Model geometries dominated by locally finite graphs

We study model geometries of finitely generated groups. We show 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 also prove the only such groups of cohomological two are surface groups and generalised Baumslag–Solitar groups.