  We study equivalence relations that arise from translation actions
$\Gamma\curvearrowright G$ which are associated to dense embeddings $\Gamma<G$
of countable groups into second countable locally compact groups. Assuming that
$G$ is simply connected and the action $\Gamma\curvearrowright G$ is strongly
ergodic, we prove that $\Gamma\curvearrowright G$ is orbit equivalent to
another such translation action $\Lambda\curvearrowright H$ if and only if
there exists an isomorphism $\delta:G\rightarrow H$ 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 $\Gamma\curvearrowright G$ has property (T), then any
cocycle $w:\Gamma\times G\rightarrow\Lambda$ with values into a countable group
$\Lambda$ is cohomologous to a homomorphism $\delta:\Gamma\rightarrow\Lambda$.
As a consequence, we deduce that the action $\Gamma\curvearrowright G$ is orbit
equivalent superrigid: any free nonsingular action $\Lambda\curvearrowright Y$
which is orbit equivalent to $\Gamma\curvearrowright G$, is necessarily
conjugate to an induction of $\Gamma\curvearrowright G$.
