Strong rigidity of II${}_1$ factors arising from malleable actions of w-rigid groups, II

We prove that any isomorphism $\theta :M_0\simeq M$ of group measure space II${}_1$ factors, $M_0=L^{\mathrm{\infty }}(X_0,{\mu }_0){⋊}_{{\sigma }_0}{G}_0$, $M=L^{\mathrm{\infty }}(X,\mu ){⋊}_{\sigma }G$, with $G_0$ containing infinite normal subgroups with the relative property (T) of Kazhdan-Margulis (i.e. $G_0$ {\it w-rigid}) and $G$ an ICC group acting by Bernoulli shifts $\sigma$, essentially comes from an isomorphism of probability spaces which conjugates the actions. Moreover, any isomorphism $\theta$ of $M_0$ onto a corner'' $pMp$ of $M$, for $p\in M$ an idempotent, forces $p=1$. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and all isomorphisms between such factors come from isomorphisms of the corresponding groups. This settles a group measure space version’’ of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations.