Translation Actions on Non-Unimodular Groups and Strong Ergodicity

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 $Γ\curvearrowright G$ given by a countable dense subgroup $Γ$ 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 $Γ\curvearrowright G$ for an almost unimodular lcsc group $G$ is equivalent to the strong ergodicity of $Γ\cap \ker(Δ_G)\curvearrowright \ker(Δ_G)$, where $Δ_G$ 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.

Theorems & Definitions (19)

• Theorem A
• Definition 1.1
• Theorem 1.2
• proof
• Theorem 1.3
• proof
• Proposition 2.1
• proof
• Corollary 2.2
• proof