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29 March 2024
 
  » arxiv » math.OA/0305306

 Article overview


Strong Rigidity of II$_1$ Factors Arising from Malleable Actions of w-Rigid Groups, I
Sorin Popa ;
Date 21 May 2003
Subject Operator Algebras; Group Theory MSC-class: 46L55, 46L10, 46L40, 22D25, 22D40, 28D15 | math.OA math.GR
AbstractWe consider cross-product II$_1$ factors $M = N times_{sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $sigma: G o { ext{ m Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable’’ and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=Bbb Z^2 times SL(2, Bbb Z)$. We deduce that for any countable group $S subset Bbb R_+^*$ there exist II$_1$ factors $M$ with $mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.
Source arXiv, math.OA/0305306
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