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

Let $X$ be a compact metrizable space, and let $\mathrm{\Delta }$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X,\mathrm{\Delta })$. In particular, it is shown that $(X,\mathrm{\Delta })$ 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 $\mathrm{C}(X)⋊{\mathbb{Z}}^d$, 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.