Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces (Q6169585)

An essentially free, ergodic, nonsingular action \(\Gamma \curvearrowright (X, \mu)\) of a countable group \(\Gamma\) is \(\mathrm{W}^{\ast}\)-superrigid if any essentially free, ergodic, nonsingular, countable group action \(\Lambda \curvearrowright (Y, \nu)\) such that \(L^{\infty}(Y) \rtimes \Lambda\) is stably isomorphic with \(L^{\infty}(Y) \rtimes \Gamma\) must be conjugate to an induction of \(\Gamma \curvearrowright (X, \mu)\). In the paper under review, the authors prove \(\mathrm{W}^{\ast}\)-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool they use is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. They also provide 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.