Lattices in amenable groups
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 that 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.