Group actions on Banach spaces and a geometric characterization of a-T-menability

Abstract

We prove a geometric characterization of a-T-menability through proper, affine, isometric actions on the Banach spaces $L_p[0,1]$ for $1<p<2$. This answers a question of A. Valette.

This version differs significantly from both the published version and the previous arXiv versions. The reason is a long overdue correction of implication 1=>3 in the proof of the main result. In the current version the proof is fixed but with a very different argument. I owe thanks to Cornelia Drutu for helpful correspondence which led to this update