Discrete Groups, Expanding Graphs and Invariant Measures
In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit constru
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Series Editors J. Oesterle A. Weinstein
Alexander Lubotzky
Discrete Groups, Expanding Graphsand Invariant Measures Appendix by Jonathan D. Rogawski
Springer Basel AG
Author: Alexander Lubotzky Institute of Mathematics Hebrew University Jerusalem 91904 Israel
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data
Lubotzky, Alexander:
Discrete groups, expanding graphs and invariant measures I Alexander Lubotzky. Appendix by Jonathan D. Rogawski. (Progress in mathematics ; Vol. 125) ISBN 978-3-0346-0331-7 ISBN 978-3-0346-0332-4 (eBook) DOI 10.1007/978-3-0346-0332-4 NE:GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 1994 Springer Basel AG Originally published by Birkhäuser V erlag in 1994 Softcover reprint of the bardeover 1st edition 1994 Printed on acid-free paper produced of chlorine-free pulp ISBN 978-3-0346-0331-7 987654321
Table of Contents
0
Introduction 00000000000000000000000000000000000000000000000000000
1
Expanding graphs 1.0
1.1 1.2 2
2ol 202 203
1
5
Introduction 00000000000000000000000000000000000000o0000000 The Hausdorff-Banach-Tarski paradox 0000000000000000000000 Invariant measures 0000000000000000000000000000000000000000 Notes 0000000000000000000000000000000000000000000000000000
7 7 13
18
Kazhdan Property (T) and its applications 3o0 301 302 303 3.4 305
4
1
The Banach-Ruziewicz problern 200
3
Introduction Expanders and their applications 000000000000000000000000000 Existence of expanders 000000000000000000000000000000000000
ix
lntroduction 0000000000000000000000000000000000000000000000 Kazhdan property (T) for semi-simple groups 000000000000000 Lattices and arithmetic subgroups 00000000000000000000000000 Explicit construction of expanders using property (T) 00000000 Solution of the Ruziewicz problern for sn, n ~ 4 using property (T) 0000000000000000000000000000000000000000000000 Notes 0000000000000000000000000000000000000000000000000000
19 19 27 30 34 39
The Laplacian and its eigenvalues 4o0
41
401
lntroduction 0000000000000000000000000000000000000000000000 The geometric Laplacian 0000000000000000000000000000000000
402
The combinatorial Laplacian 0000000000000000000000000000000
44
403
Eigenvalues, isoperimetric inequalities and representations 000 Setberg Theorem AJ ~ 6 and expanders 0000000000000000000
49
4.4 405 406
?
Random walk on k-regular graphs; Ramanujan graphs 0000000 Notes 0000000000000000000000000000000000000000000000000000
41
52 55 59
Vl
5
TABLE OF CONTENTS
The representation theory of PGLz 500
5o1
502
Introduction Representations and spherical functions Irreducihle representations of PSLz(~) and eigenvalues of the Laplacian The tree associ
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