Isometric group actions on Banach spaces and representations vanishing at infinity

Details Isometric group actions on Banach spaces and representations vanishing at infinity Isometric group actions on Banach spaces and representations vanishing at infinity

2006-12-14

arxiv.org/abs/math/0612398v1

Transform. Groups 13 (2008), no. 1, 125--147

Mathematics

Representation Theory

28 pages

Scientific paper

10.1007/s00031-008-9006-0

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.

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