[Paper Review] Cocycle and Orbit Equivalence Superrigidity for Bernoulli Actions of Kazhdan Groups

Sorin Popa|arXiv (Cornell University)|Dec 30, 2005

Advanced Operator Algebra Research11 references9 citations

TL;DR

This paper establishes cocycle and orbit equivalence superrigidity for Bernoulli actions of Kazhdan groups with infinite normal subgroups possessing relative property (T). It shows that measurable cocycles into certain unitary groups are cohomologous to group homomorphisms, and that orbit equivalences between such Bernoulli actions and free ergodic actions arise from group isomorphisms and conjugacies, proving strong rigidity in the absence of measurable conjugacy beyond group isomorphism.

ABSTRACT

We prove that if a countable discrete group Γ contains an infinite normal subgroup with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary) and V is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a Bernoulli Γ-action is cohomologous to a group morphism of Γ into V. We use this result to prove that if in addition Γ is infinite conjugacy class, then any orbit equivalence between a Bernoulli Γ-action and a free ergodic measure preserving action of some group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ.

Motivation & Objective

• To establish cocycle superrigidity for Bernoulli actions of countable discrete groups containing an infinite normal subgroup with relative property (T).
• To extend this result to orbit equivalence rigidity under the additional assumption that the group has infinite conjugacy class.
• To demonstrate that orbit equivalences between Bernoulli Γ-actions and free ergodic measure-preserving actions of another group Λ must arise from group isomorphisms and conjugacies.
• To generalize rigidity results to groups such as SL(2, Z) ⋉ Z² and products H × H′ with H an infinite Kazhdan group.
• To unify cocycle and orbit equivalence rigidity under a single framework using relative property (T) and unitary cocycle analysis.

Proposed method

• Use the relative property (T) of an infinite normal subgroup to constrain the structure of measurable cocycles with values in unitary groups of finite von Neumann algebras.
• Apply techniques from operator algebras and ergodic theory to show that such cocycles are cohomologous to group homomorphisms into the unitary group V.
• Utilize the infinite conjugacy class condition to strengthen the rigidity of orbit equivalence relations.
• Leverage the structure of Bernoulli actions to analyze the cohomology class of cocycles and their equivalence to homomorphisms.
• Establish that any orbit equivalence between a Bernoulli Γ-action and a free ergodic Λ-action must be implemented by a conjugacy via a group isomorphism Γ ≃ Λ.
• Work within the framework of finite von Neumann algebras to ensure the unitary groups V are well-behaved (e.g. countable or separable compact).

Experimental results

Research questions

• RQ1 Under what conditions is a measurable cocycle with values in a unitary group of a finite von Neumann algebra cohomologous to a group homomorphism for a Bernoulli Γ-action?
• RQ2 Can orbit equivalence between a Bernoulli Γ-action and a free ergodic action of another group Λ be realized through a group isomorphism and conjugacy?
• RQ3 How does the presence of a normal subgroup with relative property (T) affect the rigidity of cocycles and orbit equivalence in Bernoulli actions?
• RQ4 What role does the infinite conjugacy class condition play in enforcing superrigidity of orbit equivalence relations?
• RQ5 To what extent do groups like SL(2, Z) ⋉ Z² or H × H′ with H Kazhdan exhibit cocycle and orbit equivalence superrigidity?

Key findings

• Any V-valued measurable cocycle for a Bernoulli Γ-action is cohomologous to a group homomorphism when Γ has an infinite normal subgroup with relative property (T) and V is a closed subgroup of the unitaries of a finite von Neumann algebra.
• If Γ is also infinite conjugacy class, then any orbit equivalence between a Bernoulli Γ-action and a free ergodic measure-preserving action of Λ is implemented by a conjugacy of the actions via a group isomorphism Γ ≃ Λ.
• The result applies to groups such as SL(2, Z) ⋉ Z² and H × H′ with H an infinite Kazhdan group and H′ arbitrary.
• The unitary group V can be countable discrete or separable compact, ensuring broad applicability of the cocycle superrigidity result.
• The proof relies on the interplay between relative property (T), ergodic theory, and the structure of von Neumann algebras to constrain cocycle cohomology.
• The orbit equivalence superrigidity result shows that no non-trivial measurable conjugacy exists beyond the group isomorphism, establishing strong rigidity.