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Strong rigidity of II$_1$ factors arising from malleable actions of w-rigid groups, II | Sorin Popa
; | Date: |
7 Jul 2004 | Subject: | Operator Algebras; Group Theory MSC-class: 46L55, 46L10, 46L40, 22D25, 22D40, 28D15 | math.OA math.GR | Abstract: | We prove that any isomorphism $ heta:M_0simeq M$ of group measure space II$_1$ factors, $M_0=L^infty(X_0, mu_0)
times_{sigma_0} G_0$, $M=L^infty(X, mu)
times_{sigma} G$, 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 $ heta$ of $M_0$ onto a ``corner’’ $pMp$ of $M$, for $pin M$ 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. | Source: | arXiv, math.OA/0407103 | Services: | Forum | Review | PDF | Favorites |
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