On groupoids and $C^{\ast }$-algebras from self-similar actions

Given a self-similar groupoid action $(G,E)$ on the path space of a finite graph, we study the associated Exel-Pardo 'etale groupoid $\mathcal{G}(G,E)$ and its $C^{\ast }$-algebra $C^{\ast }(G,E)$. We review some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids which resembles Takai duality. We also describe a general strategy to compute the $K$-theory of $C^{\ast }(G,E)$ and the homology of $\mathcal{G}(G,E)$ in certain cases and illustrate with an example.