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

An essentially free group action Γ ↷ ( X , μ ) is called W ∗ -superrigid if the crossed product von Neumann algebra L ∞ ( X ) ⋊ Γ completely remembers the group Γ and its action on ( X , μ ) . 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.