Orbit inequivalent actions of non-amenable groups

TL;DR

The paper develops a method to construct many non-amenable group actions that are orbit inequivalent, expanding the understanding of group action diversity in ergodic theory.

Contribution

It introduces a new construction generalizing co-induction to produce continuum many orbit inequivalent actions for non-amenable groups.

Findings

Constructed free measure preserving actions with specific orbit relations.

Proved existence of continuum many orbit inequivalent actions for non-amenable groups.

Extended co-induction concept to broader class of group actions.

Abstract

Consider two free measure preserving group actions $\Gamma \actson (X, \mu), \Delta \actson (X, \mu)$, and a measure preserving action $\Delta \actson^a (Z, \nu)$ where $(X, \mu), (Z, \nu)$ are standard probability spaces. We show how to construct free measure preserving actions $\Gamma \actson^c (Y, m)$, $\Delta \actson^d (Y, m)$ on a standard probability space such that $E_{\Delta}^d \subset E_{\Gamma}^c$ and $d$ has $a$ as a factor. This generalizes the standard notion of co-induction of actions of groups from actions of subgroups. We then use this construction to show that if $\Gamma$ is a countable non-amenable group, then $\Gamma$ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.