Extensions of generic measure-preserving actions

TL;DR

This paper proves that for certain groups, generic measure-preserving actions of a subgroup can be extended to free actions of the larger group, generalizing previous results for cyclic subgroups.

Contribution

It extends Ageev's result by showing that for countable abelian groups and finitely-generated subgroups, generic actions extend to free actions of the entire group.

Findings

Generic measure-preserving actions of finitely-generated subgroups extend to free actions of the larger group

Extension results apply to countable abelian groups and their subgroups

Generalizes previous cyclic subgroup extension results

Abstract

We show that, whenever Gamma is a countable abelian group and Delta is a finitely-generated subgroup of Gamma, a generic measure-preserving action of Delta on a standard atomless probability space (X,mu) extends to a free measure-preserving action of Gamma on (X,mu). This extends a result of Ageev, corresponding to the case when Delta is infinite cyclic.