The Ergodic Theory of Lattice Subgroups

Preface vii

0.1 Main objectives vii

0.2 Ergodic theory and amenable groups viii

0.3 Ergodic theory and nonamenable groups x

Chapter 1. Main results: Semisimple Lie groups case 1

1.1 Admissible sets 1

1.2 Ergodic theorems on semisimple Lie groups 2

1.3 The lattice point-counting problem in admissible domains 4

1.4 Ergodic theorems for lattice subgroups 6

1.5 Scope of the method 8

Chapter 2. Examples and applications 11

2.1 Hyperbolic lattice points problem 11

2.2 Counting integral unimodular matrices 12

2.3 Integral equivalence of general forms 13

2.4 Lattice points in S-algebraic groups 15

2.5 Examples of ergodic theorems for lattice actions 16

Chapter 3. Definitions, preliminaries, and basic tools 19

3.1 Maximal and exponential-maximal inequalities 19

3.2 S-algebraic groups and upper local dimension 21

3.3 Admissible and coarsely admissible sets 21

3.4 Absolute continuity and examples of admissible averages 23

3.5 Balanced and well-balanced families on product groups 26

3.6 Roughly radial and quasi-uniform sets 27

3.7 Spectral gap and strong spectral gap 29

3.8 Finite-dimensional subrepresentations 30

Chapter 4. Main results and an overview of the proofs 33

4.1 Statement of ergodic theorems for S-algebraic groups 33

4.2 Ergodic theorems in the absence of a spectral gap: overview 35

4.3 Ergodic theorems in the presence of a spectral gap: overview 38

4.4 Statement of ergodic theorems for lattice subgroups 40

4.5 Ergodic theorems for lattice subgroups: overview 42

4.6 Volume regularity and volume asymptotics: overview 44

Chapter 5. Proof of ergodic theorems for S-algebraic groups 47

5.1 Iwasawa groups and spectral estimates 47

5.2 Ergodic theorems in the presence of a spectral gap 50

5.3 Ergodic theorems in the absence of a spectral gap, I 56

5.4 Ergodic theorems in the absence of a spectral gap, II 57

5.5 Ergodic theorems in the absence of a spectral gap, III 60

5.6 The invariance principle and stability of admissible averages 67

Chapter 6. Proof of ergodic theorems for lattice subgroups 71

6.1 Induced action 71

6.2 Reduction theorems 74

6.3 Strong maximal inequality 75

6.4 Mean ergodic theorem 78

6.5 Pointwise ergodic theorem 83

6.6 Exponential mean ergodic theorem 84

6.7 Exponential strong maximal inequality 87

6.8 Completion of the proofs 90

6.9 Equidistribution in isometric actions 91

Chapter 7. Volume estimates and volume regularity 93

7.1 Admissibility of standard averages 93

7.2 Convolution arguments 98

7.3 Admissible, well-balanced, and boundary-regular families 101

7.4 Admissible sets on principal homogeneous spaces 105

7.5 Tauberian arguments and H¨older continuity 107

Chapter 8. Comments and complements 113

8.1 Lattice point-counting with explicit error term 113

8.2 Exponentially fast convergence versus equidistribution 115

8.3 Remark about balanced sets 116

Bibliography 117

Index 121