W∗-superrigidity of mixing Gaussian actions of rigid groups

We generalize W∗-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action Γ↷X is W∗-superrigid. More precisely, if Λ↷Y is another free ergodic action such that the crossed-product von Neumann algebras are isomorphic L∞(X)⋊Γ≃L∞(Y)⋊Λ, then the actions are conjugate. We prove a similar statement whenever Γ is a non-amenable ICC product of two infinite groups.