Uniform property $Γ$ and the small boundary property

Abstract: We prove that, for a free action $\alpha \colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $\Gamma$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_\alpha G)$. The reverse implication has been demonstrated by Kerr and Szab\'o for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $\alpha$ is also minimal, then almost finiteness of $\alpha$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_\alpha G)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $\alpha \colon G \curvearrowright X$ and $\beta \colon H \curvearrowright Y$ are free actions and $\alpha$ has the small boundary property, then $\alpha \times \beta \colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $\alpha$ and $\beta$ are free minimal actions.