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Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups

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Abstract

We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. \(\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2\), or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and \(\mathcal{V}\) is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. \(\mathcal{V}\) countable discrete, or separable compact), then any \(\mathcal{V}\)-valued measurable cocycle for a measure preserving action \(\Gamma\curvearrowright X\) of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action \(\Gamma\curvearrowright[0,1]^{\Gamma}\)) is cohomologous to a group morphism of Γ into \(\mathcal{V}\). We use the case \(\mathcal{V}\) discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between \(\Gamma\curvearrowright X\) and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

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  1. Math.Dept., UCLA, University of California, Los Angeles, CA, 90095-155505, USA

    Sorin Popa

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Popa, S. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. math. 170, 243–295 (2007). https://doi.org/10.1007/s00222-007-0063-0

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