论文标题
自由组的存在封闭的措施保护行动
Existentially closed measure-preserving actions of free groups
论文作者
论文摘要
本文是由使用连续模型理论的概率措施保护（PMP）动作的研究激发的。这种行动被视为由概率度量空间的衡量代数组成的度量结构，其自动形态家族扩展了。我们证明，给定自由组的存在封闭的PMP动作形成一个基本类，因此PMP $ \ Mathbb {f} _K $ -Actions的理论具有模型伴侣。我们表明该模型伴侣是稳定的，并且具有量化器消除。我们还证明，$ \ mathbb {f} _k $在其使用HAAR测量的情况下完成的操作是指标通用的，因此，正如我们所显示的，它已存在。 我们从更一般的定理中得出我们的主要结果，该定理为存在的模型伴侣提供了一组足够的条件。
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.
