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

Abstract

Abstract We study equivalence relations that arise from translation actions Γ ↷ G \Gamma\curvearrowright G which are associated to dense embeddings Γ < G \Gamma<G of countable groups into second countable locally compact groups. Assuming that G is simply connected and the action Γ ↷ G \Gamma\curvearrowright G is strongly ergodic, we prove that Γ ↷ G \Gamma\curvearrowright G is orbit equivalent to another such translation action Λ ↷ H \Lambda\curvearrowright H if and only if there exists an isomorphism δ : G → H \delta:G\to H such that δ ⁢ ( Γ ) = Λ \delta(\Gamma)=\Lambda . 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 G/\Sigma , where Σ < G \Sigma<G is either a discrete or an algebraic subgroup. We also prove that if G is simply connected and the action Γ ↷ G \Gamma\curvearrowright G has property (T), then any cocycle w : Γ × G → Λ w:\Gamma\times G\to\Lambda with values in a countable group Λ is cohomologous to a homomorphism δ : Γ → Λ \delta:\Gamma\to\Lambda . As a consequence, we deduce that the action Γ ↷ G \Gamma\curvearrowright G is orbit equivalent superrigid: any free nonsingular action Λ ↷ Y \Lambda\curvearrowright Y which is orbit equivalent to Γ ↷ G \Gamma\curvearrowright G , is necessarily conjugate to an induction of Γ ↷ G \Gamma\curvearrowright G .