Inventiones mathematicae

, Volume 182, Issue 2, pp 371–417

Group measure space decomposition of II1 factors and W*-superrigidity

Article

Abstract

We prove a “unique crossed product decomposition” result for group measure space II1 factors L (X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family \(\mathcal{G}\), which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if Tn denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of \(\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z})\) is W-superrigid, i.e. any isomorphism between L (X)⋊Γ and an arbitrary group measure space factor L (Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family \(\mathcal{G}\), the Bernoulli actions of Γ are W-superrigid.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of California at Los AngelesLos AngelesUSA
  2. 2.Department of MathematicsK.U. LeuvenLeuvenBelgium

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