Kazhdan’s property

Bachir Bekka, Pierre de la de la Harpe, Alain Valette0521887208, 9780521887205, 9780511395116

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).

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Table of contents :
Cover……Page 1
Half-title……Page 0
Series-title……Page 2
Title……Page 3
Copyright……Page 4
Contents……Page 5
List of figures……Page 9
List of symbols……Page 10
Introduction……Page 15
First appearance of Property (T)……Page 18
Property (T) for the groups Sp(n, 1) and F4(-20)……Page 21
Construction of expanding graphs and Property (T) for pairs……Page 22
Normal subgroups in lattices……Page 24
The Ruziewicz problem……Page 25
Fundamental groups of II 1 factors……Page 26
Property (T) in ergodic theory……Page 27
Further examples of groups with Property (T)……Page 29
Finite presentations with Property (T), examples beyond locally compact groups, and other new examples……Page 31
Kazhdan constants……Page 32
Product replacement algorithm……Page 34
Action of Kazhdan groups on manifolds of dimensions ≤ 2……Page 35
Variations……Page 36
Part I Kazhdan’s Property (T)……Page 39
1.1 First definition of Property (T)……Page 41
1.2 Property (T) in terms of Fell’s topology……Page 46
1.3 Compact generation and other consequences……Page 50
Some general facts……Page 54
Some facts about unitary representations of SL2(K)……Page 58
1.5 Property (T) for Sp2………Page 64
1.6 Property (T) for higher rank algebraic groups……Page 72
Property (T) is inherited by lattices……Page 74
Behaviour under short exact sequences……Page 77
Covering groups……Page 78
1.8 Exercises……Page 81
2 Property (FH)……Page 87
2.1 Affine isometric actions and Property (FH)……Page 88
2.2 1-cohomology……Page 89
Constructing affine isometric actions……Page 94
Actions on trees and Property (FA)……Page 95
2.4 Consequences of Property (FH)……Page 99
2.5 Hereditary properties……Page 102
2.6 Actions on real hyperbolic spaces……Page 107
2.7 Actions on boundaries of rank 1 symmetric spaces……Page 114
2.8 Wreath products……Page 118
2.9 Actions on the circle……Page 121
1-cocycles associated to actions on the circle……Page 122
A cohomological criterion for the existence of invariant measures……Page 124
Geodesic currents……Page 125
Groups acting freely on S1……Page 130
Proof of Theorem 2.9.1……Page 131
2.10 Functions conditionally of negative type……Page 133
2.11 A consequence of Schoenberg’s Theorem……Page 136
1-cohomology and weak containment……Page 141
The Delorme–Guichardet Theorem……Page 143
Another characterisation of Property (T)……Page 145
2.13 Concordance……Page 146
2.14 Exercises……Page 147
3 Reduced cohomology……Page 150
3.1 Affine isometric actions almost having fixed points……Page 151
3.2 A theorem by Y. Shalom……Page 154
Gelfand pairs……Page 165
A mean value property……Page 169
Harmonicity……Page 172
The case of a non-compact semisimple Lie group……Page 175
Growth of harmonic mappings on rank 1 spaces……Page 177
3.4 The question of finite presentability……Page 185
3.5 Other consequences of Shalom’s Theorem……Page 189
3.6 Property (T) is not geometric……Page 193
3.7 Exercises……Page 196
4.1 Bounded generation of………Page 198
4.2 A Kazhdan constant for………Page 207
Property (T) for………Page 212
Property (T) for………Page 215
Property (T) for SLn(R)……Page 221
Property (T) for the loop group of SLn(C)……Page 224
4.4 Exercises……Page 227
5 A spectral criterion for Property (T)……Page 230
5.1 Stationary measures for random walks……Page 231
5.2 Laplace and Markov operators……Page 232
5.3 Random walks on finite sets……Page 236
5.4 G-equivariant random walks on quasi-transitive free sets……Page 238
5.5 A local spectral criterion……Page 250
5.6 Zuk’s criterion……Page 255
5.7 Groups acting on A0365A2-buildings……Page 259
5.8 Exercises……Page 264
Expander graphs……Page 267
Examples of expander graphs……Page 273
6.2 Norm of convolution operators……Page 276
6.3 Ergodic theory and Property (T)……Page 278
Orbit equivalence and measure equivalence……Page 284
6.4 Uniqueness of invariant means……Page 290
6.5 Exercises……Page 293
Open examples of groups……Page 296
Properties of Kazhdan groups……Page 297
Kazhdan subsets of amenable groups……Page 298
Fundamental groups of manifolds……Page 299
Part II Background on Unitary Representations……Page 301
A.1 Unitary representations……Page 303
A.2 Schur’s Lemma……Page 310
A.3 The Haar measure of a locally compact group……Page 313
A.4 The regular representation of a locally compact group……Page 319
A.5 Representations of compact groups……Page 320
A.6 Unitary representations associated to group actions……Page 321
A.7 Group actions associated to orthogonal representations……Page 325
A.8 Exercises……Page 335
B.1 Invariant measures……Page 338
B.2 Lattices in locally compact groups……Page 346
B.3 Exercises……Page 351
C.1 Kernels of positive type……Page 354
C.2 Kernels conditionally of negative type……Page 359
C.3 Schoenberg’s Theorem……Page 363
C.4 Functions on groups……Page 365
C.5 The cone of functions of positive type……Page 371
C.6 Exercises……Page 379
D.1 The Fourier transform……Page 383
D.2 Bochner’s Theorem……Page 386
D.3 Unitary representations of locally compact abelian groups……Page 387
D.4 Local fields……Page 391
D.5 Exercises……Page 394
E.1 Definition of induced representations……Page 397
E.2 Some properties of induced representations……Page 403
E.3 Induced representations with invariant vectors……Page 405
E.4 Exercises……Page 407
F.1 Weak containment of unitary representations……Page 409
F.2 Fell topology on sets of unitary representations……Page 416
F.3 Continuity of operations……Page 421
F.4 The C*-algebras of a locally compact group……Page 425
F.5 Direct integrals of unitary representations……Page 427
F.6 Exercises……Page 431
Appendix G Amenability……Page 434
G.1 Invariant means……Page 435
G.2 Examples of amenable groups……Page 438
G.3 Weak containment and amenability……Page 441
G.4 Kesten’s characterisation of amenability……Page 447
G.5 Følner’s property……Page 454
G.6 Exercises……Page 459
Bibliography……Page 463
Index……Page 482