Model geometries of finitely generated groups

TL;DR

This paper characterizes the model geometries of finitely generated groups, identifying conditions under which they are dominated by symmetric spaces or graphs, and classifying groups based on their embedding properties in locally compact groups.

Contribution

It provides a comprehensive classification of finitely generated groups' model geometries, linking algebraic properties to geometric and topological structures.

Findings

Groups without non-trivial finite rank free abelian subgroups are dominated by symmetric spaces or graphs.

Groups with certain commensurated subgroups can embed as lattices in non-compact locally compact groups.

Surface groups and Baumslag-Solitar groups are the only cohomological two groups with specific lattice embeddings.

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.