Title: Model geometries of finitely generated groups
Abstract: The central theme of geometric group theory is to study groups via their actions on metric spaces. A model geometry of a finitely generated group is a proper geodesic metric space admitting a proper, cocompact and isometric group action. Every finitely generated group has a model geometry that is a locally finite graph, namely its Cayley graph with respect to a finite generating set. I will introduce the notion of domination of metric spaces and give necessary and sufficient conditions for all model geometries of a finitely generated group to be dominated by a locally finite graph.  Equivalently, this characterises the class of finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). Among groups of cohomological dimension two, the only such groups are surface groups and generalised Baumslag-Solitar groups.