  Let $G$ be a locally compact amenable group. We say that G has property (M)
if every closed subgroup of finite covolume in G is cocompact. A classical
theorem of Mostow ensures that connected solvable Lie groups have property (M).
We prove a non-Archimedean extension of Mostow's theorem by showing the
amenable linear locally compact groups have property (M). However property (M)
does not hold for all solvable locally compact groups: indeed, we exhibit an
example of a metabelian locally compact group with a non-uniform lattice. We
show that compactly generated metabelian groups, and more generally
nilpotent-by-nilpotent groups, do have property (M). Finally, we highlight a
connection of property (M) with the subtle relation between the analytic
notions of strong ergodicity and the spectral gap.
