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

Abstract

We prove that any isomorphism $\theta :M_0\simeq M$ of group measure space II${}_1$ factors, $M_0=L^{\infty }(X_0,{\mu }_0){⋊}_{{\sigma }_0}{G}_0$, $M=L^{\infty }(X,\mu ){⋊}_{\sigma }G$, 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.

Sequel Part II of the paper with the same title math.OA/0305306. 45 pages