Elementary Number Theory, Group Theory and Ramanujan Graphs

154 Pages·2003·1.61 MB·English

by  Giuliana Davidoff

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LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
Managingeditor:ProfessorC.M.Series,MathematicsInstitute
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
3 Localfields,J.W.S.CASSELS
4 Anintroductiontotwistortheory:Secondedition,S.A.HUGGETT&K.P.TOD
5 Introductiontogeneralrelativity,L.P.HUGHSTON&K.P.TOD
7 Thetheoryofevolutionanddynamicalsystems,J.HOFBAUER&K.SIGMUND
8 SummingandnuclearnormsinBanachspacetheory,G.J.O.JAMESON
9 AutomorphismsofsurfacesafterNielsenandThurston,A.CASSON&S.BLEILER
11 Spacetimeandsingularities,G.NABER
12 Undergraduatealgebraicgeometry,MILESREID
13 AnintroductiontoHankeloperators,J.R.PARTINGTON
15 Presentationsofgroups:Secondedition,D.L.JOHNSON
17 Aspectsofquantumfieldtheoryincurvedspacetime,S.A.FULLING
18 Braidsandcoverings:Selectedtopics,VAGNLUNDSGAARDHANSEN
19 Stepsincommutativealgebra,R.Y.SHARP
20 Communicationtheory,C.M.GOLDIE&R.G.E.PINCH
21 RepresentationsoffinitegroupsofLietype,FRANC¸OISDIGNE&JEANMICHEL
22 Designs,graphs,codes,andtheirlinks,P.J.CAMERON&J.H.VANLINT
23 Complexalgebraiccurves,FRANCESKIRWAN
24 Lecturesonellipticcurves,J.W.S.CASSELS
25 Hyperbolicgeometry,BIRGERIVERSEN
26 AnintroductiontothetheoryofL-functionsandEisensteinseries,H.HIDA
27 Hilbertspace:Compactoperatorsandthetracetheorem,J.R.RETHERFORD
28 Potentialtheoryinthecomplexplane,T.RANSFORD
29 Undergraduatecommutativealgebra,M.REID
31 TheLaplacianonaRiemannianmanifold,S.ROSENBERG
32 LecturesonLiegroupsandLiealgebras,R.CARTER,G.SEGAL,&
I.MACDONALD
33 AprimerofalgebraicD-modules,S.C.COUTINHO
34 Complexalgebraicsurfaces,A.BEAUVILLE
35 Youngtableaux,W.FULTON
37 Amathematicalintroductiontowavelets,P.WOJTASZCZYK
38 Harmonicmaps,loopgroups,andintegrablesystems,M.GUEST
39 Settheoryfortheworkingmathematician,K.CIESIELSKI
40 Ergodictheoryanddynamicalsystems,M.POLLICOTT&M.YURI
41 Thealgorithmicresolutionofdiophantineequations,N.P.SMART
42 Equilibriumstatesinergodictheory,G.KELLER
43 Fourieranalysisonfinitegroupsandapplications,AUDREYTERRAS
44 Classicalinvarianttheory,PETERJ.OLVER
45 Permutationgroups,P.J.CAMERON
46 Riemannsurfaces:Aprimer,A.BEARDON
47 Introductorylecturesonringsandmodules,J.BEACHY
48 Settheory,A.HAJNA´L,P.HAMBURGER,&A.MATE
49 K-theoryforC∗-algebras,M.RORDAM,F.LARSEN,&N.LAUSTSEN
50 Abriefguidetoalgebraicnumbertheory,H.P.F.SWINNERTON-DYER
51 Stepsincommutativealgebra:Secondedition,R.Y.SHARP
52 FiniteMarkovchainsandalgorithmicapplications,O.HAGGSTROM
53 Theprimenumbertheorem,G.J.O.JAMESON
54 Topicsingraphautomorphismsandreconstruction,J.LAURI&R.SCAPELLATO
55 Elementarynumbertheory,grouptheory,andRamanujangraphs,G.DAVIDOFF,
P.SARNAK,&A.VALETTE
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ELEMENTARY NUMBER
THEORY, GROUP THEORY,
AND RAMANUJAN GRAPHS
GIULIANA DAVIDOFF
MountHolyokeCollege
PETER SARNAK
PrincetonUniversity&NYU
ALAIN VALETTE
UniversitedeNeuchatel
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  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521824262
© Giuliana Davidoff, Peter Sarnak, Alain Valette 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
- ---- eBook (NetLibrary)
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Cambridge University Press has no responsibility for the persistence or accuracy of
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Contents
Preface pageix
AnOverview 1
Chapter1. GraphTheory 8
1.1.TheAdjacencyMatrixandItsSpectrum 8
1.2.InequalitiesontheSpectralGap 12
1.3.AsymptoticBehaviorofEigenvaluesinFamilies
ofExpanders 18
1.4.ProofoftheAsymptoticBehavior 20
1.5.IndependenceNumberandChromaticNumber 30
1.6.LargeGirthandLargeChromaticNumber 32
1.7.NotesonChapter1 36
Chapter2. NumberTheory 38
2.1.Introduction 38
2.2.SumsofTwoSquares 39
2.3.QuadraticReciprocity 48
2.4.SumsofFourSquares 52
2.5.Quaternions 57
2.6.TheArithmeticofIntegerQuaternions 59
2.7.NotesonChapter2 70
Chapter3. PSL (q) 72
2
3.1.SomeFiniteGroups 72
3.2.Simplicity 73
3.3.StructureofSubgroups 76
3.4.RepresentationTheoryofFiniteGroups 85
3.5.DegreesofRepresentationsofPSL (q) 102
2
3.6.NotesonChapter3 107
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vi Contents
Chapter4. TheGraphs Xp,q 108
4.1.CayleyGraphs 108
4.2.Constructionof Xp,q 112
4.3.GirthandConnectedness 115
4.4.SpectralEstimates 122
4.5.NotesonChapter4 130
Appendix 4.RegularGraphswithLargeGirth 132
Bibliography 138
Index 143
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Preface
Thesenotesareintendedforageneralmathematicalaudience.Inparticular,
wehaveinmindthattheycouldbeusedasacourseforundergraduates.They
containanexplicitconstructionofhighlyconnectedbutsparsegraphsknown
asexpandergraphs.Besidestheirinterestincombinatoricsandgraphtheory,
thesegraphshaveapplicationstocomputerscienceandengineering.Ouraim
has been to give a self-contained treatment. Thus, the relevant background
material in graph theory, number theory, group theory, and representation
theory is presented. The text can be used as a brief introduction to these
modern subjects as well as an example of how such topics are synthesized
in modern mathematics. Prerequisites include linear algebra together with
elementaryalgebra,analysis,andcombinatorics.
GiulianaDavidoff
DepartmentofMathematics
MountHolyokeCollege
SouthHadley,MA
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