Strong Rigidity of II₁ Factors Arising from Malleable Actions of w-Rigid Groups, I

2003/05/21 by Sorin Popa, Popa, Sorin · 9 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Spectral Theory in Mathematical Physics #math.GR #math.OA #msc:22D25 #msc:22D40 #msc:28D15 #msc:46L10 #msc:46L40 #msc:46L55

paper · pdf · doi:10.48550/arxiv.math/0305306

revised version 40 pages

arxiv created 2005/06/02 · arxiv updated 2009/11/30

Abstract

We consider cross-product II₁ factors M = N\rtimes_σ G, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ: G → \rm AutN 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 Z² \rtimes SL(2, \Bbb Z). We deduce that for any countable group S ⊂ \Bbb R₊^* there exist II₁ factors M with \mycal F(M)=S, thus bringing new light to a longstanding problem of Murray and von Neumann.

Citations

• On Groups of Measure Preserving Transformations. I

Cited by

• McDuff superrigidity for group II₁ factors
• Strong solidity classification of Coxeter groups
• Property (T) group factors whose Jones index set equals all positive integers
• The impact of Schur multipliers in harmonic analysis and operator algebras
• Compressible subalgebras in II₁ factors
• Relative solidity results and their applications to computations of some II₁ factor invariants
• Discrete measured groupoid von Neumann algebras via the Gaussian deformation
• Some computations of 1-cohomology groups and construction of non orbit equivalent actions
• W*-correlations of II₁ factors and rigidity of tensor products and graph products

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• Strong rigidity of II₁ factors arising from malleable actions of w-rigid groups, II
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• McDuff superrigidity for group II₁ factors
• Measure equivalence rigidity via s-malleable deformations
• Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces