, Volume 182, Issue 2, pp 371-417

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

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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 T n 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.

S. Popa was partially supported by NSF Grant DMS-0601082.
S. Vaes was partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0231.07 of the Research Foundation—Flanders (FWO) and K.U. Leuven BOF research Grant OT/08/032.