  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.
