An essentially free group action of a discrete group $\Gamma$ on a measure space $(X,\mu)$ 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 $(X,\mu)$. After a brief survey of earlier W*-superrigidity theorems, I will focus on a recent joint work with Daniel Drimbe in which we prove W*-superrigidity for certain dense subgroups of PSL(2,R) acting isometrically on the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation.

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