Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups
By Sorin Popa. 2005
206 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/df3ae7aadecf22e3f0af18d3717fbeec72cb4dce
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Citations in the corpus
- Cartan Subalgebras in $C^*$-Algebras (2008, background): “Popa has recently (see [37, 38]) constructed and studied a large class of type II1 factors (from Bernoulli actions of groups with property T ) which have a distinguished Cartan subalgebra, unique up to inner conjugacy; moreover, these factors satisfy remarkable rigidity properties: isomorphisms of t”
- Group measure space decomposition of II1 factors and W*-superrigidity (2009, methodology): “Theorem 6.7 will be proven as a consequence of Theorem 5.2 and a number of cocycle superrigidity theorems from [ 47 , 49] that we recall here.”
- Rigidity for von Neumann algebras and their invariants (2010, background): “ certain amalgamated free products Γ = Γ1∗ΣΓ2 over a common amenable subgroup Σ which is sufficiently non-normal inside Γ (see [PV09]). In combination with Popa’s OE superrigidity for Bernoulli actions [Po05] or Kida’s OE superrigidity [Ki09], we obtained the following examples of W∗-superrigid action”
- Orbit Equivalence and Measured Group Theory (2010, background): “And along the years, various rigidity results entailed some specific families of groups to admit continously 7 many non-OE actions (see for instance [BG81, Zim84, GG88b, MS06, Pop06b, Pop07a]).”
- On cocycle superrigidity for Gaussian actions (2009, methodology): “One of the seminal results using these techniques is Popa’s cocycle superrigidity theorem [38, 40] (see also [15, 51] for more on this), which obtains orbit equivalence superrigidity results by means of untwisting cocycles into a finite von Neumann algebra.”
- Integrable measure equivalence and rigidity of hyperbolic lattices (2010, background): “The closed convergence group Hµ is a locally compact subgroup of Homeo(S 1); the classification of all such groups is well known, and the only ones with convergence property are conjugate to PGL2(R) [16, pp. 51–54; 24, pp. 345–348].”
- W*-superrigidity for Bernoulli actions of property (T) groups (2010, methodology): “In combination with Popa’s OE superrigidity result [37] this proves Theorem A.”