一致性质 Γ 与小边界性质
Uniform property $Γ$ and the small boundary property
AI总结：
本文证明自由顺从群作用的小边界性质等价于相应 Cartan 子代数的一致性质 Γ，极小情形下几乎有限性等价于迹 Z-稳定性，并证明了这两类性质在乘积作用下的保持性。
摘要：
我们证明，对于紧度量空间上可数无限离散顺从群 G 的自由作用 α: G ↷ X，小边界性质由 Cartan 子代数 (C(X) ⊆ C(X) ⋊_α G) 的一致性质 Γ 所蕴含。Kerr 和 Szabó 已对自由作用证明了反向蕴含，由此我们得到这两个条件是等价的。我们进一步证明，如果 α 还是极小的，那么 α 的几乎有限性由子代数 (C(X) ⊆ C(X) ⋊_α G) 的迹 Z-稳定性所蕴含。反向蕴含由 Kerr 给出，从而也得到这两个性质的等价性。作为一个应用，我们证明如果 α: G ↷ X 和 β: H ↷ Y 是自由作用且 α 具有小边界性质，则 α × β: G × H ↷ X × Y 具有小边界性质。在 α 和 β 为自由极小作用的情形下，对几乎有限性也得到了类似的保持性质。
英文摘要：
We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions.
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