Existentially closed measure-preserving actions of free groups

2022/03/18 by Berenstein, Alexander, Henson, C. Ward, Ibarlucía, Tomás · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2203.10178

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 \mathbbF_k-actions has a model companion. We show this model companion is stable and has quantifier elimination. We also prove that the action of \mathbbF_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 \mathbbF_k-actions on a separably categorical, stable metric structure.

Cited by

• Model Theory of General von Neumann Algebras II: Group Actions and Crossed Products

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