Poincaré inequalities and rigidity for actions on Banach spaces

  • Piotr W. Nowak

    Polskiej Akademii Nauk, Warszawa, Poland

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 GG on a reflexive Banach space XX has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that H1(G,π)=0H^1(G,\pi)=0 for every isometric representation π\pi of GG on XX. The condition is expressed in terms of pp-Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which H1(G,π)H^1(G,\pi) vanishes for every isometric representation π\pi on an LpL_p space for some p>2p>2. Our methods allow to estimate such a pp 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.

Cite this article

Piotr W. Nowak, Poincaré inequalities and rigidity for actions on Banach spaces. J. Eur. Math. Soc. 17 (2015), no. 3, pp. 689–709

DOI 10.4171/JEMS/514