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RIGIDITY and ARITHMETICITY Marc BURGER

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RIGIDITY and ARITHMETICITY Marc BURGER 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 ◦ '. 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. 0 1 x z 1 2) G = f@ 0 1 y A : x; y; z 2 Rg is a 2-step nilpotent Lie group { called the Heisenberg 0 0 1 0 1 x z 1 group { and the subgroup Γ = f@ 0 1 y A : x; y; z 2Zg 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 2R 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 := fΛ 2 R : Vol(R =Λ) = 1g, 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 0 J 1 @ F0 A ; 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 0 λ1 1 B ::: C B C B λp C B C S = fB In−2p C : λi 6= 0g B −1 C B λp C @ ::: A −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 0 1 1 0 1 1 t GF = SO(F ) = fg 2GL3 : g @ 1 A g = @ 1 A ; detg = 1g. −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) 2 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 ' : H=Γ ! G=p(Γ) which equivariant, i.e., '(hx) = p(h)'(x), for all h 2 H and x 2 H=Γ. If α denotes an H-invariant measure on H=Γ, the direct image '∗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 α2R n 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.
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