Orbit inequivalent actions of non-amenable groups

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.