Valette, Alain

2008, De Cornulier, Yvan, Tessera, Romain, Valette, Alain

Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

1995, Bekka, Bachir, Valette, Alain

Let Gamma be a lattice in a non-compact simple Lie group G. We prove that the canonical map from the full C*-algebra C*(Gamma) to the multiplier algebra M(C*(G)) is not injective in general (it is never injective if G has Kazhdan's property (T), and not injective for many lattices either in SO(n, 1) or SU(n, 1)). For a locally compact group G, Fell introduced a property (WF3), stating that for any closed subgroup H of G, the canonical map from C*(H) to M(C*(G)) is injective. We prove that, for an almost connected G, property (WF3) is equivalent to amenability.

2005, Bourdon, Marc, Martin, Florian, Valette, Alain

We prove two results on the first L-P-cohomology (H) over bar (1)((p))(Gamma) of a finitely generated group Gamma: 1) If N subset of H subset of Gamma is a chain of subgroups, with N non-amenable and normal in Gamma, then (H) over bar (1)((P))(Gamma) = 0 as soon as (H) over bar (1)((P))(H) = 0. This allows for a short proof of a result of W. Luck: if N < Gamma, N is infinite, finitely generated as a group, and Gamma/N contains an element of infinite order, then (H) over bar (1)((2))(Gamma) = 0. 2) If Gamma acts isometrically, properly discontinuously on a proper CAT(- 1) space X, with at least 3 limit points in theta X, then for p larger than the critical exponent e(Gamma) of Gamma in X, one has (H) over bar (1)((p)) not equal A 0. As a consequence we extend a result of Y Shalom: let G be a cocompact lattice in a rank 1 simple Lie group; if G is isomorphic to Gamma, then e(G) < e(Gamma).

1993, De La Harpe, Pierre, Robertson, Guyan, Valette, Alain

Let GAMMA be a finitely generated group. In the group algebra C[T], form the average h of a finite set S of generators of GAMMA. Given a unitary representation pi of GAMMA, we relate spectral properties of the operator pi(h) to ProPerties Of GAMMA and pi. For the universal representation pi(un) of GAMMA, we prove in particular the following results. First, the spectrum Sp(pi(un) (h)) contains the complex number , of modulus one iff Sp(pi(un) (h)) is invariant under multiplication by z, iff there exists a character x: GAMMA --> T such that chi(S) = {z}. Second, for S-1 = S, the group GAMMA has Kazhdan's proPertY (T) if and only if 1 is isolated in Sp(pi(un) (h)); in this case, the distance between 1 and other point, of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.

2004, Valette, Alain