2014/06/25 by Adrian Ioana, Ioana, Adrian
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #math.DS #math.GR #math.OA
paper · pdf · doi:10.48550/arxiv.1406.6628
arxiv created 2014/06/25 · arxiv updated 2014/06/26
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→ 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× G→Λ with values into a countable group Λ is cohomologous to a homomorphism δ:Γ→Λ. 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.