Uniform property $\Gamma$ and the small boundary property

Abstract

We prove that, for a free action $\alpha :G\curvearrowright X$ 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 $(C(X)\subseteq C(X){⋊}_{\alpha }G)$. 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 $\alpha$ is also minimal, then almost finiteness of $\alpha$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X)\subseteq C(X){⋊}_{\alpha }G)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $\alpha :G\curvearrowright X$ and $\beta :H\curvearrowright Y$ are free actions and $\alpha$ has the small boundary property, then $\alpha \times \beta :G\times H\curvearrowright X\times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $\alpha$ and $\beta$ are free minimal actions.

24 pages; minor edits. This version has been accepted on Transactions of the AMS