Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

We prove that any isomorphism θ:M0≃M of group measure space II1 factors, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_0=L^\infty(X_0, \mu_0) \rtimes_{\sigma_0} G_0$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M=L^\infty(X, \mu)\rtimes_{\sigma} G$\end{document}, with G0 an ICC group containing an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (i.e. G0w-rigid) and σ a Bernoulli action of some ICC group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G0≃G. Moreover, any isomorphism θ of M0 onto a “corner” pMp of M, for p∈M an idempotent, forces p=1. In particular, all group measure space factors associated with Bernoulli actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism 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.