Synergodic actions of product groups

Abstract

This article studies a structural aspect of measure-preserving actions of products of countable discrete groups, involving a so-called 'synergodic decomposition' in terms of the ergodic components of the actions of the two factor groups. We show that this construction provides a canonical way to detect whether the action is built as a product of actions on independent measure spaces, and we use it to prove a result about convergence of ergodic averages on product groups which are highly imbalanced between the factors. Defining an action to be synergodic if it is isomorphic to its synergodic decomposition, we show that if a countable group $G$ is amenable then every action of $G \times G$ can be approximated by synergodic actions and that this statement fails if $G$ is a nonabelian free group. The last result relies on the refutation of Connes' embedding conjecture.