Existentially closed measure-preserving actions of free groups

This paper is motivated by the study of probability-measure-preserving (pmp) actions of free groups using continuous model theory. Such an action is treated as a metric structure that consists of the measure algebra of the probability measure space expanded by a family of its automorphisms. We prove that the existentially closed pmp actions of a given free group form an elementary class, and therefore the theory of pmp $\mathbb {F}_k$-actions has a model companion. We show this model companion is stable and has quantifier elimination. We also prove that the action of $\mathbb {F}_k$ on its profinite completion with the Haar measure is metrically generic and therefore, as we show, it is existentially closed.

We deduce our main result from a more general theorem, which gives a set of sufficient conditions for the existence of a model companion for the theory of $\mathbb {F}_k$-actions on a separably categorical, stable metric structure.