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∈>∈ 4n∈+∈2. To prove this, we introduce Property\left( {\text{BP}_\text{0}\text{V} } \right), with V being a Banach space: a locally compact group G has Property\left( {\text{BP}_\text{0}\text{V} } \right)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\left( {\text{BP}_\text{0}\text{V} } \right). 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 L2-Betti number is nonzero.