Valette, Alain

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

Our main result is that the simple Lie group G = Sp( n; 1) acts metrically properly isometrically on L-p( G) if p > 4 n + 2. To prove this, we introduce Property (BP0V), with V being a Banach space: a locally compact group G has Property (BP0V) if every affine isometric action of G on V, such that the linear part is a C-0- representation of G, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have Property ( BP V 0). As a consequence, for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on L-2(G) is nonzero; and we characterize uniform lattices in those groups for which the first L-2- Betti number is nonzero.

2004, 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.

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).