Coamenability and strong ergodicity

TL;DR

This paper explores the relationship between coamenability and strong ergodicity in measure-preserving relations and group actions, establishing conditions under which strong ergodicity is preserved or characterized.

Contribution

It extends existing methods to show equivalences of strong ergodicity under coamenable inclusions and generalizes results to non-ergodic relations and group actions.

Findings

Strong ergodicity of $ R$ is equivalent to that of $ S$ under coamenable inclusion when both are ergodic.

In coamenable group inclusions, strongly ergodic actions have countably many ergodic components, each strongly ergodic.

Generalized results apply even when the smaller relation or subgroup is not ergodic.

Abstract

Following methods of Bannon-Marrakchi-Ozawa, we show that for coamenable inclusion $\mathcal{S}\leq \mathcal{R}$ of ergodic, probability measure-preserving relations, we have that $\mathcal{R}$ is strongly ergodic if and only if $\mathcal{S}$ is strongly ergodic. More general results are given when $\mathcal{S}\leq \mathcal{R}$ is coamenable, $\mathcal{R}$ is strongly ergodic, but we do not assume ergodicity of $\mathcal{S}$. As a consequence, if $\Lambda\leq \Gamma$ is a coamenable inclusion of groups, then any strongly ergodic $\Gamma$ action has countably many ergodic components for the $\Lambda$ action, each of which is strongly ergodic.