Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations

TL;DR

This paper investigates the conditions under which the first cohomology vanishes for uniformly bounded representations on Hilbert spaces, using spectral criteria related to L^p-cohomology of Banach spaces.

Contribution

It introduces a spectral condition that ensures the vanishing of 1-cohomology for certain bounded representations, linking geometric and algebraic properties.

Findings

Spectral condition guarantees vanishing of 1-cohomology

Connects L^p-cohomology with representation theory

Provides criteria for rigidity in Banach space representations

Abstract

We consider L^p-cohomology of reflexive Banach spaces and give a spectral condition implying the vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space.