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 $\mathrm{\Gamma }\curvearrowright G$ given by a countable dense subgroup $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }\curvearrowright G$ for an almost unimodular lcsc group $G$ is equivalent to the strong ergodicity of $\mathrm{\Gamma }\cap \ker ({\mathrm{\Delta }}_G)\curvearrowright \ker ({\mathrm{\Delta }}_G)$, where ${\mathrm{\Delta }}_G$ 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.