Localized cohomology and some applications of Popa's cocycle super-rigidity theorem

TL;DR

This paper demonstrates that orbit equivalence for certain measure-preserving group actions with property (T) is a highly complex, complete analytic equivalence relation, highlighting deep connections between ergodic theory and descriptive set theory.

Contribution

It establishes the complexity of orbit equivalence for ergodic actions with property (T), linking cohomological methods to descriptive set theory.

Findings

Orbit equivalence is a complete analytic relation.

Connections between property (T) and orbit equivalence complexity.

Advances understanding of ergodic actions and their classification.

Abstract

We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.