Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces

An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove W*-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type $II_1$ equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction.