Isometric group actions on banach spaces and representations vanishing at infinity

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.