha,b(Z) and Ha,b(Z).x = Rx Ha,b(Z) are lattices in the additive group Ha,b(R). Hence Vol Ha,b(R)/Ha,b(Z).x 2 2 #Ha,b(Z)/Ha,b(Z).x = =|detRx |= N(x) = m . Vol Ha,b(R)/Ha,b(Z) 2 Now there are only ﬁnitely many subgroups of index m ∈Ha,b(Z). Let

Ha,b(Z).x1,... Ha,b(Z).xr(m) m be the set of distincts subgroups so obtained. For any x∈Ha,b(Z) there is an i between 1 and r(m), such that

14 Ha,b(Z).x = Ha,b(Z).xi. −1 1 Setting y = xix we have that y ∈Ha,b(Z) and N(y) = 1, hence y ∈Ha,b(Z). This shows that r(m) m G 1 Ha,b(Z) = Ha,b(Z).xi. i=1 1 1 1 Proof of theorem 3.4.— We have to show that Ha,b(R)/Ha,b(Z) is compact. Let g ∈Ha,b(R), and Lg the left multiplication by g. Then Λ := Lg Ha,b(Z) is a lattice in Ha,b(R). and 2 Vol(Ha,b(R)/Λ) =|det(Lg)|= N(g) = 1. It follows from Minkowski’s lemma (proposition 3.1) that the closed convex balanced set

V = {g ∈Ha,b(R): |yi |≤ 1, 1 ≤ i ≤ 4}, 4 whose volume is 2 , contains a nonzero element of λ. In other words, there exists x∈Ha,b(Z) \ {0} with gx ∈ V . Since Ha,b(Q) is a division algebra, it folows that N(x) is nonzero. In particular × gx∈V ∩ Ha,b(R) .

Observe that N(gx) = N(x) = m and |m|=|N(x)|=|N(gx)|≤ (a + 1)(b + 1) since |(gx)i |≤ 1 for each coordinate of gx. The set

Vm = {x∈Ha,b(R):|xi |≤ 1,N(x) = m} × is a compact set contained in Ha,b(R) and it follows from the above discussion that gx is in 0 0 1 Vm. Applying lemma 3.5, we have x = x xi for some x ∈Ha,b(Z), 1 ≤ i ≤ r(m), so that 0 −1 [ −1 gx ∈ Vmxi ⊂ Vmxi . 1≤|m|≤(a+1)(b+1) 1≤i≤r(m) 1 [ −1 1 Let C := Ha,b(R) ∩ Vmxi . This is a compact subset of Ha,b(R) and we have 1≤|m|≤(a+1)(b+1) 1≤i≤r(m) 1 0 1 ﬁnally shown that for any g ∈Ha,b(R) there exists x ∈Ha,b(Z) such that gx is in C.

Finally we show that Γa,b is an arithmetic lattice in SL2(R). 1 1 For every x∈Ha,b(C), let lx be the matrix of the endomorphism Lx of Ha,b(C) w.r.t. the basis {1; i; j; k}. We obtain an algebra homomorphism 1 l : Ha,b(C) → M4,4(C) x 7→ lx which is injective. Likewise, let rx be the matrix of Rx in the basis {1; i; j; k}. One veriﬁes that {ri; rj; rk} is in M4,4(Z). Here is a characterisation of the image of l. LEMMA 3.6.— l Ha,b(C) = {A∈M4,4(C): Ari = riA, Arj = rjA, Ark = rkA}. Proof.— It is clear that l Ha,b(C) satisﬁes these properties. Conversely, if A ∈ M4,4(C) commutes with ri, rj, rk then it commutes with rx for all x∈Ha,b(C). Hence

A(y) = A(1.y) = Ary(1) = ryA(1) = A(1).y = lA(1)(y).

15 1 Next we characterize l Ha,b(C) . For every matrix A∈M4,4(C), let

4 3 2 CA(λ) := λ − λ c1(A) + λ c2(A) − λc3(A) − c4(A).

Then the ci(A)’s are polynomials in the matrix entries of A with coeﬃcints in Z. Now

2 clx (λ) = det(λ − lx) = N(λ − lx) = λ4 − 2Tr(x)λ3 + [2N(x) + Tr(x)2]λ2 − 2Tr(x)N(x)λ + N(x)2,

2 where Tr(x) = x + x. Hence 2N(x) = c2(lx) − c1(lx) .

2 Now set G := {A∈GL4(C): A commutes with ri, rj, rk andc2(A) − c1(A) = 2}. Then G is a 1 connected algebraic subgroup of GL4(C) deﬁned over Q, isomorphic as a Lie group to Ha,b(C) via l−1. Composing with l−1 we get a Lie group isomorphism

p : GR → SL2(R) such that p G(Z) = Γa,b.

Observe that in this example GR is connected and p is injective.

3.C Restriction of scalars and arithmetic lattices

Throughout this section, G will denote an algebraic group.

DEFINITION.— The algebraic group G is called almost simple if the adjoint representation Ad is irreducible. If G is deﬁned over k, it is called almost k-simple if Adkis irreducible.

Here is now a connection between algebraic and Lie groups.

PROPOSITION 3.7.— Let H be a connected semisimple Lie group with trivial center. There ◦ exists an algebraic group G < GLn(C) deﬁned over Q such that, as Lie groups, H and GR are isomorphic.

Proof.— Let h be the Lie algebra of H and Ad be its adjoint representation. The AdH = (Auth)◦, where Auth is the group of automorphisms of the real Lie algebra h. Since H has ∼ ◦ trivial center, it is isomorphic to AdH, so that H = (Auth) . Now let X1,... Xn be a basis of h. The coordinates of the vectors [Xi,Xj] in this basis

n X k [Xi,Xj] = aijXk k=1 are the structure constants of h. Via this basis, we identify h with Rn. The group

G := {g ∈GLn(C):[gXi, gXj] = g[Xi,Xj], 1 ≤ i, j ≤ n}

k is an algebraic group, deﬁned by quadratic equations with coeﬃcients in the ﬁeld Q({aij}) ⊂ R. ∼ ◦ We have obviously that GR = Auth, hence H = GR. According to a Chevalley’s theorem, any real semisimple algebra has a basis whose structure constants are in Q. Using such a basis, one obtains G deﬁned over Q.

16 Restriction of scalars.— In this section K = C, k is a ﬁnite extension of Q of degree d and O is the ring of integers of k. We are going to describe an operation which to any algebraic group G deﬁned over k associates an algebraic group RK/QG deﬁned over Q and a group homomorphism ∼ j : G(k) −→ RK/QG(Q) ∼ such that j G(O) = RK/QG(Z). We describe this operation in the more general context of aﬃne varieties V ⊂ CN . We ﬁrst recall a few basic facts from Galois theory. d M a) There is a basis α1,... αd of k over Q such that O = Zαi. i=1 b) There are d distinct ﬁeld embeddings σi : k → C. They are linearly independant in the C-vector space of all maps k → C. In particular, the matrix [σi(αj)]i,j is invertible. We set α1 := id. c) Q = {x∈k : σi(x) = x, 1 ≤ i ≤ d}. d) Any ﬁeld imbedding σ : k → C can be extended to an automorphism of C. First we describe restriction of scalars for the case of the aﬃne variety CN deﬁned over k. Every ﬁeld imbedding σi : k → C gives rise to a Q-linear map N N N σi : k → C w = (w1, ...wN ) 7→ (σiw1, ...σiwN ) from which one deduces a Q-linear map N ι : k → Md,N (C) σ1(w) w = (w1, ...wN ) 7→ ... σN (w) Explicitly σ1w1 ... σ1wN w 7→ ...... σdw1 ... σdwN N N It is plain that ι identiﬁes the Q-vector space k with ι(k ), a Q-subspace of Md,N (C). Expanding wk ∈k in the Q-basis α1,... αd d X wk = wjkαj, wjk ∈ Q, j=1 one gets that the k-th column of the above matrix is σ1wk w1k ... = [σi(αj)]i,j ... . σdwk wdk

Hence if T : Md,N (C) → Md,N (C) denotes the C-linear map obtained from left multiplication by [σi(αj)]i,j, we have

17 N ι(k ) = T Md,N (Q) . Therefore j := T −1ι identiﬁes the k-points of the k-variety CN with the Q-points of the N Q-variety Md,N (C). It follows from a) that j(O ) = Md,N (Z). N Now let V ⊂ C be an aﬃne variety deﬁned over k and I = I(V ) ⊂ C[X1, ...XN ] its deﬁning σ ideal. For σ ∈ AutC and p ∈ C[X1, ...XN ], we denote by p the polynomial whose coeﬃcients are obtained by applying σ to those of p. This way AutC acts by Q-algebra automorphisms σ on C[X1, ...XN ]. For σ ∈ AutC we let V be the aﬃne variety corresponding to the ideal σ σ σ N d I = {p }p∈I . Since V is deﬁned over k, V is deﬁned over σ(k). Identifying (C ) with Md,N (C), we consider Q V σi = V σ1 × ... × V σd

Q σi as an aﬃne subvariety of Md,N (C). The variety V can be described as the set of common zeroes of the polynomial mappings

pˇ: Md,N (C) → Md,N (C)

σ1 X11 ... X1N p (X11, ..., X1N ) 0 ... 0 ...... 7→ ...... σ Xd1 ... XdN p d (Xd1, ..., XdN ) 0 ... 0

−1 Q σi where p∈Ik(V )(:= I(V ) ∩ k[X1, ...XN ]). Therefore the aﬃne variety T ( V ) ⊂ Md,N (C) −1 is the set of common zeroes of the polynomial mappingsp ˜ := T ◦ pˇ◦ T , p∈Ik(V ). The point of this construction is that the polynomial mappingsp ˜ are with coeﬃcients in Q. Indeed −1 N p˜ Md,N (Q) = T pˇ ι(k ) σ1(w) and if ... ∈ι(kN ), w ∈kN , then σd(w) σ1(w) σ1.p(w) 0 ... 0 σ1.p(w) σ1.0 ... σ1.0 pˇ ... = ...... = ...... σd(w) σd.p(w) 0 ... 0 σd.p(w) σd.0 ... σd.0 σ1(w) −1 so that T pˇ. ... is in Md,N (Q). Hencep ˜ Md,N (Q) ⊂ Md,N (Q), from which one σd(w) deduces easily thatp ˜ has coeﬃcients in Q. Hence the aﬃne variety

−1 Q σi RK/QV := T ( V ) ⊂ Md,N (C) is deﬁned over Q. Moreover the map j = T −1ι

Q σi V (k) → V → RK/QV (Q) σ1(w) σ1(w) w 7→ ... 7→ T −1 ... σd(w) σd(w) identiﬁes the k-points of V with the Q-points of RK/QV . Besides j V (O) = RK/QV (Z). Applying this construction to an algebraic group G deﬁned over k, one obtains

18 PROPOSITION 3.8.— k is a ﬁnite extension of Q of degree d and O is the ring of integers of k. Let σi : k → C be the d distinct ﬁeld imbeddings of k ∈ C, and L be the smallest ﬁeld containing the σi(k)’s. Consider the group homomorphism d Y ι : G → Gσi i=1 g 7→ (σ1(g), ...σd(g). d Y σi Then the L-algebraic group G is isomorphic over L to a Q-algebraic group RK/QG. This i=1 isomorphism identiﬁes ι G(k) with RK/QG(Q) and ι G(O) with RK/QG(Z). Restriction of scalars is most useful when combined with Borel-Harish Chandra’s theorem. It also serves to illustrate the deﬁnition of arithmetic lattices. √ √ √ √ EXAMPLE.—k = Q( 2), O = Z[ 2], σ = id and σ2(x + 2y) = x − 2y. Set σ√= σ2. Let 2 2 2 G = SO(F ), where√ F : C → C is the quadratic form F (x1, x2, x3) = x1 + x2 − 2x3. G is deﬁned over Q( 2). Observe that k ∪ σ(k) ⊂ R. According to proposition 3.8, RK/QG is σ ∼ σ isomorphic over R to the R-group G × G . Hence RK/QGR −→ GR × G (R), and under this isomorphism RK/QG(Z) is sent to the subgroup √ Γ := {(g, σ.g): g ∈SO(F, Z[ 2])}. σ σ Observe that GR is isomorphic to SO2,1(R) and G (R) is compact since F is positive deﬁnite. It follows from Borel-Harish-Chandra that RK/QG(Z) is a lattice in RK/QGR, and hence Γ σ σ is a lattice in SO(F, R) × SO(F , R). Since SO(F√ , R) is compact the projection of Γ to SO(F, R) is a lattice in this group. Hence SO(F, Z[ 2]) is a lattice in SO(F, R).

4. AMENABLE GROUPS

DEFINITION.— A Fr´echet space is a topological vector space V such that (i) V is Hausdorﬀ; (ii) the topology of V is generated by a family of seminorms k − kα, α∈I. ∗ ∗ EXAMPLE.— Let B be a Banach space, B its dual and Bσ the topological vector space deﬁned by the weak-∗-topology on B∗. Then V is a Fr´echet space, the family of seminorms being ∗ kλkv=|λ(v)|, v ∈B, λ∈B . We will frequently use Banach-Alaoglu’s theorem which asserts that the unit ball in B∗ {λ∈B∗ :kλk≤ 1} ∗ is a compact subset of V = Bσ.

DEFINITION.— A topological group L is amenable if for any continuous linear action L × E → E of L on a Fr´echet space E, and for any convex compact nonvoid L-invariant set I ⊂ E, there is an L-ﬁxed point in I. A group L is amenable if as a discrete group it is amenable.

19 PROPOSITION 4.1.— An abelian group is amenable. Proof.— Let A × E → E and I ⊂ E be as in the deﬁnition above. For T ∈ A, denote by ET the closed subspace of E consisting of all T -ﬁxed points and set IT := I ∩ ET . For y ∈I and n an integer, the convex combination 1 C (y) := [y + T y + ... + T ny] n n + 1 is in I. For a seminorm k − kα we have 1 k T C (y) − C (y) k ≤ k (T n+1y − y) k n n α n + 1 α 2 ≤ max k x kα n + 1 x∈I T Hence any accumulation point of the sequence Cn(y) N∈N is T -invariant: I 6= ∅. The group A is abelian and hence acts on the Fr´echet space ET , leaving the convex compact nonvoid set IT invariant. In particular, we have for all S ∈A

(IT )S = IT ∩ IS 6= ∅. \ T \ T By induction we get I 6= ∅ for any ﬁnite subset F ∈A, and hence I 6= ∅ since I is T ∈F T ∈A compact.

PROPOSITION 4.2.— An extension R of topological groups 1 → A → R → B → 1 with A and B amenable, is amenable. Proof.— Let R × E → E and I ⊂ E be as in the deﬁnition of amenability. The space EA of A-ﬁxed points is a Fr´echet space on which B = R/A acts. Since A is amenable, IA = EA ∩ I 6= ∅. Since I is R-invariant, IA ⊂ EA is B-invariant. Now B being amenable, R A R we have I = (I ) 6= ∅. COROLLARY 4.3.— A solvable group is amenable.

PROPOSITION 4.4.— A compact group is amenable. Proof.— Let K be a compact group, α a Haar measure on K and K × E → E, I ⊂ E be as Z in the deﬁnition of amenability. For v ∈I, k.v dα(k) is a K-ﬁxed point in I. K COROLLARY 4.5.— An extension L of topological groups 1 → R → L → K → 1 with R solvable and K compact, is amenable.

COROLLARY 4.6.— Let G be a connected semisimple algebraic group deﬁned over R and P be a minimal parabolic subgroup of G deﬁned over R. The topological group P = P(R) is amenable.

Proof.— P is the semidirect product P = Z(S) n U where U is the unipotent radical of P and Z(S) is the centralizer of a maximal R-split torus S. Besides, Z(S) = M · S where M is the biggest connected anisotropic subgroup of Z(S) and M ∩ S is ﬁnite. The subgroup B = S · U is a solvable normal subgroup of P . Furthermore P/B is compact since so is M. The result follows from corollary 4.5.

20 n EXAMPLE.— G = SO(F ), the special orthogonal group of a quadratic form F : R → R of index p. In a suitable basis {e1, ... en}, F has matrix

J F0 J

n−2p where J := [δi+j,p+1] is a p × p matrix and F0 : R → R is positive deﬁnite. The stabilizer P∈SO(F ) of the ﬂag (e1) ⊂ (e1, e2) ⊂ ... ⊂ (e1, ...ep) is a minimal parabolic subgroup deﬁned over R. An element of this subgroup has the form A11 A12 A13 0 A21 A22 0 0 A33

t −1 where A11 and A33 are upper triangular matrices and A33 = J A11 J, A21 ∈ SO(F0). As maximal R-split torus, one may choose

λ1 ... λp S = { In−2p : λi 6= 0} −1 λp ... −1 λ1 whose centralizer is SO(F0). The unipotent radical U of P is the set of matrices A11 A12 A13 0 In−2p A22 0 0 A33 where A11 and A33 are unipotent. One has the decomposition P = SO(F0).S.U, and the compact group SO(F0) is the quotient of P by the normal solvable subgroup S.U. A remarkable property shared by amenable groups is the following.

PROPOSITION 4.7.— Let L × X → X be a continuous action of a topological amenable group L on a compact space X. Then, there exists an L-invariant probability measure on X. Proof.— The group L acts continuously on C(X), the Banach space of continuous functions f with norm

kf ksup:= sup |f(x)|. x∈X The dual of C(X) is M(X), the Banach space of bounded measures on X, with norm hµ, fi kµkmass:= sup . 06=f∈C(X) kf ksup

Then L acts continuously on the Fr´echet space M(X)σ (weak topology), and leaves invariant the nonvoid compact convex subset M1(X) consisting of probability measures. The proposi- tion then follows from the assumption that L is amenable.

21 We use now proposition 4.7 to prove that F2, the free group on two generators, is not amenable. 1 We realize F2 as a subgroup of SL2(R) and show that on P R there is no F2-invariant prob- ability measure.

(1) Let η, γ ∈SL2(R) be hyperbolic elements, and η± and γ± be the corresponding attracting and repelling ﬁxed points. Assume that

{γ+; γ−} ∩ {η+; η−}= 6 ∅. n n Claim: there is an integer n such that the subgroup of SL2(R) generated by η and γ is free. This is an easy consequence of Ping-Pong lemma. Set a := γn and b := ηn. (2) Let µ be a probability measure on P1R which is invariant under hai, the subgroup generated by a. For any closed neighborhood V of γ−, we have \ k 1 a P R \ V = {γ+}, k≥1 1 1 hence µ(γ+) = lim µ a.( R \ V ) = µ R \ V , which shows that supp(µ) is contained in n→∞ P P {γ+} ∪ {γ−}. Assuming that µ is ha, bi-invariant, we get

supp(µ) ⊂ ({γ+} ∪ {γ−}) ∩ ({η+} ∪ {η−}), which contradicts the assumption that µ is a probability measure.

5. THE FURSTENBERG MAP

Let G be a Lie group and Γ < G be a lattice. Let

ρ :Γ → GL(Vk) be a linear representation of Γ ∈ Vk, a ﬁnite dimensional vector space over k. Here k is R, C or a ﬁnite extension of Qp. The projective space PVk is a compact space on which Γ acts via the representation ρ:

Γ × PVk → PVk (γ, x) 7→ ρ(γ)(x).

The group Γ acts by isometries on C(PVk), the Banach space of continuous functions with norm

kf ksup:= sup |f(x)|, x∈PVk via −1 ρ(γ)∗f(x) := f(ρ(γ) x).

The dual of C(PVk) is M(PVk), the Banach space of bounded measures on PVk, with norm hµ, fi kµkmass:= sup . 06=f∈C(PVk) kf ksup

Then Γ acts on M(PVk) via

22 Z ρ(γ)∗µ(f) := f(ρ(γ).x) dµ(x), PVk and the action Γ × M(PVk) → M(PVk) is continuous for the weak topology on M(PVk). This may be veriﬁed directly. It also follows from the following general fact we will use later. Let L be a topological group and L × B → B a continuous action on a Banach space B. Then L acts by isometries on B∗ the dual of B and the action ∗ ∗ L × Bσ → Bσ ∗ of L on Bσ, the dual endowed with the weak topology, is continuous. THEOREM 5.1.— Let P < G be a closed amenable subgroup. There exists a measurable map 1 ψ : G/P → M (PVk) which is Γ-equivariant.

REMARKS. —1) Let Y be a topological space. A map ψ : G/P → Y is measurable if ψ, seen as a map G → Y , is measurable (i.e., for every Borel set B ⊂ Y , ψ−1(B) ⊂ G is measurable w.r.t. Haar measure). 2) M(PVk) is endowed with the Borel structure deduced from the weak topology. One veriﬁes that it coincides with the Borel structure deﬁned by the norm topology. Proof.— Let α be the G-invariant probability measure on Γ \ G. We deﬁne 1 F := LΓ G, C(PVk) to be the Banach space of all Γ-equivariant measurable maps f : G → C(PVk) such that Z kf ksup,1:= kf(g)ksup dα(g) < ∞. Γ\G The dual of F is the Banach space ∞ E := LΓ G, M(PVk) of all Γ-equivariant measurable maps m : G → M(PVk) such that

kmkmass,∞:= ess sup km(g)kmass< ∞. g∈G The pairing F × E → R Z (f, m) 7→ hf(g), m(g)i dα(g) Γ\G realizes the duality.

The formula f∗h(g) := f(gh) deﬁnes a continuous action of G on F by isometries. Therefore G acts continuously on Eσ, Eσ being the dual of F with weak topology, via the formula m∗h(g) := m(gh), m∈E. Consider 1 I = {m∈E : m(g)∈M (PVk) for almost every g ∈G}. One veriﬁes that I is G-invariant, convex, compact and non void. The last assertion follows from the fact that Γ is a discrete subgroup of G. Since P is amenable, there is a P -ﬁxed point in I.

23 1 Connection with Teichm¨uller theory. —The group PSL2(C) acts on P C in the obvious way: 1 1 PSL2(C) × P C → P C a b az + b ( , z) 7→ , c d cz + d

2 1 and PSL2(R) leaves H := {z ∈ C : Iz > 0} ⊂ P C invariant. Observe that the boundary of 2 1 H is P R. Now, PSL2(R) is the group of orientation preserving isometries of the hyperbolic plane dx2 + dy2 2, ds2 = . H y2 0 Let Γ, Γ be lattices in PSL2(R) such that (1) The surfaces S := Γ \ H2 and S := Γ0 \ H2 are compact 0 (equivalently, Γ \ PSL2(R) and Γ \ PSL2(R) are compact); (2) Γ and Γ0 act without ﬁxed point on H2; (3) S and S0 are homeomorphic. ∼ ∼ A homeomorphism f : S −→ S0 lifts to a homeomorphism F : H2 −→ H2 which is equivariant w.r.t. an isomorphism θ :Γ −→∼ Γ0, that is

F (γ.z) = θ(γ)F (z) ∀z ∈H2 ∀γ ∈Γ. Now F is easily seen to be a quasi-isometry of H2, and therefore extends via Mostow’s con- struction to a quasi-symmetric homeomorphism q.s. ϕ : P1R −→ P1R which is again Γ-equivariant. On the other hand, we have associated to the representation 0 θ :Γ → Γ < PSL2(R) a Furstenberg map 1 1 1 ψ : P R → M P R which is Γ-equivariant measurable. It can be shown that in this situation, ψ takes its values 1 1 in the Dirac measures on P R, and in fact that ψ(x) = δϕ(x) ∀x∈P R.

6. THE ZARISKI SUPPORT MAP

Let k be a local ﬁeld, i.e., R, C or a ﬁnite extension of Qp. Let Vk be ﬁnite dimensional k-vector space, V := Vk ⊗ K where K is an algebraic closure of k and Vark(PV ) be the space of projective subvarieties of PV deﬁned over k – see below. In this chapter, we study the map 1 suppZ M (PVk) −→ Vark(PV ) Z µ 7→ supp(µ)

24 which to a probability measure µ associates the Zariski closure of its support. We will prove that for a natural topology on Vark(PV ), this map is Borel.

The map suppZ is the composition of the following two maps. supp M1( V ) −→ P ( V ) (a) P k F P k µ 7→ supp(µ) the map that associates to every measure µ its support supp(µ). Here PF (PV ) is the space of closed subspaces of PVk.

ClZ PF (PVk) −→ Vark(PV ) (b) Z F 7→ F the map which to every closed subset F ⊂ PV associates its Zariski closure. 1 6.A supp : M (PVk) → PF (PVk)

We endow the space PF (PVk) of closed subspaces of PVk with a topology which has the following two equivalent descriptions.

(1) Let d be a metric on PVk which generates its topology and set for F1, F2 ∈PF (PVk):

D(F1,F2) := max{d(f1,F2), d(f2,F1): f1 ∈F1, f2 ∈F2}.

One veriﬁes that D is a metric on PF (PVk): the Hausdorﬀ metric. One can show that PF (PVk) is a compact metric space.

(2) For any open set U ⊂ PVk, deﬁne

TU := {F ∈PF (PVk): F ∩ U 6= ∅}; IU := {F ∈PF (PVk): F ⊂ U}.

Taking unions and ﬁnite intersections of TU ’s and IU ’s provides a topology on PF (PVk). It is clear that open sets for the topology (2) are open for the topology (1). Conversely, let F ∈ PF (PVk) and B(F, ε) an open ε-ball with center F in the Hausdorﬀ metric. Cover F by ﬁnitely many ε/2-balls: n [ i F ⊂ Bε/2 i=1 and let Fε/2 be the open ε/2-neighborhood of F . One veriﬁes that n \ IF ∩ TBi ⊂ B(F, ε). ε/2 ε/2 i=1

LEMMA 6.1.— supp is a PGL(Vk)-equivariant Borel map.

Proof.— The map is clearly PGL(Vk)-equivariant.

(a) We have for U ⊂ PVk open:

{µ : supp(µ)∈TU } = {µ : supp(µ) ∩ U 6= ∅} = {µ : µ(U) > 0}. It suﬃces to verify that the evaluation map 1 M (PVk) → R µ 7→ µ(U)

25 is Borel. Indeed, let {fn : PVk → [0; 1]}n be a sequence of continuous functions converging pointwise to χU , the characteristic function of U. By Lebesgue’s dominated convergence 1 theorem, we have for every µ∈M ( Vk): µ(U) = lim µ(fn). Therefore the above evaluation P n→∞ map is pointwise limit of the sequence of continuous maps 1 M (PVk) → R µ 7→ µ(fn), hence is Borel. [ (b) The open set U ⊂ PVk can be written as a countable union of closed sets U = Fn. n≥1 [ [ So {µ : supp(µ)⊂U} = {µ : supp(µ)⊂Fn} = {µ : µ(PVk \ Fn) = 0} is Borel. n≥1 n≥1

6.B ClZ : PF (PVk) → Vark(PV )

Recall we are given Vk and V := Vk ⊗K, where K is an algebraic closure of k. Denote by K[V ] the space of polynomials on V , and by k[V ] the space of polynomials on V with coeﬃcients on k. Let π : V \{0} → PV be the canonical projection. −1 DEFINITION.— A projective subvariety is a set S ⊂ PV such that π (S) ∪ {0} is the set of common zeroes of an ideal of polynomials in K[V ].

−1 We denote by I(S) the set of polynomials vanishing on π (S) ∪ {0}. Let K[V ]d be the space of homogeneous polynomials of degree d on V . Then ∞ M K[V ] = K[V ]d d=0 is a graded algebra. The deﬁning ideal of a projective variety has the fundamental property: ∞ M I(S) = I(S)d, d=0 where I(S)d := I(S) ∩ K[V ]d, i.e., I(S) contains all homogeneous components of it elements. −1 P Indeed, let P ∈I(S) and x∈π (S) ∪ {0}. Then P vanishes on λx for all λ∈K. If P = d Pd is the decomposition of P into its homogeneous components, we have P d d λ Pd(x) = 0, for all λ∈K.

Since K is inﬁnite, this implies Pd(x) = 0. DEFINITION.— A projective variety S ⊂ PV is deﬁned over k if I(S) := I(S) ∩ k[V ] generates I(S) over K. ∞ M Obviously I(S) = Id(S) where Id(S) := I(S) ∩ k[V ]d. d=0 One veriﬁes that intersections and ﬁnite unions of projective varieties are projective varieties. They form a family of closed sets of a topology on PV called Zariski topology.

The injection Vk ,→ V induces an injection PVk ,→ PV .

26 Z LEMMA 6.2.— Let F ⊂ PVk be a subset. Then the Zariski closure F ⊂ PV is deﬁned over k.

Proof.— Denote by π : V \{0} → PV and by πk : Vk \{0} → PVk the canonical projections. By deﬁnition Z I(F ) = {P ∈K[V ]: P |π−1(F )= 0} and hence Z Id(F ) = {P ∈K[V ]d : P |π−1(F )= 0} = {P ∈K[V ]d : P | −1 = 0}. πk (F ) −1 Let r = dimKK[V ]d −dimKI[V ]K. By linear algebra there exists x1, ... xr ∈πk (F ) ⊂ Vk \{0} such that

{P ∈K[V ]d : P | −1 = 0} = {P ∈K[V ]d : P (xi) = 0, 1 ≤ i ≤ r}. πk (F )

Now {P (xi)}1≤i≤r is a linear system of r equations with coeﬃcients in k, the unknown being the coeﬃcients of P . By linear algebra, the K-vector space of solutions is generated by the Z Z Z k-vector space of solutions with coordinates in k. Hence Id(F ) = Id(F ) ⊗ K, and hence F is deﬁned over k.

Let Vark(PV ) be the set of projective varieties of PV deﬁned over k. Recall that M I(S) = I(S) ∩ k[V ] and I(S) = Id(S), d for S ∈ Vark(PV ), where Id(S) = {P ∈ K[V ]d : P vanishes on S} (for a homogeneous polyno- mial, vanishing on a set in PV makes sense).

NOTATION. —For a ﬁnite dimensional k-vector space Wk, we set d G Gr(Wk) := Grl(Wk), l=0 where Grl(Wk) is the Grassmannian of l-planes in Wk. This is a compact topological space. Via the injection ∞ Y Vark(PV ) → Gr(k[V ]d) d=0 S 7→ Id(S) d∈N ∞ Y we consider Vark(PV ) as a subset of the compact topological space Gr(k[V ]d). We endow d=0 Vark(PV ) with the induced topology. In virtue of lemma 6.2, we may consider the map

ClZ : PF (PVk) → Vark(PV ) Z F 7→ F Z which will turn out to be Borel. We need the following description of Id(F ). The Pl¨ucker imbedding in degree d is the map ∗ Pld : PVk → Pk[V ]d

27 deﬁned as follows. Every vector v ∈ Vk gives by evaluation a linear form on k[V ]d, denoted by Ple d(v). The map v 7→ Ple d(v) is homogeneous of degree d, and so deﬁnes Pld, which is a homeomorphism onto its image.

∗ For each k-vector space Wk, we also have a polarity map Pol : Gr(Wk ) → Gr(Wk), which is a homeomorphism.

∗ Now the map Pld : PVk → Pk[V ]d induces a continuous map ∗ Plcd : PF (PVk) → PF Pk[V ]d F 7→ Pld(F ) ∗ ∗ ∗ Let [−]lin : PF Pk[V ]d → Gr(k[V ]d) denote the map which to a closed subset F of Pk[V ]d ∗ associates its linear space, seen as an element of Gr(k[V ]d). We have Z Id(F ) = Pol [Plcd(F )]lin . This is a straightforward veriﬁcation.

LEMMA 6.3.— Let Wk be a ﬁnite dimensional k-vector space. (i) The ﬁbers of the map

dim : PF (PWk) → N F 7→ dim[F ]lin are locally closed. (ii) For any interger l, the restriction to dim−1(l) of the map

PF (PWk) → Gr(Wk) F 7→ [F ]lin is continuous.

0 0 Proof.— Let F , F ∈ PF (PWk) such that D(F,F ) < ε. Set l := dim[F ]lin. Take x1,... xl ∈F such that

[F ]lin = [{x1,... xl}]lin. 0 Let y1,... yl ∈F such that d(xi, yi) < ε for 1 ≤ i ≤ l. For ε suﬃciently small, dim[{y1, ...yl}]lin = 0 l, hence dim[F ]lin ≥ dim[F ]lin, which shows that dim is lower semicontinuous and proves (i). 0 0 If now dim[F ]lin = dim[F ]lin, then [F ]lin is near [F ]lin ∈Gr(Wk), which proves (ii).

PROPOSITION 6.4.— The map ClZ : PF (PVk) → Vark(PV ) is Borel, PGL(Vk)-equivariant.

Proof.— PGL(Vk)-equivariance is clear. By deﬁnition of the topology on Vark(PV ), we have to verify that for all integer d, the map

PF (PVk) → Grk[V ]d Z F 7→ Id(F ) is Borel. But this map is the composition

Plcd ∗ [−]lin ∗ Pol PF (PVk) −→ PF Pk[V ]d −→ Gr(k[V ]d) −→ Gr(k[V ]d).

Now, Plcd and Pol are continuous, and it follows from lemma 6.3 that [−]lin is Borel.

28 6.C Combining the above results, we ﬁnally get:

1 THEOREM 6.5.— The Zariski support map suppZ : M (PVk) → Vark(PV ) is Borel and PGL(Vk)-equivariant.

7. ERGODIC ACTIONS AND BOREL STRUCTURES

7.A Ergodic actions

Let X be a locally compact and countable at inﬁnity topological space.

DEFINITIONS.— (i) µ, ν positive Radon measures on X are equivalent if the family of µ- negligible sets coincides with the family of ν-negligible ones. (ii) The class of a positive Radon measure µ is the set of positive Radon measures equivalent to µ. (iii) Let R be a group acting on X by homeomorphisms. The class of µ is R-invariant if for all r ∈R, r∗µ and µ are equivalent. (iv) In the previous situation, the action of R is ergodic if for any R-invariant measurable set E ⊂ X, either E or X \ E is µ-negligible.

For our purpose, we study actions on measure spaces (X, µ) of the following type: X = G/H, where G is a Lie group and H is a closed subgroup of G. Then G/H is a C∞-manifold on which G acts by diﬀeomorphisms. All Riemannian volumes (constructed via metrics) on G/H are in the same measure class, and G preserves this class, called the canonical class. For example, the Haar measure of G is in the canonical class. Besides, if G/H carries a G-invariant measure, this measure is in the canonical class. The projection p : G → G/H is a ﬁbration. From this and Fubini’s theorem follows that, for all E ⊂ G/H measurable

E is of measure 0 ⇐⇒ p−1(E) is of measure 0.

From this we get immediately

LEMMA 7.1.— Let R, H be closed subgroups of G. The R-action on G/H is ergodic if and only if the H-action on G/R is ergodic.

We now turn to the main result of this chapter.

THEOREM 7.2.— Let G be a simple connected Lie group and t be an element of G such that Adt is diagonalizable with positive real eigenvalues, not all = 1. Let A = hti. Let Γ be a lattice. The action of A on Γ \ G is ergodic.

From lemma 7.1 and theorem 7.2, we get

COROLLARY 7.3.— Let G, Γ, t∈G be as above, and let R < G be a closed subgroup containing t. The action of Γ on G/R is ergodic.

29 EXAMPLE.— G = PSL2(R); Γ < G is a lattice;

λ 0 ∗ ∗ t = ; P = { ∈ PSL (R)}. 0 λ−1 0 ∗ 2 Then hti and hence P acts ergodically on Γ \ G and therefore Γ acts ergodically on G/P , that is P1R. Theorem 7.2 follows from a theorem on unitary representations of G. DEFINITION.— A unitary representation of G in a Hilbert space H is a homomorphism π : G → U(H) of G into the group of unitary opertaors of H such that the corresponding action G × H → H (g, v) 7→ π(g)v is continuous. 2 EXAMPLE. —Let α be the G-invariant probability measure on Γ \ G and denote L (Γ \ G, α) by H. Then π(g)f(x) := f(xg), f ∈H, g ∈G deﬁnes a unitary representation of G∈H. THEOREM 7.4.— Let G, t, A be as above. Let (π, H) be a unitary representation of G. Any A-invariant vector is G-invariant.

Proof.— Let g = Lie G and g = g+ ⊕ g0 ⊕ g−, where g := {X ∈g : lim Ad(t)nX = 0} − n→∞ g := {X ∈g : lim Ad(t)nX = 0} + n→−∞ g0 := {X ∈g : Ad(t)X = X} Let HA be the closed subspace of H consisting of all A-invariant vectors. For v ∈ HA and g = exp X, X ∈g, we have:

kπ(g)v − v k = kπ(g)π(t−n)v − π(t−n)v k = kπ(tn)π(g)π(t−n)v − v k = kπexp(AdtnX)v − v k .

Hence for X ∈g−, we have kπ(g)v − v k= lim kπexp(AdtnX)v − v k= 0, n→∞ and for X ∈g+: kπ(g)v − v k= lim kπexp(AdtnX)v − v k= 0. n→−∞ In both cases π(g)v = v. A Since every element of g0 commutes with A, it leaves H invariant. So the group generated A by exp g−, exp g0 and exp g+ leaves H invariant. Since this group contains a neighborhood of e and G is connected, this group is G. Hence, we get a representation G → U(HA).

The kernel of this homomorphism contains exp g− and exp g+, and hence is of positive dimen- G A sion. Since G is simple and connected, this kernel is G, and therefore H = H .

30 Proof of theorem 7.2.— Let α be the G-invariant probability measure on Γ \ G and H := L2(Γ \ G, α), with π(g)f(x) := f(xg), f ∈H, g ∈G Since α is G-invariant, (π, H) is a unitary representation of G. Let E ⊂ Γ \ G be an A- invariant measurable set. Assume α(E) > 0. Let χE denote the characteristic function of E. Then χE is a nonzero vector in H and is A-invariant. Hence it is G-invariant – see theorem 7.4. Since G acts transitively on Γ \ G, χE coincides almost everywhere with χΓ\G. Hence α (Γ \ G) − E = 0. ◦ COROLLARY 7.5.— Let G be a simple algebraic group deﬁned over R such that G = GR is not compact. Let P be a minimal parabolic subgroup of G deﬁned over R, and P := P ∩ G. Let Γ be a lattice. The diagonal action of Γ on G/P × G/P is ergodic. Proof.— Denote by O the set of couples of opposite chambers in G/P × G/P : it is a single orbit under G, identiﬁed with G/A. The complement of O is a ﬁnite union of subvarieties of strictly lower dimension. This complement is therefore of measure zero. Hence, ergodicity of Γ on G/P × G/P is equivalent to ergodicity of Γ on G/A. But this follows from corollary 7.3. 7.B Borel structures

DEFINITION.— A Borel space is a set Y together with a σ-algebra of subsets. Every topological space has an associated Borel structure. Now let R, X, µ be as in the deﬁnition of ergodicity; in particular, assume that the R-action on X is ergodic. Let Y be a Borel space and f : X → Y be an R-invariant measurable map. From the point of view of the measure µ, the R-action behaves like a transitive action. One expects therefore f to be constant almost everywhere. In order for this assertion to be true, we need some additional hypothesis on the Borel space Y .

DEFINITION.— A Borel space Y is countably separated if there exists a countable family of Borel sets separating points of Y . A countably separated Borel space Y admits a countable coding which is Borel. More precisely, endow {0; 1} with the discrete topology, and {0; 1}N with the product topology. Now put on N {0; 1} the associated Borel structure. Let (Ai)i∈N be a family of Borel sets which separates points of Y . Then the map ϕ : Y → {0; 1}N y 7→ χAi (y) i∈N is an injective Borel map; one can show that its image is a Borel subset of {0; 1}N and that ϕ is a Borel isomorphism onto it.

LEMMA 7.6.— Let R, X, µ be as in the deﬁnition of ergodicity. Assume that R acts ergodically on X. Let Y be a countably generated Borel set and f : X → Y be an R-invariant measurable function. Then f is constant almost everywhere. Proof.— Since X is countable at inﬁnity, we may assume that µ(X) = 1. For any Borel subset A ⊂ Y , f −1(A) is an R-invariant measurable set and therefore of measure 0 or 1. Therefore,

31 there exists at most one point p ∈ Y such that µf −1(p) = 1. We have to show that there is one.

Let (An)n∈N be a family of Borel sets separating points of Y . Then for every p∈Y , the set

−1 \ Xp(n) := f Ai 1≤i≤n p∈Ai is of measure 1 or 0. Now we have ∞ −1 \ f (p) = Xp(n) n=1 −1 and hence µ(f (p)) is the limit of the stationnary sequence µ(Xp(n)) . Hence there is np such that

−1 µ f (p) = µ Xp(np) .

Clearly {Xp(np)}p∈Y is a countable family of measurable sets of X which cover X. Therefore −1 there is p∈Y such that µ Xp(np) > 0, and so µ Xp(np) = µ f (p) = 1.

LEMMA 7.7.— Let R × X → X be an ergodic action. Assume that the topological space R \ X has a countably separated Borel structure. Then there exists an R-orbit O ⊂ X such that µ(X − O) = 0.

Proof.— Consider the quotient map X → R \ X and apply lemma 7.6. This means that an ergodic action R × X → X is either essentially transitive or the quotient Borel structure is very complicated. It is clearly desirable to have criteria on an action R × X → X which guarantee that the quotient R \ X is countably separated.

LEMMA 7.8.— Let R be a group acting by homeomorphisms on a metrisable separable space Y . If all R-orbits are locally closed, then the topological space R\Y has a countably separated Borel structure. Proof.— The canonical projection p : Y → R \ Y is an open map. Indeed if O ⊂ Y is open, the set [ p−1p(O) = rO r∈R is a union of open sets and therefore open. Now Y has a countable basis of open sets, and so does R \ Y . Therefore it suﬃces to show that any two distinct points x, y ∈ R \ Y can be separated by an open set. Assume ﬁrst y 6∈ {x}. Then (R \ Y ) − {x} is an open set containing y but not x. Assume now y ∈ {x}. By assumption, {x} is open in {x} and therefore there exists V ⊂ R \ Y open with V ∩ {x} = {x}.

Clearly V contains x but not y.

32 PROPOSITION 7.9.— Let R be locally compact countable at inﬁnity acting continuously on Y metrisable separable complete. Then the R-orbits in Y are locally closed if and only if each orbit map R/StabR(y) → R.y is a homeomorphism onto its image.

Proof.— (⇐=) : R/StabR(y) is locally compact. Hence R.y is locally compct, and therefore locally closed. (=⇒) : Consider the orbit map

R/StabR(y) → R.y r 7→ r.y By assumption, R.y is open in R.y. Let U be a compact neighborhood of e ∈ R, and D ⊂ R be a countable dense set. Hence [ R = dU, d∈D [ and therefore R.y = dUy. By Baire’s category theorem there exists d ∈ D such that d∈D dUy is a neighborhood of one of its points. Therefore Uy has the same property. Now let N be a neighborhood e ∈ R and choose U such that U · U ⊂ N, U −1 = U. Now Uy is a neighborhood of one of its points, say uy and hence Ny ⊃ u−1Uy is a neighborhood of y. The map R/StabR(y) → R.y is continuous, injective and open: it is a homeomorphism.

8. THE BOUNDARY MAP. THE FIRST CASE OF DICHOTOMY

8.A Let G be a simple connected Lie group and Γ < G a lattice. Let H be a connected algebraic group deﬁned over k and such that the adjoint representation

Ad : Hk → GL(Vk) is irreducible (V = LieG).

Let π :Γ → Hk be a homomorphism such that π(Γ) is Zariski dense in H. Composing the above arrows, we obtain a representation

Γ → GL(Vk). To this situation we associated – see Chapter 5 – the Furstenberg map 1 ϕ : G/P → M (PVk) which is Γ-equivariant measurable. Recall that P is a minimal parabolic subgroup of G. We compose ϕ with the GL(Vk)-equivariant Borel map 1 SuppZ : M (PVk) → Vark(PV ) to obtain the boundary map

Φ: G/P → Vark(PV ) We are going to analyze properties of the homomorphism π in terms of properties of the boundary map Φ.

33 THEOREM 8.1.— If Φ is essentially constant, π(Γ) < Hk is relatively compact. The notion of essential image.— Let X be a locally compact space countable at inﬁnity, µ be a positive Radon measure on X and Y a topological space. Let f : X → Y be µ-measurable map. A point y ∈Y is an essential value of f if for every neighborhood V of y µf −1(V ) > 0.

The set of essential values of f, Essv(f) is called the essential image of f. It is a closed subset of Y . The map f is said essentially constant if Essv(f) is reduced to a point.

8.B First we need a key result due to H. Furstenberg, concerning the dynamics of the action 1 of PGL(Vk) on M (PVk).

1 N PROPOSITION 8.2.— Let µ, ν ∈ M (PVk) and {gn}n≥1 a sequence of PGL(Vk) such that

lim (gn)∗µ = ν. n→∞

If {gn}n≥1 is not relatively compact, there exists Ak, Bk proper linear subspaces of PVk such that

supp(ν) ⊂ Ak ∪ Bk. Furthemore, dimA + dimB = dimV − 1.

Proof.— Assume that {gn}n≥1 is not relatively compact. Up to passing to a subsequence, we may assume that lim gn = g, where g is in PEnd(Vk) and g 6∈PGL(Vk). Furthemore, we may n→∞ assume that

lim gn.Ker(g) = Ak ∈ Gr(Vk). n→∞

Let Bk := Im(g) ⊂ PVk. For every l ∈PVk \ Ker(g), we have lim gn(l)∈Bk, and for l ∈Ker(g), lim gn(l)∈Ak.

For f : PVk → R continuous with compact support in PVk \(Ak ∪Bk), the sequence (f ◦gn)n≥1 tends to 0 pointwise. By Dominated Convergence, we have ν(f) = 0. 1 COROLLARY 8.3.— Let ν ∈M (PVk) and assume that supp(ν) is not contained in the union of two proper linear subspaces. Then

StabPGL(Vk)(ν) := {g ∈PGL(Vk): g∗ν = ν} is compact.

Proof.— It follows from proposition 8.2 that StabPGL(Vk)(ν) is sequentially compact and hence compact.

34 8.C Now we assume that Φ = SuppZ ◦ ϕ is essentially constant and that Y ∈ Vark(PV ) is its essential image. Since Φ is Γ-equivariant, the projective variety Y is π(Γ)-invariant and therefore H-invariant since π(Γ) is Zariski dense in H. In particular, Yk ⊂ PVk is Hk-invariant. Therefore, ϕ takes its values in 1 MZD(Yk) := {µ∈M (PVk) : supp(µ) ⊂ Yk and suppZ (µ) ⊂ Y }. N LEMMA 8.4.— Let µ, ν ∈ MZD(Yk) and {hn}n≥1 a sequence of Hk such that

lim Ad(hn)∗µ = ν. n→∞

The sequence {hn}n≥1 is relatively compact. In particular,

(i) The Hk-orbits in MZD(Yk) are closed.

(ii) For every µ∈MZD(Yk), StabHk (µ) is compact.

Proof.— Assume that {hn}n≥1 is not relatively compact and hence {Ad(hn)}n≥1 is not rela- tively compact in AdHk < PGL(Vk). There are Ak, Bk proper linear subspaces of PVk such that

supp(ν) ⊂ Ak ∪ Bk and dimA + dimB = dimV − 1, see proposition 8.2. Since supp(ν) ⊂ Ak ∪ Bk, we have Y = suppZ (ν) ⊂ A ∪ B. The representation

Ad : Hk → GL(Vk) being irreducible, we have PVk = [Yk]lin ⊂ [Ak ∪ Bk]lin. In particular, [A ∪ B]lin = PV . It follows by dimension that A ∩ B = ∅. Therefore U := {h∈H : hA ∩ B = ∅ and A ∩ hB = ∅} is a nonvoid Zariski open subset of H. For all h∈U we have Y = Y ∩ hY ⊂ (A ∪ B) ∩ (hA ∪ hB) = (A ∩ hA) ∪ (B ∩ hB)

Since [Y ]lin = PV , it follows by dimension that A = hA and B = hB. Therefore U is also Zariski closed. Since H is connected, we have H = U. In particular, Ak is Hk-invariant. This contradicts the assumption that Ad is irreducible.

COROLLARY 8.5.— Orbits of the diagonal action of Hk ∈MZD(Yk) × MZD(Yk) are closed. Proof.— Follows immediately from lemma 8.4. Proof of theorem 8.1.— We ﬁrst show that the essential image of the map

ϕ : G/P → MZD(Yk) is reduced to a point. The map

ϕ × ϕ : G/P × G/P → MZD(Yk) × MZD(Yk) is measurable and equivariant for the diagonal action of Γ on the source and the diagonal action of Hk on the target. Let q : MZD(Yk) × MZD(Yk) → Hk \ MZD(Yk) × MZD(Yk)

35 be the canonical projection. The measurable map q ◦ϕ is Γ-invariant. It follows from corollary 8.5 and lemma 7.8 that the Borel structure of Hk \ MZD(Yk) × MZD(Yk) is countably separated. Since the action of Γ on G/P × G/P is ergodic – see corollary 7.5, it follows from lemma 7.6 that q ◦ ϕ is essentially constant. Therefore, there is an Hk-orbit

O ⊂ MZD(Yk) × MZD(Yk) such that for almost every (x, y)∈G/P ×G/P ,(ϕ×ϕ)(x, y) is in O. Since O is closed (corollary 8.5), we have: Essv(ϕ) × Essv(ϕ) = Essv(ϕ × ϕ) ⊂ O. Therefore O contains a point of the diagonal

∆ ⊂ MZD(Yk) × MZD(Yk) and O being an Hk-orbit, we have O ⊂ ∆. Therefore Essv(ϕ) × Essv(ϕ) ⊂ ∆ which shows that Essv(ϕ) is reduced to a point. Let Essv(ϕ) = {µ}. Since ϕ is Γ-equivariant,

π(Γ) < StabHk (µ). So π(Γ) is relatively compact.

8.D In the case G = SL2(R), we obtain more precise results.

THEOREM 8.6.— Let Γ < SL2(R) be a lattice and let

π :Γ → SL2(R) be a homomorphism such that π(Γ) is Zariski dense in SL2(R). There exists a unique Γ- equivariant measurable map ϕ : P1R → P1R. Proof.— Consider the boundary map 1 1 Φ: P R → VarR(P C) 1 associated to π. We can parametrize VarR(P C) as follows: 1 1 G 1 n VarR(P C) = {P C} t (P R) /Sn, n≥1 where Sn denotes the group of permutations on n letters. By ergodicity of the Γ-action on 1 1 1 n P R, we observe that the essential image of Φ is either {P C} or contained in (P R) /Sn for some n. If it were {P1C}, Φ would be essentially constant and hence by theorem 8.1, π(Γ) would be relatively compact. Hence π(Γ) would be contained in a conjugate of SO2(R), which is a proper algebraic subgroup of SL2(R), contradicting the assumption that π(Γ) is Zariski dense in SL2. To prove the theorem, it suﬃces to show that if 1 1 n ψ : P R → (P R) /Sn (n ≥ 2) 1 n is any Γ-equivariant map, its essential image is contained in the diagonal ∆n ⊂ (P R) /Sn. 1 Assume the contrary, then by ergodicity of the Γ-action on P R, ψ(x) 6∈ ∆n for almost every x∈P1R. Now consider 1 1 1 n 1 n ψ × ψ : P R × P R → (P R) /Sn × (P R) /Sn.

36 This map is Γ-equivariant for the diagonal action of Γ on the source and the diagonal action of SL2(R) on the target. One veriﬁes that the orbits of the latter action are locally closed so that there exists an SL2(R)-orbit 1 n 1 n O ⊂ (P R) /Sn × (P R) /Sn such that (ψ(x), ψ(x)) is in O for almost every (x, y) ∈ P1R × P1R. Therefore we have the following: For almost every (x, y)∈P1R × P1R (1) ψ(x) and ψ(y) are not on the diagonal; (2) for all γ ∈Γ, (ψ(γx), ψ(y))∈O. Take one point (x, y) satisfying these two properties. It follows from (2) that for any γ there is an h∈SL2(R) such that (ψ(γx), ψ(y)) = h(ψ(x), ψ(y)). In particular, h∈Stabψ(y) and h−1π(γ)∈Stabψ(x). Hence π(Γ) < Stabψ(y).Stabψ(x). 1 n But ψ(x) and ψ(y)are not contained in the diagonal of (P R) /Sn, which implies that Stab ψ(y) and Stabψ(x) are contained in some conjugate of a 0 A = { : a∈R×} R 0 a−1

From this follows that π(Γ) is not Zariski dense in SL2. A contradiction.

9. THE BOUNDARY MAP. THE SECOND CASE OF THE DICHOTOMY

Under the hypothesis of 8.1, we assume now that the boundary map

Φ: G/P → Vark(PV ) is not essentially constant. Recall that ∞ Y Vark(PV ) ⊂ Gr k[V ]d , d=0 see Chapter 6. Let pd : Vark(PV ) → Gr k[V ]d denote the canonical projection. There exists d ≥ 1 such that pd ◦ Φ: G/P → Gr k[V ]d G is not essentially constant. Besides, we have Gr k[V ]d = Grl k[V ]d . Since Γ acts ergodi- l≥1 cally on G/P , there exists l ≥ 1 such that for almost every x∈G/P , (pd ◦ Φ)(x)∈Grl k[V ]d . At this point, we need the following fundamental result. THEOREM 9.1.— Let k be a ﬁeld, W an algebraic variety deﬁned over k, H an algebraic group deﬁned over k and H × W → W an algebraic action of H on W , deﬁned over k. Then the orbits of Hk ∈Wk are locally closed. 0 Reference.— This is [A C-B, theorem 6.1].

37 From this, it follows that the orbits of Hk ∈Grl k[V ]d are locally closed. It follows from the fact that Γ operates ergodically on G/P that there exists an Hk-orbit O ⊂ Grl k[V ]d such that

(pd ◦ Φ)(x)∈O, for almost every x∈G/P .

Choose p∈O. Then StabH (p) = StabH (p) ∩ Hk. Setting L := StabH (p), we see that L is an k algebraic subgroup of H deﬁned over k, and StabHk (p) = Lk. Since all Hk-orbits in Grl k[V ]d are locally closed, the orbit map

Hk/Lk → O hLk 7→ h(p) is a homeomorphism (proposition 7.3).

We can therefore consider pd ◦ Φ as a Γ-equivariant measurable map

ϕ : G/P → Hk/Lk. Since ϕ is not essentially constant, L 6= H. Summarizing the results of chapter 8 and 9.1, we obtain

THEOREM 9.2.— Let G be a connected simple Lie group, Γ < G be a lattice, H a k-almost simple connected group and

π :Γ → Hk a homomoprhism such that π(Γ) is Zariski dense in H. If π(Γ) is not relatively compact in Hk, there exists L < H an algebraic subgroup deﬁned over k with L 6= H, and a measurable map

ϕ : G/P → Hk/Lk, where P is a minimal parabolic subgroup of G. Recall that our goal is to extend the homomorphism π to G. Without further assumptions on Γ or G, theorem 9.2 is as far as we can go. In order to proceed further, one can make one of the following assumptions.

(1) CommGΓ is dense in G. (2) G has rank ≥ 2. Case (1) will be treated in the chapter 10, case (2) in chapter 11.

10. COMMENSURATOR SUPERRIGIDITY

Let now G be a connected simple Lie group, Γ < G a lattice, Λ a subgroup such that

Γ < Λ < CommGΓ, H a k-almost simple connected group and

π :Λ → Hk a homomorphism such that π(Λ) is Zariski dense in H.

38 10.1. Our ﬁrst goal is to construct a Λ-equivariant measurable map from G/P into a space of the form Hk/Lk. This does not follow directly from theorem 9.2 since we have not made any assumption on the topological nature of Λ. In fact, we will eventually assume that Λ is dense in G. We consider now the family F consisting of all pairs (ϕ, M) where

(1) M is an Hk-homogeneous space of the form Hk/Lk, where L is an algebraic subgroup of H deﬁned over k, such that L 6= H, (2) there exists Γ0 < Λ commensurable with Γ such that ϕ : G/P → M is Γ0-equivariant measurable.

THEOREM 10.1.— Assume that π(Γ) < Hk is not relatively compact. Then F= 6 ∅. Furthe- more, there exists (ψ, N) ∈ F, unique up to Hk-equivariant isomorphism, such that for all (ϕ, M)∈F, there exists an Hk-equivariant map p : N → M such that the diagram

ψ G/P −→ N & ↓ p ϕ M commutes. Proof.— (a) F= 6 ∅: this follows from theorem 9.2 provided we can show that π(Γ) is Zariski Z dense in H. Our assumption is that π(Λ) is dense in H. Let L = π(Γ) be the Zariski closure −1 of π(Γ)∈H. For all λ∈π(Λ), π(Γ)/ π(Γ) ∩ λ π(Γ)λ is ﬁnite (recall Λ < CommGΓ). Hence, the image of π(Γ) ∈ H/(L ∩ λ−1Lλ) is ﬁnite. Therefore π(Γ) is contained in a Zariski closed subset of the form n G −1 hi(λ Lλ), i=1 and so is L. This shows that L/(L ∩ λ−1Lλ) is ﬁnite and hence L◦ = (L ∩ λ−1Lλ)◦ < L◦ ∩ λ−1L◦λ. This shows that π(Λ) normalizes L◦ and hence L◦ /H since π(Λ) is Zariski dense. But H is k-almost simple, so L◦ = H.

(b) On F, we deﬁne the following preorder: (ϕ1,M1) ≥ (ϕ2,M2) if there exists an Hk- equivariant map p : M1 → M2 such that

ϕ1 G/P −→ M1 & ↓ p

ϕ2 M2 commutes.

Let F0 = F/ ∼ be the quotient of F under the equivalence relation (ϕ1,M1) ∼ (ϕ2,M2) if (ϕ1,M1) ≥ (ϕ2,M2) and (ϕ2,M2) ≥ (ϕ1,M1). Then, ≥ deﬁnes an order on F0. Let H be the set of homogeneous spaces of the form (1) above with the obvious preorder structure, and let H0 be the associated ordered set. The map

39 F : F → H (ϕ, M) 7→ M induces an increasing map of ordered spaces F : F0 → H0. The theorem will follow if we can show that F0 has a maximal element.

First, we need the following observation: if M1 ≥ M2 and M2 ≥ M1, then any Hk-equivariant map M1 → M2 is an isomorphism. Let pi : Mi → M3−i be Hk-equivariant maps. It clearly suﬃces to show that q := p2◦p1 : M1 → M1 is an isomorphism. Now identify M1 with Hk/Lk. Let q(eLk) = hLk. Since q is Hk-equivariant, we have for all l ∈Lk:

lhLk = lq(eLk) = q(lLk) = q(eLk) = hLk,

n −1 \ −i i and hence h Lkh < Lk. Consider Un = h Lh ; {Un}n≥0 is a decreasing sequence of i=0 algebraic subgroups of H. It stabilizes: there exists n such that Un = Un+1; and hence

−n n −(n+1) −n h Lkh = (Un)k = (Un+1)k = h Lkh ,

−1 from which follows h Lkh = Lk, and so h∈N Lk . From this follows that q is invertible.

0 0 Let C ⊂ F0 be a chain, and set C = F (C). To a chain C ∈ H0 corresponds a sequence of algebraic subgroups of H which stabilizes. Therefore, C0 is ﬁnite. Now we can observe that the restriction of F to C is injective. Indeed, assume (ϕ1,M1) ≥ (ϕ2,M2), M1 ≥ M2 and M2 ≥ M1. Let p : M1 → M2 be the Hk-equivariant map such that ϕ2 = p ◦ ϕ1. We have observed that p −1 must be an isomorphism: p ◦ ϕ2 = ϕ1, which shows that (ϕ2,M2) ≥ (ϕ1,M1). Finally, we conclude that C is ﬁnite, and therefore has a maximal element. By Zorn’s lemma, we conclude that F0 has a maximal element.

Now we show that for c1, c2 ∈ F0, there exists a ∈ F0 with a ≥ c1 and a ≥ c2. Let (ϕ2,M2), (ϕ1,M1) ∈ F0. Let Γi < Λ be such that ϕi is Γi-equivariant and Γi is commen- surable with Γ. Set Γ3 := Γ1 ∩ Γ2: this group is commensurable with Γ. Consider the map

ϕ3 : G/P → M1 × M2 x 7→ ϕ1(x), ϕ2(x) which is Γ3-equivariant measurable. It follows from theorem 9.1 and from the ergodicity of Γ3 on G/P that the essential image of ϕ3 is contained in one Hk-orbit, say M3, contained in M1 × M2. Using the projections on each factor, we see that (ϕ3,M3) ≥ (ϕ1,M1) and (ϕ3,M3) ≥ (ϕ2,M2)

0 Let θ : G/P → N, N = Hk/Lk be a versal element of F given by theorem 10.1. Let L be the 0 normalizer in H of the Zariski closure of Lk, and q : Hk/Lk → Hk/Lk the canonical projection.

0 COROLLARY.— The map θ : G/P → Hk/Lk is Λ-equivariant and measurable.

−1 Proof.— For λ ∈ Λ and (ϕ, M) ∈ F, one deﬁnes via the formula λ∗ϕ(x) := π(λ)ϕ(λ x), an action of Λ on F which preserves the preorder on F. Now (θ, N) being maximal, the maps λ∗θ and θ diﬀer by an Hk-equivariant map N → N. This implies that θ is Λ-equivariant.

40 0 Proof of the Commensurator Superrigidity. Let θ : G/P → Wk, W = H/L , be the map given Z Z by corollary 10.2. Since π(Λ) = H and θ is Λ-equivariant, we have W = Essv(θ) . Deﬁne θ : G → F (G, Wk) by θ(g)(x) := θ(gxP ).

One veriﬁes that for the topology of convergence in measure on F (G, Wk), the map θ is continuous. It is also Λ-equivariant. Since Λ is dense in G, it acts ergodically on G, and therefore there exists an Hk-orbit O ∈ F (G, Wk) such that θ(g) ∈ O for almost every g ∈ G. One deduces then from the fact that O is open in O and θ is continuous, that θ(G) ⊂ O. In particular, O = Hk∗θ. Since Essv(θ) is Zariski dense in W , StabH (θ) ﬁxes pointwise W and k thus is contained in Z Hk , since H is k-almost simple. We thus get a well deﬁned continuous map h : G → Hk/Z Hk such that θ(g) = h(g)∗θ for all g ∈ G. One veriﬁes that h is a homomorphism; being Λ- equivariant, it provides the desired extension.

11. HIGHER RANK SUPERRIGIDITY

◦ Let G be a connected R-almost simple group with real rank ≥ 2, Γ a lattice in G := GR, H a connected k-almost simple group and π :Γ → Hk a homomorphism with unbounded and Zariski dense image. Then there exists a proper k-subgroup L < H and a Γ-equivariant measurable map

ϕ : G/P → Hk/Lk which is not essentially constant. Here, P = P(R), where P is a minimal parabolic subgroup of G. The main point of the proof consists in showing that under these conditions, one has k = R or C and ϕ is essentially rational. Now, let P− be an R-parabolic opposite P, S < P∩P− a maximal R-split torus, t∈S := S(R), u − − with t 6= 1. Ct is the centraliser of t ∈ G, and Ct is N ∩ Ct, where N is the unipotent radical of P−.

PROPOSITION 11.1.— For almost every g ∈G, the map u Ct (R) → Hk/Lk n 7→ ϕ(gnP ) is essentially continuous. If k = R or C, this map is essentially rational; otherwise, it is essentially constant.

Proof.— Set C := Ct(R). Composing the map G × C → G/P (g, c) 7→ ϕgcP with ϕ, and using Fubini’s theorem, we deduce that for almost every g ∈G, the map

ϕg : C → Hk/Lk c 7→ ϕ(gc)

41 is measurable, and φ : G/hti → F C,Hk/Lk g 7→ ϕg is Γ-equivariant measurable. Since Γ acts ergodically on G/hti (theorem 7.2), we deduce that there exists ψ ∈F C,Hk/Lk such that

φ(g)∈Hk∗ψ, for almost every g ∈G. Using again Fubini’s theorem, we get that for almost every g ∈G, and almost every c∈C

φ(gc), φ(g)∈Hk∗ψ, and thus there exists hg(c)∈Hk such that

(?) φ(gc) = hg(c)∗φ(g). Fixing g, c such that (?) holds, we get 0 0 0 (??) ϕg(cc ) = hg(c)ϕg(c ), for almost every c ∈C. Deﬁne Z Vg := Essv(ϕg) ; Ng := {h∈H : hVg = Vg};

Zg := {h∈H : h|Vg = idVg } and Hg := Ng/Zg.

Using (?) and (??), we see that hg : C → Hg(k) is a well deﬁned homomorphism, and thus coincides almost everywhere with a continuous one. This proves the continuity assertion. u If now k 6= R, C, then the map is constant since Ct (R) is connected. If now k = R or C, the ∞ u homomorphism hg is C . Furthemore, since Ct is a unipotent subgroup of the semisimple u part of C , h (C ) < Hg is unipotent and hence h | u is polynomial. In particular, the map t g t g Ct is essentially rational. To conclude rationality of ϕ on all of G/P , we need the following two lemmas. u LEMMA 11.2.— There are t , ... t ∈S \{e}; and connected (unipotent) R-subgroups U < C 1 n i ti such that n Y − (i) The product map Ui → N is an R-isomorphism of varieties. i=1 n n Y Y − (ii) Ui / Ui < N . i=r i=r−1 m n LEMMA 11.3.— Let k = R, C, V be a k-variety and f : R × R → V be a measurable map such that for almost every x∈Rm n fx : R → V y 7→ f(x, y) is essentially k-rational, and so is f y : Rm → V x 7→ f(x, y) for almost every y ∈Rn. Then f is essentially rational. Reference.— This is [Z, theorem 3.4.4].

42 COROLLARY 11.4.— The ﬁeld k is R or C and ϕ : G/P → Hk/Lk is essentially rational. Proof.— Take a parametrization of N− given by lemma 11.2. We prove by induction on r ≤ n, that for almost every g ∈G, the map n Y Ui(R) → Hk/Lk i=r n 7→ ϕ(gnP ) is essentially rational if k = R or C, and is essentially constant otherwise. For r = n, this is the content of proposition 11.1. Assuming it true for r, consider n Y Ur−1(R) × Ui(R) → Hk/Lk i=r (u, v) 7→ ϕ(guvP ) which by proposition 11.1 is essentially rational if k = R or C; essentially constant otherwise, n Y in variable v, for almost all g ∈ G and u ∈ Ur−1(R). Since Ur−1(R) normalizes Ui(R), we i=r get the same assertion in the variable u ∈ Ur−1(R). Thus, if k = R or C, lemma 11.3 implies the rationality of n Y Ui(R) → Hk/Lk i=r−1 n 7→ ϕ(gnP ) and, if k 6= R or C, Fubini’s theorem implies that the map above is essentially constant. Thus, we get that − N (R) → Hk/Lk n 7→ ϕ(gnP ) is essentially constant if k 6= R and C. Since N−(R) parametrizes the big cell in G/P which is of full measure, we get that ϕ is essentially constant. A contradiction. Thus k = R or C, and the last map is essentially rational. Proof of the superrigidity theorem.— We have now a rational map ϕ : G/P → H/L which is Γ-equivariant. Consider the graph Gr(π) := { γ, π(γ) ∈G × H}γ∈Γ, Z and Gr(π) its Zariski closure in G × H. Since Γ is Zariski dense in G and π(Γ) is Zariski Z dense in H, the projection of Gr(π) on each factor is surjective. Now let R be the space R(G/P, H/L) of rational maps, endowed with the action of G × H deﬁned by −1 (g, h)∗ψ(x) := hψ(g x), ψ ∈R. Z Since StabG×H (ϕ) is an algebraic subgroup, we have that Gr(π) ﬁxes ϕ. Furthemore, if Z (e, h1) and (e, h2) are in Gr(π) , then h1ϕ(x) = h2ϕ(x) for all x∈G/P . Since the image of ϕ −1 is Zariski dense in H/L, we get that h2 h1 ﬁxes pointwise H/L and hence −1 \ −1 h2 h1 ∈ hLh = {e}, h∈H

43 Z since H is assumed to be adjoint. Thus, Gr(π) is the graph of a morphism of algebraic groups πext : G → H, deﬁned over k and extending π.

12. ARITHMETICITY

In this section, we prove the higher rank arithmeticity theorem. Thus, let Γ be a lattice in G := G(R)◦, where G is connected, adjoint, R-simple of real rank ≥ 2. We consider G as a subgroup of Aut(g), where g = Lie(G), and ﬁx a Chevalley basis of g, so that G is deﬁned over Q (proposition 3.7). For g ∈Aut(g), let Tr(g) denote the trace of g. We ﬁrst show Tr(Γ) ⊂ Q. Indeed, any σ ∈Aut(C) deﬁnes a homomorphism σ :Γ → G(C) with Zariski dense image. Superrigidity implies that – either σ(Γ) < G(C) is bounded, so that |Trσ(γ)| ≤ dim(g) for any γ ∈Γ;

– or there exists θσ ∈Aut(g) with θσ|Γ = σ |Γ. In particular, we have Tr(γ) = Tr θσ(γ) = Tr σ(γ) for any γ ∈ Γ. Thus, for all γ ∈ Γ, Aut(C)Tr(γ) is a bounded subset of C, which implies that Tr(Γ) is contained in Q. Now let k = Q Tr(Γ) be the ﬁeld generated by {Tr(γ)}γ∈Γ, let C[G] be the space of regular functions on G, and let ρ : G → GL C[G] be the regular representation. Let TC be the linear Γ span of {ρ(g)Tr}g∈G and let TΓ ⊂ k be the linear span of the set of right Γ-translates of Γ Γ Tr |Γ∈ k . Since Γ is Zariski dense in G, the restriction map R : C[G] → C is injective and induces an isomorphism ∼ R : TG −→ TΓ ⊗ C. Z Thus, we get a Γ-invariant k-structure on the vector space TG; the group H = ρ(Γ) is therefore deﬁned over k ⊂ R, ρ : G → H is an isomorphism deﬁned over k, and ρ(Γ) < Hk. Thus, replacing G by H, we may assume that G is deﬁned over k and Γ < G(k). Next, we claim that if O denotes the ring of integers of k,Γ ∩ G(O) is of ﬁnite index in Γ: let k ,→ kv be a non Archimedean place of k, and ιv : G(k) ,→ G(kv) be the corresponding injection. Let Ov be the ring of integers of kv. Since Γ < G(k) is ﬁnitely generated, the set P of places for which ιv(Γ) 6⊂ G(Ov) is ﬁnite. Moreover, by superrigidity, ιv(Γ) < G(kv) is bounded for all non Archimidean places. Thus, ιv(Γ)/ ιv(Γ) ∩ G(Ov) is ﬁnite, which implies since P is ﬁnite, that Γ ∩ G(O) is ﬁnite index in Γ. Thus, replacing Γ by a subgroup of ﬁnite index, we may assume Γ < G(O). Under the canonical isomorphism ∼ D : G(k) −→ Resk/Q G(Q), the image D(Γ) lies in Resk/Q G(Z). Furthemore, using that k is the trace ﬁeld of Γ, one shows easily that D(Γ) is Zariski dense in Resk/Q G. Let Resk/Q G = L × U be the decomposition into R-factors, such that U(R) is compact and L(R) is without compact factor. Let ∆ be the image in L(R) of Resk/Q G(Z); ∆ is a lattice in L(R). Let q be the composition of D with

44 the projection on L(R). Observe that the projection of q(Γ) on every R-simple factor of L(R) is unbounded. Thus, q extends to an isomorphism of R-groups ∼ qext : G −→ L, such that qext(Γ) < ∆.

Thus, if p denotes the composition of the projection Resk/Q G(R) → L(R) with the inverse of qext, we get

p : Resk/Q G(R) → G, continuous surjective with compact kernel and such that Γ < p Resk/Q G(Z) .

REFERENCES 0 0 [A C-B] N. A CAMPO, M. BURGER, R´eseauxarithm´etiqueset commensurateur, d’apr`esG.A. Margulis, Inventiones Mathematicæ 116 (1994), 1-25. [R] M. RAGHUNATHAN, Discrete subgroups of Lie groups, Springer, 1972. [Z] R.J. ZIMMER, Ergodic theory and semisimple groups, Birkh¨auser,1984.

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