Modular actions and amenable representations

Consider a measure-preserving action $\Gamma \curvearrowright (X, \mu)$ of a countable group $\Gamma$ and a measurable cocycle $\alpha \colon X \times \Gamma \to \mathrm{Aut}(Y)$ with countable image, where $(X, \mu)$ is a standard Lebesgue space and $(Y, \nu)$ is any probability space. We prove that if the Koopman representation associated to the action $\Gamma \curvearrowright X$ is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action $\Gamma \curvearrowright^\alpha X \times Y$ to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones.