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Garland's method with Banach coefficients

Izhar Oppenheim

Vol. 16 (2023), No. 3, 861–890
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 Lp 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.

Keywords
group cohomology, random groups, stability, Garland's method, vanishing of cohomology, Banach property (T)
Mathematical Subject Classification
Primary: 20J06
Secondary: 20F67, 20P05, 46B20, 46M20
Milestones
Received: 13 May 2021
Revised: 29 July 2021
Accepted: 8 September 2021
Published: 25 May 2023
Authors
Izhar Oppenheim
Department of Mathematics
Ben-Gurion University of the Negev
Be’er Sheva
Israel

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