标题： 存在性闭合的保测度的近似树可分群作用 显示英文标题

标题： Existentially closed measure-preserving actions of approximately treeable groups

摘要： 给定一个可数群$\Gamma$，令$\mathcal{K}_\Gamma$表示{\pmp }的$\Gamma$的作用类，我们研究当$\mathcal{K}_\Gamma$的模型伴随存在时的条件。 Berenstein、Henson 和 Ibarlucía 表明，当$\Gamma$是一个可数生成元的非交换自由群时，$\mathcal{K}_\Gamma$的模型伴随存在。 我们通过证明当$\Gamma$是一个近似树形群时，$\cal K_\Gamma$的模型伴随存在，从而显著推广了他们的结果。 近似树形群类包含树形群类以及普遍自由群类，即与非交换自由群具有相同全称理论的群类。 我们使用模型伴随存在的开映射特征来证明这一结果；此外，这种开映射特征在模型伴随存在时提供了具体的、遍历论的公理。 我们展示了如何在树形群的情况下简化这些公理，为自由群的模型伴随提供了一种替代的公理化方法，该方法最初由Berenstein、Henson和Ibarlucía利用模型论稳定性理论的技术进行公理化。 在此过程中，我们证明了一个纯粹的遍历论结果，具有独立兴趣，即有限生成的普遍自由群（也称为极限群）具有Kechris的MD性质。 我们还证明了对于具有Kechris的EMD性质的群，其紧致完成作用是存在闭的，而对于没有性质(T)的群，泛在的存在闭作用是弱混合的，推广了Berenstein、Henson和Ibarlucía针对非交换自由群的情况所得出的结果。 显示英文摘要

摘要： Given a countable group $\Gamma$, letting $\mathcal{K}_\Gamma$ denote the class of {\pmp} actions of $\Gamma$, we study the question of when the model companion of $\mathcal{K}_\Gamma$ exists. Berenstein, Henson, and Ibarluc\'ia showed that the model companion of $\mathcal{K}_\Gamma$ exists when $\Gamma$ is a nonabelian free group on a countable number of generators. We significantly generalize their result by showing that the model companion of $\cal K_\Gamma$ exists whenever $\Gamma$ is an approximately treeable group. The class of approximately treeable groups contain the class of treeable groups as well as the class of universally free groups, that is, the class of groups with the same universal theory as nonabelian free groups. We prove this result using an open mapping characterization of when the model companion exists; moreover, this open mapping characterization provides concrete, ergodic-theoretic axioms for the model companion when it exists. We show how to simplify these axioms in the case of treeable groups, providing an alternate axiomatization for the model companion in the case of the free group, which was first axiomatized by Berenstein, Henson, and Ibarluc\'ia using techniques from model-theoretic stability theory. Along the way, we prove a purely ergodic-theoretic result of independent interest, namely that finitely generated universally free groups (also known as limit groups) have Kechris' property MD. We also show that for groups with Kechris' EMD property, the profinite completion action is existentially closed, and for groups without property (T), the generic existentially closed action is weakly mixing, generalizing results of Berenstein, Henson, and Ibarluc\'ia for the case of nonabelian free groups.