Kazhdan's Property

488 Pages·2008·2.06 MB·English

by  Bachir Bekka

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Kazhdan’sProperty(T)
Property(T)isarigiditypropertyfortopologicalgroups,firstformulatedby
D.Kazhdaninthemid-1960swiththeaimofdemonstratingthatalargeclassof
latticesarefinitelygenerated.LaterdevelopmentshaveshownthatProperty(T)plays
animportantroleinanamazinglylargevarietyofsubjects,includingdiscrete
subgroupsofLiegroups,ergodictheory,randomwalks,operatoralgebras,
combinatorics,andtheoreticalcomputerscience.
Thismonographoffersacomprehensiveintroductiontothetheory.Itdescribesthe
twomostimportantpointsofviewonProperty(T):thefirstusesaunitarygroup
representationapproach,andthesecondafixedpointpropertyforaffineisometric
actions.Viathesetheauthorsdiscussarangeofimportantexamplesandapplications
toseveraldomainsofmathematics.Adetailedappendixprovidesasystematic
expositionofpartsofthetheoryofgrouprepresentationsthatareusedtoformulate
anddevelopProperty(T).
Bachir BekkaisProfessorofMathematicsattheUniversitédeRennes1,France.
Pierre de la HarpeisProfessorofMathematicsattheUniversitédeGenève,
Switzerland.
Alain ValetteisProfessorofMathematicsattheUniversitédeNeuchâtel,
Switzerland.
NEWMATHEMATICALMONOGRAPHS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.
Foracompleteserieslistingvisithttp://www.cambridge.org/uk/series/sSeries.asp?code=NMM
1 M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2 J.B.GarnettandD.E.MarshallHarmonicMeasure
3 P.M.CohnFreeIdealRingsandLocalizationinGeneralRings
4 E.BombieriandW.GublerHeightsinDiophantineGeometry
5 Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6 S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7 A.ShlapentokhHilbert’sTenthProblem
8 G.O.MichlerTheoryofFiniteSimpleGroups
9 A.BakerandG.WüstholzLogarithmicFormsandDiophantineGeometry
10 P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11 B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
Kazhdan’s Property (T)
BACHIR BEKKA, PIERRE DE LA HARPE AND
ALAIN VALETTE
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521887205
© B. Bekka, P. de la Harpe and A. Valette 2008
This publication 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 2008
ISBN-13 978-0-511-39377-8 eBook (EBL)
ISBN-13 978-0-521-88720-5 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Listoffigures pageix
Listofsymbols x
Introduction 1
HistoricalIntroduction 4
PART I: KAZHDAN’S PROPERTY (T) 25
1 Definitions,firstconsequences,andbasicexamples 27
1.1 FirstdefinitionofProperty(T) 27
1.2 Property(T)intermsofFell’stopology 32
1.3 Compactgenerationandotherconsequences 36
1.4 Property(T)forSL (K),n≥3 40
n
1.5 Property(T)forSp (K),n≥2 50
2n
1.6 Property(T)forhigherrankalgebraicgroups 58
1.7 Hereditaryproperties 60
1.8 Exercises 67
2 Property(FH) 73
2.1 AffineisometricactionsandProperty(FH) 74
2.2 1-cohomology 75
2.3 Actionsontrees 80
2.4 ConsequencesofProperty(FH) 85
2.5 Hereditaryproperties 88
2.6 Actionsonrealhyperbolicspaces 93
2.7 Actionsonboundariesofrank1symmetricspaces 100
2.8 Wreathproducts 104
2.9 Actionsonthecircle 107
2.10 Functionsconditionallyofnegativetype 119
v
vi Contents
2.11 AconsequenceofSchoenberg’sTheorem 122
2.12 TheDelorme–GuichardetTheorem 127
2.13 Concordance 132
2.14 Exercises 133
3 Reducedcohomology 136
3.1 Affineisometricactionsalmosthaving
fixedpoints 137
3.2 AtheorembyY.Shalom 140
3.3 Property(T)forSp(n,1) 151
3.4 Thequestionoffinitepresentability 171
3.5 OtherconsequencesofShalom’sTheorem 175
3.6 Property(T)isnotgeometric 179
3.7 Exercises 182
4 Boundedgeneration 184
4.1 BoundedgenerationofSL (Z)forn ≥ 3 184
n
4.2 AKazhdanconstantforSL (Z) 193
n
4.3 Property(T)forSL (R) 201
n
4.4 Exercises 213
5 AspectralcriterionforProperty(T) 216
5.1 Stationarymeasuresforrandomwalks 217
5.2 LaplaceandMarkovoperators 218
5.3 Randomwalksonfinitesets 222
5.4 G-equivariantrandomwalksonquasi-transitive
freesets 224
5.5 Alocalspectralcriterion 236
5.6 Zuk’scriterion 241
(cid:1)
5.7 GroupsactingonA -buildings 245
2
5.8 Exercises 250
6 SomeapplicationsofProperty(T) 253
6.1 Expandergraphs 253
6.2 Normofconvolutionoperators 262
6.3 ErgodictheoryandProperty(T) 264
6.4 Uniquenessofinvariantmeans 276
6.5 Exercises 279
7 Ashortlistofopenquestions 282
Contents vii
PART II: BACKGROUND ON UNITARY
REPRESENTATIONS 287
A Unitarygrouprepresentations 289
A.1 Unitaryrepresentations 289
A.2 Schur’sLemma 296
A.3 TheHaarmeasureofalocallycompactgroup 299
A.4 Theregularrepresentationofalocallycompactgroup 305
A.5 Representationsofcompactgroups 306
A.6 Unitaryrepresentationsassociatedtogroupactions 307
A.7 Groupactionsassociatedtoorthogonalrepresentations 311
A.8 Exercises 321
B Measuresonhomogeneousspaces 324
B.1 Invariantmeasures 324
B.2 Latticesinlocallycompactgroups 332
B.3 Exercises 337
C FunctionsofpositivetypeandGNSconstruction 340
C.1 Kernelsofpositivetype 340
C.2 Kernelsconditionallyofnegativetype 345
C.3 Schoenberg’sTheorem 349
C.4 Functionsongroups 351
C.5 Theconeoffunctionsofpositivetype 357
C.6 Exercises 365
D UnitaryRepresentationsoflocallycompact
abeliangroups 369
D.1 TheFouriertransform 369
D.2 Bochner’sTheorem 372
D.3 Unitaryrepresentationsoflocallycompactabelian
groups 373
D.4 Localfields 377
D.5 Exercises 380
E Inducedrepresentations 383
E.1 Definitionofinducedrepresentations 383
E.2 Somepropertiesofinducedrepresentations 389
E.3 Inducedrepresentationswithinvariantvectors 391
E.4 Exercises 393
viii Contents
F WeakcontainmentandFell’stopology 395
F.1 Weakcontainmentofunitaryrepresentations 395
F.2 Felltopologyonsetsofunitaryrepresentations 402
F.3 Continuityofoperations 407
∗
F.4 TheC -algebrasofalocallycompactgroup 411
F.5 Directintegralsofunitaryrepresentations 413
F.6 Exercises 417
G Amenability 420
G.1 Invariantmeans 421
G.2 Examplesofamenablegroups 424
G.3 Weakcontainmentandamenability 427
G.4 Kesten’scharacterisationofamenability 433
G.5 Følner’sproperty 440
G.6 Exercises 445
Bibliography 449
Index 468