  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.
