Orbit inequivalent actions of non-amenable groups

Consider two free measure preserving group actions $\Gamma \actson (X, \mu), \Delta \actson (X, \mu)$,andameasurepreservingaction$\Delta \actson^a (Z, \nu)$where$(X, \mu), (Z, \nu)$ are standard probability spaces. We show how to construct free measure preserving actions $\mathrm{\Gamma }{\text{actson}}^c(Y,m)$, $\Delta \actson^d (Y, m)$onastandardprobabilityspacesuchthat$E_{\Delta}^d \subset E_{\Gamma}^c$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 $\mathrm{\Gamma }$ is a countable non-amenable group, then $\mathrm{\Gamma }$ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.