Strict comparison for C^* -algebras arising from almost finite groupoids

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