On extensions of typical group actions

Abstract

For every countable abelian group $G$ we find the set of all its subgroups $H$ ($H\leq G$) such that a typical measure-preserving $H$-action on a standard atomless probability space $(X,\mathcal{F}, \mu)$ can be extended to a free measure-preserving $G$-action on $(X,\mathcal{F}, \mu)$. The description of all such pairs $H\leq G$ was made in purely group terms, in the language of the dual $\hat{G}$, and $G$-actions with discrete spectrum. As an application, we answer a question when a typical $H$-action can be extended to a $G$-action with some dynamic property, or to a $G$-action at all. In particular, we offer first examples of pairs $H\leq G$ satisfying both $G$ is countable abelian, and a typical $H$-action is not embeddable in a $G$-action.