On Popa's Cocycle Superrigidity Theorem

TL;DR

This paper provides an ergodic-theoretic explanation of Popa's Cocycle Superrigidity Theorem, including a relative version, applications to measurable equivalence relations, and conditions under which Gaussian actions meet the theorem's criteria.

Contribution

It offers a new ergodic-theoretic perspective on Popa's theorem, extends it to a relative version, and explores applications to measurable equivalence relations and Gaussian actions.

Findings

Established a relative version of Popa's Cocycle Superrigidity Theorem.

Demonstrated that Gaussian actions of rigid groups satisfy the theorem's assumptions.

Connected the theorem's implications to measurable equivalence relations.

Abstract

These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of ``rigid'' groups) satisfy the assumptions of Popa's theorem.