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On property (τ ) -preliminary version-

Alexander Lubotzky

Andrzej Żuk

September 5, 2003

2

Contents

I Properties and examples 11

1 Property (T) and property (τ) 13

1.1 Fell topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Property (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Examples of groups with property (T) . . . . . . . . . 14

1.3 Combinatorial approach to property (T) . . . . . . . . . . . . 15

1.4 Property (τ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.1 Profinite topologies . . . . . . . . . . . . . . . . . . . . 19

1.4.2 Lattices in semi-simple groups . . . . . . . . . . . . . . 20

1.5 Properties of groups with (τ) . . . . . . . . . . . . . . . . . . 23

1.5.1 Finite generation . . . . . . . . . . . . . . . . . . . . . 23

1.5.2 Abelian quotients . . . . . . . . . . . . . . . . . . . . . 25

2 Various equivalent forms of (τ) 29

2.1 Combinatorial reformulation . . . . . . . . . . . . . . . . . . . 30

2.1.1 Expanders and isoperimetric inequalities . . . . . . . . 30

2.1.2 The spectral gap . . . . . . . . . . . . . . . . . . . . . 30

2.1.3 Equivalent definitions . . . . . . . . . . . . . . . . . . . 31

2.2 Analytic and geometric reformulation . . . . . . . . . . . . . . 33

2.2.1 Isoperimetric inequalities . . . . . . . . . . . . . . . . . 33

2.2.2 The Laplace operator . . . . . . . . . . . . . . . . . . . 34

2.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.4 Equivalent definitions . . . . . . . . . . . . . . . . . . . 35

2.3 Measure theoretic reformulation . . . . . . . . . . . . . . . . . 37

2.3.1 Uniqueness of invariant measures . . . . . . . . . . . . 37

2.4 Cohomological interpretation . . . . . . . . . . . . . . . . . . . 37

2.5 Fixed point property . . . . . . . . . . . . . . . . . . . . . . . 43

3

4 CONTENTS

3 Quantitative (τ) and abelian quotients 45

3.1 Quantitative (τ) . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Small isoperimetric constant implies positive β1 . . . . . . . . 47

3.3 Quantitative bounds on abelian quotients . . . . . . . . . . . . 51

3.4 Bounds on the number of representations . . . . . . . . . . . . 55

4 The Selberg property 59

4.1 Selberg’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 S-arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 An equivalent formulation . . . . . . . . . . . . . . . . . . . . 64

4.4 Property (τ) and the congruence subgroup property . . . . . . 65

4.5 The Ramanujan conjecture . . . . . . . . . . . . . . . . . . . . 66

4.6 The spectrum: from the infinite to the finite . . . . . . . . . . 70

II Applications of (τ) 77

5 Expanders 79

5.1 Expanders and Ramanujan graphs . . . . . . . . . . . . . . . . 79

5.2 Dependence on generators . . . . . . . . . . . . . . . . . . . . 81

5.3 Finite simple groups as expanders . . . . . . . . . . . . . . . . 89

5.4 Diameter of finite simple groups . . . . . . . . . . . . . . . . . 92

5.5 Ramanujan groups . . . . . . . . . . . . . . . . . . . . . . . . 93

6 The product replacement algorithm 97

6.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Congruence subgroups of Aut(H) . . . . . . . . . . . . . . . . 98

6.3 The PRA and the Selberg property . . . . . . . . . . . . . . . 100

6.4 The PRA and dependence on generators . . . . . . . . . . . . 104

7 Hyperbolic manifolds 107

7.1 Thurston’s conjecture for arithmetic lattices . . . . . . . . . . 108

7.2 Arithmetic lattices in SO(n, 1) . . . . . . . . . . . . . . . . . . 110

7.3 The Lubotzky-Sarnak conjecture . . . . . . . . . . . . . . . . 112

7.4 3-manifolds, Heegaard splittings and property (τ) . . . . . . . 114

7.5 Pro-p groups and 3-manifolds . . . . . . . . . . . . . . . . . . 117

7.6 Lattices in other rank one groups . . . . . . . . . . . . . . . . 123

CONTENTS 5

8 Uniqueness of invariant measures 127 8.1 Open compact subgroups of local adelic groups . . . . . . . . 127 8.2 Uniqueness and non-uniqueness with respect to dense subgroups128

9 Property (τ) and C∗ algebras 131 9.1 Separating C∗(Γ) by finite dimensional representations: neg-

ative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.2 Seperating C∗(Γ) by finite dimensional representations: posi-

tive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 CONTENTS

Introduction

Property (T) was introduced in a seminal paper of Kazhdan [104] in 1967. A group G has this property if the trivial one dimensional representation of G is ”bounded away” from all the other irreducible unitary representations of G. Kazhdan property (T) turned out to be a powerful representation theoretic method to study discrete subgroups of Lie groups.

The current notes are about a baby version of property (T), which is called property (τ). It asserts, for a discrete group Γ, that trivial representation is bounded away fom the non-trivial irreducible finite representations (i.e., those with finite images). In many applications, it is even useful to look at a smaller subclass of representations: Let L = {Ni}i∈I be a family of finite index subgroups of Γ. Γ is said to have τ with respect to L (τ(L) for short) if the non-trivial irreducible Γ subrepresentations of l2(Γ/Ni), i ∈ I, are bounded away from the trivial representation. An important case which will be dealt in details is when Γ is an arithmetic group and L is its family of congruence subgroups.

It turns out that (τ) - being weaker than (T) - is sometimes even more useful as it holds for a wider class of groups. Moreover it can be presented in various equivalent forms (for simplicity, we assume that L is large enough to define a topology on Γ):

(a) Representation theoretical - the original definition;

(b) Combinatorial - Γ has (τ(L)) if and only if the quotient Schreier graphs form a family of expanding graphs (”expanders”);

(c) Measure theoretic - Γ has (τ(L)) if and only if the Haar measure is the unique Γ-invariant finitely additive measure on Γ̂L (the profinite group obtained from Γ by completing it with respect to the topology determined by L).

(d) Cohomological - Γ has (τ(L)) if and only if H1(Γ, L2(Γ̂L)) = 0.

7

8 CONTENTS

If in addition Γ = π1(M) where M is a compact Riemannian manifold and Mi is the finite sheeted covering of M corresponding to Ni, i ∈ I then we also have

(e) Analytic - Γ has (τ(L)) if and only if there is an ε > 0 such that λ1(Mi) > ε, for every i ∈ I, where λ1(Mi) is the smallest positive eigenvalue of the Laplace-Beltrami operator on Mi.

(f) Geometric - Γ has (τ(L)) if and only if there is an ε > 0 such that h(Mi) ≥ ε for every i, where h(Mi) is the Cheeger constant (the isoperimetric constant) of the manifold Mi.

The fact that (τ) can be expressed in so many different ways opens the door to applications in several directions. The main goal of these notes is to describe these applications which look quite unrelated, from a unified perspective. Some of these applications are, by now, quite well known and some of them are more recent. There are only few new results in this book (e.g. Sections 2.4, 5.2 and 8.2).

These applications include:

(i) The constructions of expanders. These graphs are of fundamental importance in computer science and combinatorics;

(ii) The analysis of a popular algorithm in computational group theory called ”the product replacement algorithm” (PRA - for short). This algo- rithm provides pseudo random elements from a finite group given by its generators. In practice, it turns out to have outstanding performances, but its theoretical analysis does not, as yet, explains why. Property (τ) or more precisely a ”non-commutative Selberg Theorem” can give the proper expla- nation. This connection suggests some problems and conjectures regarding (τ) for the automorphism group of the free group.

(iii) The uniqueness of the Haar measure as the only finitely additive invariant measure of some local and adélic profinite groups.

(iv) Applications to C∗ algebras and in particular to the question when the C∗ algebra of a discrete group Γ is seperated by its finite dimensional representations.

The most surprising applications are for

(v) Hyperbolic manifolds. In this regard we present a proof for arithmetic manifolds of Thurston’s conjecture on non-vanishing of the first Betti number

CONTENTS 9

of finite covers of such manifolds. Moreover the recent work of Lackenby [113] suggests a path how the Lubotzky-Sarnak conjecture (asserting that fundamental groups of compact hyperbolic 3-manifolds do not have (τ)) can lead to a proof of the famous ”virtual Haken conjecture” for hyperbolic 3- manifolds.

As said the main goal of these notes is to present these applications from a unified point of view. Along the way we discuss the connections with questions on automorphic forms, finite groups, discrete subgroups in Lie groups, 3-manifolds, pro-p groups and more. A number of open problems for further research are also presented.

In a way the current book

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