Model geometries of finitely generated groups

Abstract

We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of non-compact type, an infinite locally finite vertex-transitive graph, or a product of such spaces. We also prove that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We show the only such groups of cohomological two are surface groups and generalised Baumslag-Solitar groups, and we obtain an analogous characterisation for groups of cohomological dimension three.