Translation Actions on Non-Unimodular Groups and Strong Ergodicity

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.