Discrete Groups, Expanding Graphs and Invariant Measures

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Cover Title page 0 Introduction 1 Expanding graphs 1.0 Introduction 1.1 Expanders and their applications 1.2 Existence of expanders 2 The Banach-Ruziewicz problem 2.0 Introduction 2.1 The Hausdorff-Banach-Tarski paradox 2.2 Invariant measures 2.3 Notes 3 Kazhdan Property (T) and its applications 3.0 Introduction 3.1 Kazhdan property (T) for semi-simple groups 3.2 Lattices and arithmetic subgroups 3.3 Explicit construction of expanders using property (T) 3.4 Solution of the Ruziewicz problem for S^n, n>4 using property (T) 3.5 Notes 4 The Laplacian and its eigenvalues 4.0 Introduction 4.1 The geometric Laplacian 4.2 The combinatorial Laplacian 4.3 Eigenvalues, isoperimetric inequalities and representations 4.4 Selberg Theorem λ₁ > 3/16 and expanders 4.5 Random walk on k-regular graphs; Ramanujan graphs 4.6 Notes 5 The representation theory of PGL₂ 5.0 Introduction 5.1 Representations and spherical functions 5.2 Irreducible representations of PSL₂(R) and eigenvalues of the Laplacian 5.3 The tree associated with PGL2(Q_p) 5.4 Irreducib1e representations of PGL₂(Q_p) and eigenvalues of the Hecke operator 5.5 Spectral decomposition of Γ\G 6 Spectral decomposition of L²(G(Q)\G(A)) 6.0 Introduction 6.1 Deligne's Theorem; adèlic formulation 6.2 Quaternion algebras and groups 6.3 The Strong Approximation Theorem and its applications 6.4 Notes 7 Banach-Ruziewicz problem for n = 2,3; Ramanujan graphs 7.0 Introduction 7.1 The spectral decomposition of G'(...Q_p) 7.2 The Banach-Ruziewicz prob1em for n = 2,3 7.3 Ramanujan graphs and their extremal properties 7.4 Explicit constructions 7.5 Notes 8 Some more discrete mathematics 8.0 Introduction 8.1 The diameter of finite simple groups 8.2 Characters and eigenvalues of finite groups 8.3 Some more Ramanujan graphs (of unbounded degrees) 8.4 Ramanujan diagrams 9 Distributing points on the sphere 9.0 Introduction 9.1 Hecke operators of group action 9.2 Distributing points on S² (and S³) 10 Open problems 10.1 Expanding Graphs 10.2 The Banach-Ruziewicz Problem 10.3 Kazhdan Property (T) and its applications 10.4 The Laplacian and its eigenvalues 10.5 The representation theory of PGL₂ 10.6 Spectral decomposition of L²(G(Q)/G(A)) 10.7 Banach-Ruziewitz Problem for n = 2,3; Ramanujan Graphs 10.8 Some more discrete mathematics 10.9 Distributing points on the sphere Appendix by Jonathan D. Rogawski: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence A.0 Preliminaries A.1 Representation theory and modular forms A.2 Classification of irreducible representations A.3 Quaternion algebras A.4 The Selberg trace formula References to the Appendix References Index