相对性质（T）、不变测度的单形与存在封闭模型
Relative Property (T), simplices of invariant measures, and existentially closed models
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中文总结 AI 辅助
该研究证明了与置换群相关的不变测度单形的Bauer-Poulsen二分定理，解决了Austin的相关问题，并从模型论角度开启了仿射逻辑中存在封闭模型的研究。
中文摘要 AI 辅助
我们证明了与置换群相关的不变测度单形的Bauer-Poulsen二分定理。更准确地说，设G是可数集合S的传递置换群，H是S中某点的稳定子群，$\u00af{G}$和$\u00af{H}$分别为它们在逐点收敛拓扑下的闭包。假设波兰群$\u00af{H}$在$\u00af{G}$中具有相对性质（T），则诱导作用$G\ni 2^\u00af{S}$的不变概率测度单形$\u0026M_\text{inv}(2^\u00af{S})$是Bauer单形当且仅当$\u00af{G}$具有性质（T），否则为Poulsen单形。这解决了Austin考虑的若干例子和问题。我们从一个具有独立意义的更一般的模型论命题推导出该结果，为此我们开启了仿射逻辑中存在封闭模型的研究。
英文摘要
We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.
