Strict comparison for $C^{\ast }$-algebras arising from Almost finite groupoids

In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced ${\mathrm{C}}^{\ast }$-algebra $C_r^{\ast }(G)$ has real rank zero and strict comparison even though $C_r^{\ast }(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup $\mathrm{C}\mathrm{u}(C_r^{\ast }(G))$ is almost divisible and ${\rm Cu}(C_r^(G))$and${\rm Cu}(C_r^(G)\otimes $\mathcal{Z}$)$ are canonically order-isomorphic, where $\mathcal{Z}$ denotes the Jiang-Su algebra.