Existentially closed measure-preserving actions of free groups

TL;DR

This paper studies measure-preserving actions of free groups using continuous model theory, establishing the existence of a stable model companion with quantifier elimination and identifying a metrically generic, existentially closed action.

Contribution

It proves the theory of pmp free group actions has a model companion that is stable and admits quantifier elimination, advancing the understanding of their model-theoretic properties.

Findings

Existentially closed pmp actions form an elementary class.

The model companion of the theory is stable and has quantifier elimination.

The action on the profinite completion is metrically generic and existentially closed.

Abstract

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.