Coamenability and strong ergodicity

Abstract

Following methods of Bannon-Marrakchi-Ozawa, we show that for coamenable inclusion $\mathcal{S}\le \mathcal{R}$ 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 $\mathcal{S}\le \mathcal{R}$ is coamenable, $\mathcal{R}$ is strongly ergodic, but we do not assume ergodicity of $\mathcal{S}$. As a consequence, if $\hat{I}›\le \hat{I}“$ is a coamenable inclusion of groups, then any strongly ergodic $\hat{I}“$ action has countably many ergodic components for the $\hat{I}›$ action, each of which is strongly ergodic.

36 pages, no figures. Comments welcome!