Lattices in amenable groups

TL;DR

This paper investigates the property (M) in amenable groups, extending classical results to non-Archimedean cases, and explores which classes of groups have this property, revealing nuanced differences and connections to ergodic theory.

Contribution

It extends Mostow's theorem to non-Archimedean amenable linear groups and characterizes classes of groups with property (M), including nilpotent-by-nilpotent groups.

Findings

Amenable linear groups have property (M).

Not all solvable locally compact groups have property (M).

Property (M) relates to strong ergodicity and spectral gap.

Abstract

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.