Discrete Groups, Expanding Graphs and Invariant Measures

200 Pages·1994·1.12 MB·English

by  Alexander Lubotzky (auth.)

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Alexander Lubotzky
Discrete Groups,
Expanding Graphs and
Invariant Measures
Appendix by Jonathan D. Rogawski
Ferran Sunyer i Balaguer
Award winning monograph
Reprint of the 1994 Edition
Birkhäuser Verlag
Basel · Boston · Berlin
Author:
Alexander Lubotzky
Einstein Institute of Mathematics
Hebrew University
Jerusalem 91904
Israel
e-mail: [e-mail]
Originally published under the same title as volume 125 in the Progress in
Mathematics series by Birkhäuser Verlag, Switzerland, ISBN 978-3-7643-5075-8
© 1994 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
1991 Mathematics Subject Classification 22E40, 43A05, 11F06, 11F70, 05C25,
28C10
Library of Congress Control Number: 2009937803
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Table of Contents
0 Introduction ..................................................... ix
1 Expanding graphs
1.0 Introduction .............................................. 1
1.1 Expanders and their applications ........................... 1
1.2 Existence of expanders .................................... 5
2 The Banach-Ruziewicz problem
2.0 Introduction .............................................. 7
2.1 The Hausdorff-Banach-Tarski paradox ...................... 7
2.2 Invariant measures ........................................ 13
2.3 Notes .................................................... 18
3 Kazhdan Property (T) and its applications
3.0 Introduction .............................................. 19
3.1 Kazhdan property (T) for semi-simple groups ............... 19
3.2 Lattices and arithmetic subgroups .......................... 27
3.3 Explicit construction of expanders using property (T) ........ 30
3.4 Solution of the Ruziewicz problem for Sn,n≥4 using
property (T) .............................................. 34
3.5 Notes .................................................... 39
4 The Laplacian and its eigenvalues
4.0 Introduction .............................................. 41
4.1 The geometric Laplacian .................................. 41
4.2 The combinatorial Laplacian ............................... 44
4.3 Eigenvalues, isoperimetric inequalities and representations ... 49
4.4 Selberg Theorem λ ≥ 3 and expanders ................... 52
1 16
4.5 Random walk on k-regular graphs; Ramanujan graphs ....... 55
4.6 Notes .................................................... 59
vi Table of Contents
5 The representation theory of PGL2
5.0 Introduction .............................................. 61
5.1 Representations and spherical functions ..................... 62
5.2 Irreducible representations of PSL2((cid:2)) and eigenvalues of
the Laplacian ............................................. 65
5.3 The tree associated with PGL2((cid:3)p) ........................ 68
5.4 Irreducible representations of PGL2((cid:3)p) and eigenvalues
of the Hecke operator ..................................... 70
5.5 Spectral decomposition of Γ\G ............................ 72
6 Spectral decomposition of L2(G((cid:3))\G((cid:4)))
6.0 Introduction .............................................. 77
6.1 Deligne’s Theorem; ade`lic formulation ...................... 77
6.2 Quaternion algebras and groups ............................ 79
6.3 The Strong Approximation Theorem and its applications ..... 81
6.4 Notes .................................................... 83
7 Banach-Ruziewicz problem for n=2,3; Ramanujan graphs
7.0 Introduction .............................................. 85
7.1 The spectral decomposition of G(cid:2)((cid:5)[1])\G(cid:2)((cid:2))×G(cid:2)((cid:3) ) .... 86
p p
7.2 The Banach-Ruziewicz problem for n=2,3 ................ 86
7.3 Ramanujan graphs and their extremal properties ............. 88
7.4 Explicit constructions ..................................... 94
7.5 Notes .................................................... 99
8 Some more discrete mathematics
8.0 Introduction .............................................. 101
8.1 The diameter of finite simple groups ....................... 101
8.2 Characters and eigenvalues of finite groups ................. 106
8.3 Some more Ramanujan graphs (of unbounded degrees) ...... 112
8.4 Ramanujan diagrams ...................................... 115
9 Distributing points on the sphere
9.0 Introduction .............................................. 119
9.1 Hecke operators of group action ........................... 119
9.2 Distributing points on S2 (and S3) ......................... 121
Table of Contents vii
10 Open problems
10.1 Expanding Graphs ........................................ 125
10.2 The Banach-Ruziewicz Problem ............................ 125
10.3 Kazhdan Property (T) and its applications .................. 126
10.4 The Laplacian and its eigenvalues .......................... 128
10.5 The representation theory of PGL2 ......................... 129
10.6 Spectral decomposition of L2(G((cid:3))/G((cid:4))) ................. 129
10.7 Banach-Ruziewitz Problem for n=2,3; Ramanujan Graphs .. 130
10.8 Some more discrete mathematics ........................... 131
10.9 Distributing points on the sphere ........................... 133
Appendix: by Jonathan D. Rogawski
Modular forms, the Ramanujan conjecture and the
Jacquet-Langlands correspondence
A.0 Preliminaries ............................................. 136
A.1 Representation theory and modular forms ................... 139
A.2 Classification of irreducible representations ................. 149
A.3 Quaternion algebras ....................................... 159
A.4 The Selberg trace formula ................................. 164
References to the Appendix ...................................... 175
References .......................................................... 177
Index ............................................................... 193

0 Introduction
Inthelastfifteenyearstwoseeminglyunrelatedproblems,oneincomputerscience
andtheotherinmeasuretheory,weresolvedbyamazinglysimilartechniquesfrom
representation theory and from analytic number theory. One problem is the ex-
plicitconstructionofexpandinggraphs(«expanders»).Thesearehighlyconnected
sparsegraphswhoseexistencecanbeeasilydemonstrated butwhoseexplicit con-
struction turns out to be a difficult task. Since expanders serve as basic building
blocks for various distributed networks, an explicit construction is highly desir-
able. The other problem is one posed by Ruziewicz about seventy years ago and
studied by Banach[Ba]. It asks whether the Lebesgue measure is the only finitely
additive measure of total measure one, defined on the Lebesgue subsets of the
n-dimensional sphere and invariant under all rotations.
Thetwoproblemsseem,atfirstglance,totallyunrelated.Itisthereforesome-
what surprising that both problems were solved using similar methods: initially,
Kazhdan’sproperty(T)fromrepresentationtheoryofsemi-simple Liegroupswas
applied in both cases to achieve partial results, and later on, both problems were
solved using the (proved) Ramanujan conjecture from the theory of automorphic
forms. The fact that representation theory and automorphic forms have anything
to do with these problems is a surprise and a hint as well that the two questions
are strongly related.
The main goal of these notes is to present the two problems and their solu-
tions from a unified point of view. We will explore how both solutions are just
two different aspects of thesame phenomenon: for somegroup G,the trivial one-
dimensional representation is isolated from some subclass of irreducible unitary
representations ofG.Kazhdan’s property(T)isprecisely apropertyoftheabove.
The Ramanujan conjecture also has such an interpretation, from which the two
solutionscanbededuced.Infact,thehighlightofthesenotesisasinglearithmetic
group Γ embedded naturally in a direct product G =G1×G2 where G1 is a real
Lie group and G2 a p-adic Lie group. The Ramanujan conjecture, in its represen-
tation theoretic formulation, controls the G-irreducible representations appearing
in L2(Γ\G). Using this result for the projection of Γ into G1 =SO(3) yields an
affirmative solution to the Banach-Ruziewicz problem (for the sphere S2 – but
also more – see Chapter 7). Projecting Γ to the second factor G2 = PGL2((cid:3)p)
andapplyingthesame result,oneobtainsexpandinggraphs(asquotient graphsof
the tree associated with PGL2((cid:3)p) modulo the action of congruence subgroups
of Γ).
There are very few new results in these notes. The main intention is to
reproducetheexisting solutions ofthetwoproblems inawaywhich stresses their
unity. We also elaborate on the connection between the above two problems and
othertopics,e.g.,eigenvaluesoftheLaplacianofRiemannianmanifolds,Selberg’s
Theorem λ ≥ 3 for arithmetic hyperbolic surfaces, the combinatorics of some
1 16
finite simple groups, numerical analysis on the sphere and more.
x 0 Introduction
Hereisabriefchapter-by-chapterdescriptionofthecontents;amoredetailed
one may be found at the beginning of each chapter.
In Chapter 1 we survey and illustrate the importance of expanders and prove
their existence via counting arguments. In the second chapter we present the
Hausdorff-Banach-Tarski paradox, which is a motivation for the Ruziewicz prob-
lem(andalsoplaysaroleinitssolution)andshowthattheRuziewiczproblemfor
n=1 has a negative answer. In Chapter 3, after introducing property (T) for Lie
groupsandtheirlattices, weapplyittosolveaffirmatively theRuziewiczproblem
for n ≥ 4. In addition, we use it to give an explicit construction of expanders
which, however, are not as good as the ones whose existence is established by
counting arguments. In the nextchapter we connectour material with eigenvalues
of the Laplacian operator of Riemannian manifolds and of graphs. We also bring
to the fore Selberg’s Theorem, which can be used to give interesting explicit ex-
panders.Theproblemofexpandersistranslatedintoaproblemabouteigenvalues,
and the important notion of Ramanujan graphs is introduced.
Chapter 5 is an introduction to the representation theory of PGL2 over the
reals and the p-adics. This is a fairly well-known topic, but we stress the unified
treatment of both cases as well as the close connection with the eigenvalues of
thecorrespondingLaplacians.WealsopresentthetreeassociatedwithPGL2((cid:3)p)
analogous to the hyperbolic upper half-plane associated with PGL2((cid:2)). The ma-
terial of this chapter is a necessary background for the sequel. We continue in
Chapter 6, where we quote the works of Deligne and Jacquet-Langlands. The
next chapter merges all the ingredients to completely solve Ruziewicz’s problem,
as well as to construct Ramanujan graphs. We also present there some of the
remarkable properties of these graphs.
The next two chapters, Chapters 8 and 9, bring some miscellaneous topics
relatedtotheabove:Chapter8containsapplicationstofinitesimplegroupsaswell
as some other methods to construct Ramanujan graphs (but of unbounded degree)
and Ramanujan diagrams, while Chapter 9 brings a pseudo-random method to
distribute points on the sphere, which is an application of the above methods
and uses the same group Γ from Chapter 7. This method is of importance for
numericalanalysisonthesphereandelsewhere.OnlythecasesofS2 andS3 have
satisfactory results, while the higher-dimensional cases (which can be thought as
«quantified»Ruziewiczproblems)arestillopen.Thisproblemandmanymoreare
described in Chapter 10.
The Appendix, by Jonathan Rogawski, gives more details and elaborates on
the material described in Chapters 5 and 6. While in the body of the book we
emphasise the representation theoretic formulation of the Ramanujan (-Petersson)
conjecture,theAppendixexplainsitsclassicalformandtheconnectiontomodular
forms.RogawskialsoexplainstheJacquet–Langlandstheoryandgivesindications
of how that conjecture was proved by Deligne and how the Jacquet–Langlands
theory enables us to apply it for quaternion algebras.