Garland's method with Banach coefficients

TL;DR

This paper extends Garland's method to Banach spaces, providing new criteria for cohomology vanishing, fixed point properties, and stability in various Banach space contexts, impacting geometric group theory and functional analysis.

Contribution

It introduces a Banach space version of Garland's method applicable to all reflexive Banach spaces, enabling new cohomology vanishing criteria and applications to group fixed points and stability.

Findings

Established vanishing of cohomology for groups acting on complexes with Banach coefficients.

Derived lower bounds for conformal dimensions of group boundaries.

Provided criteria for group stability in p-Schatten norms.

Abstract

We prove a Banach version of Garland's method of proving vanishing of cohomology for groups acting on simplicial complexes. The novelty of this new version is that our new condition applies to every reflexive Banach space. This new version of Garland's method allows us to deduce several criteria for vanishing of group cohomology with coefficients in several classes of Banach spaces (uniformly curved spaces, Hilbertian spaces and $L^p$ spaces). Using these new criteria, we improve recent results regarding Banach fixed point theorems for random groups in the triangular model and give a sharp lower bound for the conformal dimension of the boundary of such groups. Also, we derive new criteria for group stability with respect to p-Schatten norms.