%0 Journal Article
%T Isometric group actions on Banach spaces and representations vanishing at infinity
%A Yves de Cornulier
%A Romain Tessera
%A Alain Valette
%J Mathematics
%D 2006
%I arXiv
%R 10.1007/s00031-008-9006-0
%X Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property $({\BP}_0^V)$, for $V$ be a Banach space: a locally compact group $G$ has property $({\BP}_0^V)$ 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 $({\BP}_0^V)$. 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 non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.
%U http://arxiv.org/abs/math/0612398v1