Higman-Thompson groups from self-similar groupoid actions

TL;DR

This paper explores the properties of groupoids derived from self-similar groupoid actions on finite graphs, connecting Higman-Thompson groups with topological full groups and discussing related algebraic and homological aspects.

Contribution

It introduces a new framework linking self-similar groupoid actions to Higman-Thompson groups and analyzes their properties via groupoid and algebraic structures.

Findings

Properties of the ample groupoid of germs $\

Relation between the Higman-Thompson group analogue and the topological full group of $\

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 $C^*(G,E)$. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for $\mathcal G(G,E)$ in some particular cases.