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

We study equivalence relations that arise froixi translation actions Gamma curved right arrow G which arc associated to dense embcddings Gamma < G of countable groups into second countable locally compact groups. Assuming that G is simply connected and the action Gamma curved right arrow G is strongly ergodic, we prove that Gamma curved right arrow G is orbit equivalent to another such translation action Lambda curved right arrow H if and only if there exists an isomorphism delta : G -> H such that delta(Gamma) = Lambda If G is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of Gamma on homogeneous spaces of the form G/Sigma, where Sigma < G is either a discrete or an algebraic subgroup. We also prove that if G is simply connected and the action Gamma curved right arrow G has property (T), then any cocycle w : Gamma x G -> A with values in a countable group Lambda is cohomologous to a homomorphism delta : Gamma -> Lambda As a consequence, we deduce that the action Gamma curved right arrow G is orbit equivalent superrigid: any free nonsingular action Lambda curved right arrow Y which is orbit equivalent to Gamma curved right arrow G, is necessarily conjugate to an induction of Gamma curved right arrow G.