Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

Abstract

We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of $Γ$ on homogeneous spaces of the form $G/Σ$, where $Σ<G$ is either a discrete or an algebraic subgroup. We also prove that if $G$ is simply connected and the action $Γ\curvearrowright G$ has property (T), then any cocycle $w:Γ\times G\rightarrowΛ$ with values into a countable group $Λ$ is cohomologous to a homomorphism $δ:Γ\rightarrowΛ$. As a consequence, we deduce that the action $Γ\curvearrowright G$ is orbit equivalent superrigid: any free nonsingular action $Λ\curvearrowright Y$ which is orbit equivalent to $Γ\curvearrowright G$, is necessarily conjugate to an induction of $Γ\curvearrowright G$.