Property (T) is a rigidity property for topological groups, first formulated by D. Kazhdan in the mid 1960's with the aim of demonstrating that a large class of lattices are finitely generated. Later developments have shown that Property (T) plays an important role in an amazingly large variety of subjects, including discrete subgroups of Lie groups, ergodic theory, random walks, operator algebras, combinatorics, and theoretical computer science. This monograph offers a comprehensive introduction to the theory. It describes the two most important points of view on Property (T): the first uses a unitary group representation approach, and the second a fixed point property for affine isometric actions. Via these the authors discuss a range of important examples and applications to several domains of mathematics. A detailed appendix provides a systematic exposition of parts of the theory of group representations that are used to formulate and develop Property (T).
Author(s): Bachir Bekka, Pierre de la Harpe, Alain Valette
Year: 2008
Language: English
Pages: 486
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
List of figures......Page 11
List of symbols......Page 12
Introduction......Page 17
First appearance of Property (T)......Page 20
Property (T) for the groups Sp(n, 1) and F4(-20)......Page 23
Construction of expanding graphs and Property (T) for pairs......Page 24
Normal subgroups in lattices......Page 26
The Ruziewicz problem......Page 27
Fundamental groups of II 1 factors......Page 28
Property (T) in ergodic theory......Page 29
Further examples of groups with Property (T)......Page 31
Finite presentations with Property (T), examples beyond locally compact groups, and other new examples......Page 33
Kazhdan constants......Page 34
Product replacement algorithm......Page 36
Action of Kazhdan groups on manifolds of dimensions ≤ 2......Page 37
Variations......Page 38
Part I Kazhdan's Property (T)......Page 41
1.1 First definition of Property (T)......Page 43
1.2 Property (T) in terms of Fell's topology......Page 48
1.3 Compact generation and other consequences......Page 52
Some general facts......Page 56
Some facts about unitary representations of SL2(K)......Page 60
1.5 Property (T) for Sp2…......Page 66
1.6 Property (T) for higher rank algebraic groups......Page 74
Property (T) is inherited by lattices......Page 76
Behaviour under short exact sequences......Page 79
Covering groups......Page 80
1.8 Exercises......Page 83
2 Property (FH)......Page 89
2.1 Affine isometric actions and Property (FH)......Page 90
2.2 1-cohomology......Page 91
Constructing affine isometric actions......Page 96
Actions on trees and Property (FA)......Page 97
2.4 Consequences of Property (FH)......Page 101
2.5 Hereditary properties......Page 104
2.6 Actions on real hyperbolic spaces......Page 109
2.7 Actions on boundaries of rank 1 symmetric spaces......Page 116
2.8 Wreath products......Page 120
2.9 Actions on the circle......Page 123
1-cocycles associated to actions on the circle......Page 124
A cohomological criterion for the existence of invariant measures......Page 126
Geodesic currents......Page 127
Groups acting freely on S1......Page 132
Proof of Theorem 2.9.1......Page 133
2.10 Functions conditionally of negative type......Page 135
2.11 A consequence of Schoenberg's Theorem......Page 138
1-cohomology and weak containment......Page 143
The Delorme–Guichardet Theorem......Page 145
Another characterisation of Property (T)......Page 147
2.13 Concordance......Page 148
2.14 Exercises......Page 149
3 Reduced cohomology......Page 152
3.1 Affine isometric actions almost having fixed points......Page 153
3.2 A theorem by Y. Shalom......Page 156
Gelfand pairs......Page 167
A mean value property......Page 171
Harmonicity......Page 174
The case of a non-compact semisimple Lie group......Page 177
Growth of harmonic mappings on rank 1 spaces......Page 179
3.4 The question of finite presentability......Page 187
3.5 Other consequences of Shalom's Theorem......Page 191
3.6 Property (T) is not geometric......Page 195
3.7 Exercises......Page 198
4.1 Bounded generation of…......Page 200
4.2 A Kazhdan constant for…......Page 209
Property (T) for…......Page 214
Property (T) for…......Page 217
Property (T) for SLn(R)......Page 223
Property (T) for the loop group of SLn(C)......Page 226
4.4 Exercises......Page 229
5 A spectral criterion for Property (T)......Page 232
5.1 Stationary measures for random walks......Page 233
5.2 Laplace and Markov operators......Page 234
5.3 Random walks on finite sets......Page 238
5.4 G-equivariant random walks on quasi-transitive free sets......Page 240
5.5 A local spectral criterion......Page 252
5.6 Zuk's criterion......Page 257
5.7 Groups acting on A0365A2-buildings......Page 261
5.8 Exercises......Page 266
Expander graphs......Page 269
Examples of expander graphs......Page 275
6.2 Norm of convolution operators......Page 278
6.3 Ergodic theory and Property (T)......Page 280
Orbit equivalence and measure equivalence......Page 286
6.4 Uniqueness of invariant means......Page 292
6.5 Exercises......Page 295
Open examples of groups......Page 298
Properties of Kazhdan groups......Page 299
Kazhdan subsets of amenable groups......Page 300
Fundamental groups of manifolds......Page 301
Part II Background on Unitary Representations......Page 303
A.1 Unitary representations......Page 305
A.2 Schur's Lemma......Page 312
A.3 The Haar measure of a locally compact group......Page 315
A.4 The regular representation of a locally compact group......Page 321
A.5 Representations of compact groups......Page 322
A.6 Unitary representations associated to group actions......Page 323
A.7 Group actions associated to orthogonal representations......Page 327
A.8 Exercises......Page 337
B.1 Invariant measures......Page 340
B.2 Lattices in locally compact groups......Page 348
B.3 Exercises......Page 353
C.1 Kernels of positive type......Page 356
C.2 Kernels conditionally of negative type......Page 361
C.3 Schoenberg's Theorem......Page 365
C.4 Functions on groups......Page 367
C.5 The cone of functions of positive type......Page 373
C.6 Exercises......Page 381
D.1 The Fourier transform......Page 385
D.2 Bochner's Theorem......Page 388
D.3 Unitary representations of locally compact abelian groups......Page 389
D.4 Local fields......Page 393
D.5 Exercises......Page 396
E.1 Definition of induced representations......Page 399
E.2 Some properties of induced representations......Page 405
E.3 Induced representations with invariant vectors......Page 407
E.4 Exercises......Page 409
F.1 Weak containment of unitary representations......Page 411
F.2 Fell topology on sets of unitary representations......Page 418
F.3 Continuity of operations......Page 423
F.4 The C*-algebras of a locally compact group......Page 427
F.5 Direct integrals of unitary representations......Page 429
F.6 Exercises......Page 433
Appendix G Amenability......Page 436
G.1 Invariant means......Page 437
G.2 Examples of amenable groups......Page 440
G.3 Weak containment and amenability......Page 443
G.4 Kesten's characterisation of amenability......Page 449
G.5 Følner's property......Page 456
G.6 Exercises......Page 461
Bibliography......Page 465
Index......Page 484
