Discrete Groups, Expanding Graphs and Invariant Measures. Modern Birkhäuser Classics (eBook)

Table of Contents 
6 
Introduction 10
1 Expanding Graphs 
13 
1.0 Introduction 
13 
1.1 Expanders and their applications 
13 
1.2 Existence of expanders 
17 
2 The Banach-Ruziewicz Problem 
19 
2.0 Introduction 
19 
2.1 The Hausdorff-Banach-Tarski paradox 
19 
2.2 Invariant Measures 
25 
2.3 Notes 
30 
3 Kazhdan Property (T) and its Applications 
31 
3.0 Introduction 
31 
3.1 Kazhdan property (T) for semi-simple groups 
31 
3.2 Lattices and arithmetic subgroups 
39 
3.3 Explicit construction of expanders using property (T) 
42 
3.4 Solution of the Ruziewicz problem for Sn, n = 4 
46 
3.5 Notes 
51 
4 The Laplacian and its Eigenvalues 
53 
4.0 Introduction 
53 
4.1 The geometric Laplacian 
53 
4.2 The combinatorial Laplacian 
56 
4.3 Eigenvalues, isoperimetric inequalities and representations 
61 
4.4 Selberg Theorem .1 = 3/ 
64 
4.5 Random walks on k-regular graphs Ramanujan graphs
67 
4.6 Notes 
71 
5 The Representation Theory of PGL2 73
5.0 Introduction 73
5.1 Representations and spherical functions 74
5.2 Irreducible representations of PSL2 (R) and eigenvalues of the Laplacian 
77 
5.3 The tree associated with PGL2 (Qp) 
80 
5.4 Irreducible representations of PGL2(Qp) and eigenvalues of the Hecke operator 
82 
5.5 Spectral decomposition of G/G 84
6 Spectral Decomposition of L²(G(Q)/G(A )) 
89 
6.0 Introduction 89
6.1 Deligne’s Theorem adèlic formulation
89 
6.2 Quaternion algebras and groups 91
6.3 The Strong Approximation Theorem and its applications 93
6.4 Notes 95
7 Banach-Ruziewicz Problem for n = 2, 3 Ramanujan Graphs
97 
7.0 Introduction 97
7.1 The spectral decomposition of G'(Z[1/p])/G'(R) × G'(Qp) 
98 
7.2 The Banach-Ruziewicz problem for n = 2, 3 98
7.3 Ramanujan graphs and their extremal properties 100
7.4 Explicit constructions 106
7.5 Notes 111
8 Some More Discrete Mathematics 113
8.0 Introduction 113
8.2 Characters and eigenvalues of finite groups 118
8.3 Some more Ramanujan graphs (of unbounded degrees) 124
8.4 Ramanujan Diagrams 127
9 Distributing Points on the Sphere 131
9.0 Introduction 131
9.1 Hecke operators of group action 131
9.2 Distributing points on S² (and S³) 
133 
10 Open Problems 136
10.1 Expanding graphs 136
10.2 The Banach-Ruziewicz Problem 136
10.3 Kazhdan Property (T) and its applications 137
10.4 The Laplacian and its eigenvalues 139
10.5 The representation theory of PGL2 140
10.6 Spectral decomposition of L²(G(Q) / G(A)) 
140 
10.7 Banach-Ruziewicz problem for n = 2, 3 Ramanujan graphs
10.8 Some more discrete mathematics 142
10.9 Distributing points on the sphere 144
Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands 
145 
A.0 Preliminaries 146
A.1 Representation theory and modular forms 149
A.2 Classification of unitary representations 159
A.3 Quaternion algebras 169
A.4 The Selberg trace formula 174
References to the Appendix 184
References 186