On Popa's Cocycle Superrigidity Theorem

By A. Furman. 2006

63 citations recorded by the source corpus.

Theorem 1.1 (Popa’s Cocycle Superrigity, Special Case). Let Γ be a discrete group with Kazhdan’s property (T), let Γ0 < Γ be an infinite index subgroup, (X0,μ0) an arbitrary probability space, and let Γ (X,μ) = (X0,μ0)0 be the corresponding generalized Bernoulli action. Then for any discrete countable group Λ and any measurable cocycle α : Γ × X → Λ there exist a homomorphism : Γ → Λ and a measurable map φ : X → Λ so that:

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups (2005, methodology): “ that if Γ,Λ are lattices in a higher rank semisimple Lie group G then the action Γ y G/Λ cannot be realized as a quotient of a Bernoulli Γ-action, more generally of a s-malleable weak mixing action ([Fu4]). This solves a problem posed in ([P3], see comments after 7.7) and shows that the case Γ high”
Orbit Equivalence and Measured Group Theory (2010, methodology): “See Furman’s ergodic theoretical treatment and generalizations [Fur07] for the Kazhdan property (T) case.”
On cocycle superrigidity for Gaussian actions (2009, methodology): “The existence of such s-malleable deformations (introduced by Popa in [36, 37]) actually occurs in a broader setting than the (generalized) Bernoulli shifts with diffuse core, but it was Furman [15] who first noticed that the even larger class of Gaussian actions are also s-malleable.”