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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

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