On the small boundary property, $\mathcal Z$-absorption, and Bauer simplexes

Let $X$ be a compact metrizable space, and let $\Delta$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X, \Delta)$. In particular, it is shown that $(X, \Delta)$ has the small boundary property if it has a restricted version of property Gamma. As an application, it is shown that, if $A$ is the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^d$, where $(X, \mathbb Z^d)$ is a free minimal topological dynamical system, or if $A$ is an AH algebra with diagonal maps, then, $A$ is $\mathcal Z$-stable if the set of extreme tracial states is compact, regardless of its dimension.