## Table of contents for Kazhdan's property (T) / Bachir Bekka, Pierre de la Harpe and Alain Valette.

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```Introduction                                                   1
Historical Introduction                                  4
PART I: KAZHDAN'S PROPERTY (T)                           25
1    Definitions, first consequences, and basic examples     27
1.1  First definition of Property (T)                   27
1.2  Property (T) in terms of Fell's topology           32
1.3  Compact generation and other consequences          36
1.4  Property (T) for SL,,(K), n > 3                    40
1.5  Property (T) for Sp2,, (K), n > 2                  50
1.6  Property (T) for higher rank algebraic groups      58
1.7  Hereditary properties                              60
1.8  Exercises                                          67
2    Property (FH)                                           73
2.1  Affine isometric actions and Property (FH)          74
2.2  1-cohomology                                        75
2.3  Actions on trees                                    80
2.4  Consequences of Property (FH)                       85
2.5   Hereditary properties                              88
2.6  Actions on real hyperbolic spaces                   93
2.7  Actions on boundaries of rank 1 symmetric spaces   100
2.8  Wreath products                                    104
2.9  Actions on the circle                              107
2.10 Functions conditionally of negative type           119
2.11 A consequence of Schoenberg's Theorem             122
2.12 The Delorme-Guichardet Theorem                    127
2.13 Concordance                                        132
2.14 Exercises                                          133
3    Reduced cohomology                                      136
3.1  Affine isometric actions almost having
fixed points                                      137
3.2  A theorem by Y. Shalom                             140
3.3  Property (T) for Sp(n, 1)                          151
3.4  The question of finite presentability             171
3.5  Other consequences of Shalom's Theorem             175
3.6  Property (T) is not geometric                      179
3.7  Exercises                                          182
4    Bounded generation                                      184
4.1  Bounded generation of SL, (Z) for n > 3            184
4.2  A Kazhdan constant for SLn (Z)                     193
4.3  Property (T) for SLn, (R)                          201
4.4  Exercises                                          213
5    A spectral criterion for Property (T)                  216
5.1  Stationary measures for random walks               217
5.2  Laplace and Markov operators                       218
5.3  Random walks on finite sets                        222
5.4  G-equivariant random walks on quasi-transitive
free sets                                         224
5.5  A local spectral criterion                         236
5.6  Zuk's criterion                                    241
5.7   Groups acting on4A2-buildings                     245
5.8  Exercises                                          250
6    Some applications of Property (T)                      253
6.1  Expander graphs                                    253
6.2   Norm of convolution operators                     262
6.3  Ergodic theory and Property (T)                    264
6.4   Uniqueness of invariant means                     276
6.5  Exercises                                          279
7    A short list of open questions                         282
PART II: BACKGROUND ON UNITARY
REPRESENTATIONS                               287
A    Unitary group representations                            289
A. I  Unitary representations                            289
A.2  Schur's Lemma                                       296
A.3  The Haar measure of a locally compact group         299
A.4  The regular representation of a locally compact group  305
A.5  Representations of compact groups                   306
A.6  Unitary representations associated to group actions  307
A.7  Group actions associated to orthogonal representations  311
A.8  Exercises                                           321
B    Measures on homogeneous spaces                           324
B.1  Invariant measures                                  324
B.2  Lattices in locally compact groups                  332
B.3  Exercises                                           337
C    Functions of positive type and GNS construction          340
C. 1  Kernels of positive type                           340
C.2  Kernels conditionally of negative type              345
C.3  Schoenberg's Theorem                                349
C.4  Functions on groups                                 351
C.5  The cone of functions of positive type              357
C.6  Exercises                                           365
D    Unitary Representations of locally compact
abelian groups                                      369
D. I The Fourier transform                               369
D.2  Bochner's Theorem                                   372
D.3  Unitary representations of locally compact abelian
groups                                              373
D.4  Local fields                                        377
D.5  Exercises                                           380
E    Induced representations                                  383
E.I  Definition of induced representations               383
E.2  Some properties of induced representations          389
E.3  Induced representations with invariant vectors      391
E.4  Exercises                                           393
F    Weak containment and Fell's topology                      395
F. 1  Weak containment of unitary representations         395
F.2  Fell topology on sets of unitary representations     402
E3   Continuity of operations                             407
.4    The C*-algebras of a locally compact group          411
F.5  Direct integrals of unitary representations          413
F.6  Exercises                                            417
G    Amenability                                               420
G. I  Invariant means                                     421
G.2   Examples of amenable groups                         424
G.3  Weak containment and amenability                     427
G4   Kesten's characterisation of amenability             433
G.5  Folner's property                                    440
G.6  Exercises                                            445
Bibliography                                              449

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Library of Congress subject headings for this publication: Topological groups, Mathematics