Homology of the Katsura-Exel-Pardo groupoid

We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^{\ast }$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^{\ast }$-algebras are groupoid $C^{\ast }$-algebras of this class.