Poincar\'e inequalities and rigidity for actions on Banach spaces

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 a condition 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 $X$. The condition is expressed in terms of $p$-Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,\pi)$ vanishes for every isometric representation $\pi$ on an $L_p$ space for some $p>2$. Our methods allow to estimate such a $p$ explicitly and yield several interesting applications. In particular, we obtain quantitative estimates for vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. We also give lower bounds on the conformal dimension of the boundary of a hyperbolic group in the Gromov density model.