Poincaré inequalities and rigidity for actions on Banach spaces

  • Piotr W. Nowak

    Polskiej Akademii Nauk, Warszawa, Poland


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 on a reflexive Banach space has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that for every isometric representation of on . The condition is expressed in terms of -Poincar\'{e} constants and we provide examples of groups, which satisfy such conditions and for which vanishes for every isometric representation on an space for some . Our methods allow to estimate such a 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