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 geometric 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. In this talk, I investigate which finitely generated groups G have the property that all model geometries of G are (essentially) locally finite graphs. I 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. This characterizes 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 generalized Baumslag-Solitar groups.