Alain Valette

Introduction to the Baum-Connes Conjecture

1 Idempotents in Group Algebras.- 2 The Baum-Connes Conjecture.- 3K-theory for (Group) C*-algebras.- 4 Classifying Spaces andK-homology.- 5 EquivariantKK-theory.- 6 The Analytical Assembly Map.- 7 Some Examples of the Assembly Map.- 8 Property (RD).- 9 The Dirac-dual Dirac Method.- 10 LafforguesKKBan Theory.- G. Mislin: On the Classifying Space for Proper Actions.- A.1 The topologists model.- A.2 The analysts model.- A.4 Spectra.

Proper group actions and the Baum-Connes conjecture

Equivariant K-Homology of the Classifying Space for Proper Actions.- On the Baum-Connes Assembly Map for Discrete Groups.

On the spectrum of the sum of generators of a finitely generated group, II

AbstractLet Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π.For the universal representationπun of Γ, we prove in particular the following results. First, the spectrum Sp(πun(h)) contains the complex numberz of modulus one iff Sp(πun(h)) is invariant under multiplication byz, iff there exists a character

Group Cohomology, Harmonic Functions and the First L2 -Betti Number

\chi :\Gamma \to \mathbb{T}

K-theoretic amenability for SL2(Qp), and the action on the associated tree

such that η(S)={z}. Second, forS−1=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(πun(h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.

Proper actions of wreath products and generalizations

Abstract.We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with “controlled” Følner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity

AbstractFor an infinite, finitely generated group Γ, we study the first cohomology group H1(Γ,λΓ) with coefficients in the left regular representation λΓ of Γ on ℓ2(Γ). We first prove thatH Γ(Γ, C Γ) embeds into HΓ(Γ,λΓ); as a consequence, ifH Γ(Γ,λΓ)=0, then Γ is not amenable with one end. For a Cayley graph X of Γ, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Γ is not amenable, then there is a natural isomorphism betweenH Γ(Γ,λΓ) and