Higman-Thompson groups from self-similar groupoid actions

Abstract

Given a self-similar groupoid action pG, Eq on a finite directed graph, we prove some properties of the corresponding ample groupoid of germs GpG, Eq. We study the analogue of the Higman-Thompson group associated to pG, Eq using G-tables and relate it to the topological full group of GpG, Eq, which is isomorphic to a subgroup of unitaries in the algebra C ˚pG, Eq. After recalling some concepts in groupoid homology, we discuss the Matui's AH-conjecture for GpG, Eq in some particular cases. introductionSelf-similar group actions are defined using subgroups of the automorphism group of a rooted tree which is viewed as the path space of a graph with one vertex and n edges. Given a finite directed graph E with no sources, the corresponding path space gives rise to a union of trees (or forest) T E , and it is natural to consider self-similar actions of subgroupoids G of PIsopT E q, the set of partial isomorphsims of T E .Given such a self-similar action pG, Eq, we study the properties of the generalized Exel-Pardo ample groupoid GpG, Eq. Since the source map of the graph is in general not G-equivariant, this determines certain modifications in the proofs of similar results from [5], where G is a group acting on E by graph automorphisms. We study the topological full group vGpG, Eqw and we relate it to the Higman-Thompson type group defined by G-tables for the action pG, Eq. After recalling some results about groupoid homology, we discuss the Matui's AH-conjecture for GpG, Eq and related issues.We begin by recalling the definition of a self-similar groupoid action pG, Eq from [7] and give several examples, illustrating some differences with the case when G is a group acting by automorphisms of the graph E. We recall some facts about the structure of the associated C ˚algebra C ˚pG, Eq defined using a C ˚-correspondence over C ˚pGq and 2020 Mathematics Subject Classification. Primary 46L05.