Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

Abstract

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.