On extensions of typical group actions

Abstract

For every countable abelian group G we find the set of all its subgroups H (H ≤ G) such that a typical measure-preserving H-action on a standard atomless probability space (X, F , µ) can be extended to a free measure-preserving G-action on (X, F , µ). The description of all such pairs H ≤ G was made in purely group terms, in the language of the dual 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 ≤ G satisfying both G is countable abelian, and a typical H-action is not embeddable in a G-action.