This is an announcement for the paper "Poincar'e inequalities and rigidity for actions on Banach spaces" by Piotr W. Nowak.

Abstract: The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present conditions implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,\pi)=0$ for every isometric representation $\pi$ of $G$ on $X$. We give examples of groups for which every affine isometric action on an $L_p$ space has a fixed point for certain $p>2$, and present several applications. In particular, we give a lower bound on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.

Archive classification: math.GR math.FA math.OA

Submitted from: pnowak@math.tamu.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1107.1896

or

http://arXiv.org/abs/1107.1896