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 properly isometrically on Lp(G) if p > 4n+2. To prove this, we introduce property (BP0 ), for V be a Banach space: a locally compact group G has property (BP0 ) if every affine isometric action of G on V , such that the linear part is a C0-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 (BP0 ). 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 L2(G) is non-zero; and we characterize uniform lattices in those groups for which the first L2-Betti number is non-zero. Mathematics Subject Classification: Primary 22D10; Secondary 20J06, 22D10, 22E40, 43A15.