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. Lécureux et al. 2020] for results regarding vanishing of the first cohomology, and see [Oppenheim 2017;Lubotzky and Oppenheim 2020] for results regarding vanishing of higher cohomologies. However, much less is known when one considers the less-structured setting of a group acting on a simplicial complex without assuming the extra structure of an affine building.Recently, the results for vanishing of the first cohomology of random groups with coefficients in Banach spaces were improved: First, Drut , u and Mackay [2019] proved vanishing of the first cohomologyThe author was partially supported by ISF grant no. 293/18 . MSC2020: primary 20J06; secondary 20F67, 20P05, 46B20, 46M20.