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

TL;DR

This paper extends spectral methods for property (T) to reflexive Banach spaces, providing conditions involving Poincaré constants that ensure fixed points for group actions and applications to cohomology vanishing and hyperbolic group boundaries.

Contribution

It introduces a new framework connecting Poincaré inequalities with property (T) in reflexive Banach spaces, including explicit estimates and applications to hyperbolic groups.

Findings

Groups satisfying the new conditions have vanishing first cohomology for all isometric representations on Lp spaces.

Provides explicit bounds on p for which H^1(G,π)=0.

Establishes lower bounds on the conformal dimension of hyperbolic group boundaries.

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 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.