Orbit inequivalent actions of non-amenable groups

Abstract

Consider two free measure preserving group actions $Γ\actson (X, μ), Δ\actson (X, μ)$, and a measure preserving action $Δ\actson^a (Z, ν)$ where $(X, μ), (Z, ν)$ are standard probability spaces. We show how to construct free measure preserving actions $Γ\actson^c (Y, m)$, $Δ\actson^d (Y, m)$ on a standard probability space such that $E_Δ^d \subset E_Γ^c$ 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 $Γ$ is a countable non-amenable group, then $Γ$ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.