Abstract
We prove that for any group $G$ in a fairly large class of generalized wreath product groups, the associated von Neumann algebra $\mathrm{L} G$ completely “remembers” the group $G$. More precisely, if $\mathrm{L} G$ is isomorphic to the von Neumann algebra $\mathrm{L} \Lambda$ of an arbitrary countable group $\Lambda$, then $\Lambda$ must be isomorphic to $G$. This represents the first superrigidity result pertaining to group von Neumann algebras.
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AUTHOR = {Popa, Sorin},
TITLE = {Cocycle and orbit equivalence superrigidity for malleable actions of {$w$}-rigid groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {170},
YEAR = {2007},
NUMBER = {2},
PAGES = {243--295},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {37A20 (28D15 37A15 37A35 46L55)},
MRNUMBER = {2342637},
MRREVIEWER = {Alain Valette},
DOI = {10.1007/s00222-007-0063-0},
ZBLNUMBER = {1131.46040},
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@incollection {Po06b, MRKEY = {2334200},
AUTHOR = {Popa, Sorin},
TITLE = {Deformation and rigidity for group actions and von {N}eumann algebras},
BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {I}},
PAGES = {445--477},
PUBLISHER = {Eur. Math. Soc., Zürich},
YEAR = {2007},
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@incollection {PV08, MRKEY = {2807839},
AUTHOR = {Popa, Sorin and Vaes, Stefaan},
TITLE = {Cocycle and orbit superrigidity for lattices in {${\rm SL}(n,\Bbb R)$} acting on homogeneous spaces},
BOOKTITLE = {Geometry, Rigidity, and Group Actions},
SERIES = {Chicago Lectures in Math.},
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ADDRESS = {Chicago, IL},
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TITLE = {Rigidity results for {B}ernoulli actions and their von {N}eumann algebras (after {S}orin {P}opa)},
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@misc{Va11, KEY={Vae11},
SORTYEAR={2011},
AUTHOR={Vaes, Stefaan},
TITLE={$W^\ast$-superrigidity for {B}ernoulli actions and wreath product group von~{N}eumann algebras},
NOTE={Lecture notes for a course at the Institut Henri Poincaré,
Paris, May 2011, during the program on Von Neumann Algebras and Ergodic Theory of Group Actions},
URL={www.wis.kuleuven.be/analyse/stefaan},
} -
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