Poincaré inequalities and rigidity for actions on Banach spaces

2011/07/10 by Nowak, Piotr W.
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1107.1896

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 H¹(G,π)=0 for every isometric representation π of G on X. The condition is expressed in terms of p-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which H¹(G,π) vanishes for every isometric representation π 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.

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• Rigidity of commuting affine actions on reflexive Banach spaces
• Fixed point-free isometric actions of topological groups on Banach spaces
• Uniform embeddings of bounded geometry spaces into reflexive Banach space
• Group Actions on Banach Spaces
• On fixed point property for L_p-representations of Kazhdan groups