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

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