Non-ergodic actions, cocycles and superrigidity

TL;DR

This paper extends superrigidity theorems to non-ergodic group actions by analyzing Borel cocycles through their ergodic components, ensuring measurability and generalizing previous results.

Contribution

It introduces a method to study cocycles over non-ergodic actions by reducing to ergodic components, enabling superrigidity results in a broader context.

Findings

Superrigidity theorems are extended to non-ergodic actions.

A measurable decomposition approach for cocycles over non-ergodic actions.

Framework for analyzing cocycles via ergodic component restrictions.

Abstract

This paper proves various results concerning non-ergodic actions of locally compact groups and particularly Borel cocycles defined over such actions. The general philosophy is to reduce the study of the cocycle to the study of its restriction to each ergodic component of the action, while being careful to show that all objects arising in the analysis depend measurably on the ergodic component. This allows us to prove a version of the superrigidity theorems for cocycles defined over non-ergodic actions.