Lubotzky Discrete Groups, Expanding Graphs and Invariant Measures. Modern Birkhäuser Classics

Appendix by Jonathan D. Rogawski
1. Auflage 2009
ISBN: 978-3-0346-0332-4
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 201 Seiten

The book presents the solutions to two problems: the first is the construction of expanding graphs - graphs which are of fundamental importance for communication networks and computer science, the second is the Ruziewicz problem concerning the finitely additive invariant measures on spheres. Both problems were partially solved using the Kazhdan property (T) from representation theory of semi-simple Lie groups. Later, complete soultions were obtained for both problems using the Ramanujan conjecture from analytic number theory. The author, who played an important role in these developments, explains the two problems and their solutions from a perspective which reveals why all these seemingly unrelated topics are so interconnected. The unified approach shows interrelations between different branches of mathematics such as graph theory, measure theory, Riemannian geometry, discrete subgroups of Lie groups, representation theory and analytic number theory. Special efforts were made to make the book accessible to graduate students in mathematics and computer science. A number of problems and suggestions for further research are presented. Reviews: "This exciting book marks the genesis of a new field. It is a field in which one passes back and forth at will through the looking glass dividing the discrete from the continuous. (...) The book is a charming combination of topics from group theory (finite and infinite), combinatorics, number theory, harmonic analysis." - Zentralblatt MATH "The Appendix, written by J. Rogawski, explains the Jacquet-Langlands theory and indicates Deligne`s proof of the Petersson-Ramanujan conjecture. It would merit its own review. (...) In conclusion, this is a wonderful way of transmitting recent mathematical research directly "from the producer to the consumer." - MathSciNet "The book is accessible to mature graduate students in mathematics and theoretical computer science. It is a nice presentation of a gem at the border of analysis, geometry, algebra and combinatorics. Those who take the effort to glance what happens behind the scene won`t regret it." - Acta Scientiarum Mathematicarum TOC:0 Introduction. - 1 Expanding graphs. - 2 The Banach-Ruziewicz problem. - 3 Kazhdan Property (T) and its applications. - 4 The Laplacian and its eigenvalues. - 5 The representation theory of PGL2. - 6 Spectral decomposition of L2(G(Q)G(A)). - 7 Banach-Ruziewicz problem for n = 2, 3, Ramanujan graphs. - 8 Some more discrete mathematics. - 9 Distributing points on the sphere. - 10 Open problems. - Appendix. - References. - Index.

Autoren/Hrsg.

Lubotzky, Alex

Inhaltsverzeichnis

1;Table of Contents
;6
2;Introduction;10
3;1 Expanding Graphs
;13
3.1;1.0 Introduction
;13
3.2;1.1 Expanders and their applications
;13
3.3;1.2 Existence of expanders
;17
4;2 The Banach-Ruziewicz Problem
;19
4.1;2.0 Introduction
;19
4.2;2.1 The Hausdorff-Banach-Tarski paradox
;19
4.3;2.2 Invariant Measures
;25
4.4;2.3 Notes
;30
5;3 Kazhdan Property (T) and its Applications
;31
5.1;3.0 Introduction
;31
5.2;3.1 Kazhdan property (T) for semi-simple groups
;31
5.3;3.2 Lattices and arithmetic subgroups
;39
5.4;3.3 Explicit construction of expanders using property (T)
;42
5.5;3.4 Solution of the Ruziewicz problem for Sn, n = 4
;46
5.6;3.5 Notes
;51
6;4 The Laplacian and its Eigenvalues
;53
6.1;4.0 Introduction
;53
6.2;4.1 The geometric Laplacian
;53
6.3;4.2 The combinatorial Laplacian
;56
6.4;4.3 Eigenvalues, isoperimetric inequalities and representations
;61
6.5;4.4 Selberg Theorem .1 = 3/
16 and expanders ;64
6.6;4.5 Random walks on k-regular graphs; Ramanujan graphs
;67
6.7;4.6 Notes
;71
7;5 The Representation Theory of PGL2;73
7.1;5.0 Introduction;73
7.2;5.1 Representations and spherical functions;74
7.3;5.2 Irreducible representations of PSL2 (R) and eigenvalues of the Laplacian
;77
7.4;5.3 The tree associated with PGL2 (Qp)
;80
7.5;5.4 Irreducible representations of PGL2(Qp) and eigenvalues of the Hecke operator
;82
7.6;5.5 Spectral decomposition of G\G;84
8;6 Spectral Decomposition of L²(G(Q)\G(A ))
;89
8.1;6.0 Introduction;89
8.2;6.1 Deligne’s Theorem; adèlic formulation
;89
8.3;6.2 Quaternion algebras and groups;91
8.4;6.3 The Strong Approximation Theorem and its applications;93
8.5;6.4 Notes;95
9;7 Banach-Ruziewicz Problem for n = 2, 3; Ramanujan Graphs
;97
9.1;7.0 Introduction;97
9.2;7.1 The spectral decomposition of G'(Z[1/p])\G'(R) × G'(Qp)
;98
9.3;7.2 The Banach-Ruziewicz problem for n = 2, 3;98
9.4;7.3 Ramanujan graphs and their extremal properties;100
9.5;7.4 Explicit constructions;106
9.6;7.5 Notes;111
10;8 Some More Discrete Mathematics;113
10.1;8.0 Introduction;113
10.2;8.2 Characters and eigenvalues of finite groups;118
10.3;8.3 Some more Ramanujan graphs (of unbounded degrees);124
10.4;8.4 Ramanujan Diagrams;127
11;9 Distributing Points on the Sphere;131
11.1;9.0 Introduction;131
11.2;9.1 Hecke operators of group action;131
11.3;9.2 Distributing points on S² (and S³)
;133
12;10 Open Problems;136
12.1;10.1 Expanding graphs;136
12.2;10.2 The Banach-Ruziewicz Problem;136
12.3;10.3 Kazhdan Property (T) and its applications;137
12.4;10.4 The Laplacian and its eigenvalues;139
12.5;10.5 The representation theory of PGL2;140
12.6;10.6 Spectral decomposition of L²(G(Q) \ G(A))
;140
12.7;10.7 Banach-Ruziewicz problem for n = 2, 3; Ramanujan graphs;141
12.8;10.8 Some more discrete mathematics;142
12.9;10.9 Distributing points on the sphere;144
13;Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands
correspondence;145
13.1;A.0 Preliminaries;146
13.2;A.1 Representation theory and modular forms;149
13.3;A.2 Classification of unitary representations;159
13.4;A.3 Quaternion algebras;169
13.5;A.4 The Selberg trace formula;174
13.6;References to the Appendix;184
14;References;186