Katsura--Exel--Pardo Groupoids and the AH~Conjecture

TL;DR

This paper proves Matui's AH conjecture for a class of groupoids associated with integral matrices, linking topological full groups and homology, and provides criteria for their finite generation.

Contribution

It establishes the truth of the AH conjecture for Katsura--Exel--Pardo groupoids and offers a criterion for the finite generation of their topological full groups.

Findings

AH conjecture verified for $ ext{Katsura--Exel--Pardo}$ groupoids

Criteria for finite generation of topological full groups

Connections between groupoid homology and topological full groups

Abstract

It is proven that Matui's AH~conjecture is true for Katsura--Exel--Pardo groupoids $\mathcal{G}_{A,B}$ associated to integral matrices $A$ and $B$. This conjecture relates the topological full group of an ample groupoid with the homology groups of the groupoid. We also give a criterion under which the topological full group $[[\mathcal{G}_{A,B}]]$ is finitely generated.