Garland's method with Banach coefficients

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.

arXiv comment34 pages; corrected version - only reflexive Banach spaces; sharp bound on conformal dimension for random groups

References in corpus (8)

• Almost commuting permutations are near commuting permutations
• Stability and Invariant Random Subgroups
• Strong Banach Property (T) for Simple Algebraic Groups of Higher Rank
• Towards Strong Banach property (T) for SL(3,R)
• Random groups, random graphs and eigenvalues of p-Laplacians
• Banach space actions and $L^2$-spectral gap
• Fixed point theorem for reflexive Banach spaces and uniformly convex non positively curved metric spaces
• Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations

Cited by in corpus (1)

• Ultrametric analogues of Ulam stability of groups