Vanishing of cohomology with coefficients in representations on Banach spaces of groups acting on Buildings

TL;DR

This paper proves that for groups acting on certain simplicial complexes, the cohomology vanishes when using Banach space representations, based on spectral properties of the complex's links.

Contribution

It introduces conditions under which cohomology vanishes for Banach space representations of groups acting on buildings, linking spectral properties to cohomological vanishing.

Findings

Cohomology vanishes for a large class of Banach space representations.

Spectral properties of links determine cohomological behavior.

Applicable to groups acting on buildings and similar complexes.

Abstract

We prove vanishing of cohomology with coefficients in representations on a large class of Banach spaces for a group acting "nicely" on a simplicial complexes based on spectral properties of the 1-dimensional links of the simplicial complex.