Translation Actions on Non-Unimodular Groups and Strong Ergodicity

TL;DR

This paper studies translation actions of dense subgroups on non-unimodular groups, revealing conditions for strong ergodicity and demonstrating the lack of rigidity among such actions.

Contribution

It characterizes when translation actions are strongly ergodic in non-unimodular groups and shows non-isomorphic groups can have orbit equivalent actions.

Findings

Strong ergodicity occurs only if the group is almost unimodular.

Strong ergodicity is equivalent to ergodicity on the kernel of the modular function.

Non-isomorphic groups can admit orbit equivalent translation actions.

Abstract

We investigate translation actions of countable dense subgroups of non-unimodular locally compact second countable (lcsc) groups. Using left-right actions, we show that the left translation action $\Gamma \curvearrowright G$ given by a countable dense subgroup $\Gamma$ of a locally compact second countable group $G$ can only be strongly ergodic if $G$ is almost unimodular. We show that the strong ergodicity of the action $\Gamma \curvearrowright G$ for an almost unimodular lcsc group $G$ is equivalent to the strong ergodicity of $\Gamma \cap \ker(\Delta_G)\curvearrowright \ker(\Delta_G)$, where $\Delta_G$ is the modular function. We demonstrate the absence of rigidity, by showing that non-isomorphic lcsc almost unimodular groups can admit orbit equivalent translation actions.