Higman-Thompson groups from self-similar groupoid actions

Abstract

Given a self-similar groupoid action $(G,E)$ on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs $\mathcal{G}(G,E)$. We study the analogue of the Higman-Thompson group associated to $(G,E)$ using $G$-tables and relate it to the topological full group of $\mathcal{G}(G,E)$, which is isomorphic to a subgroup of unitaries in the algebra $C^{\ast }(G,E)$. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for $\mathcal{G}(G,E)$ in some particular cases.

References in corpus (2)

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