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

Abstract

An essentially free group action \(\Gamma \curvearrowright (X,\mu )\) 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.