Ample groupoids: equivalence, homology, and Matui's HK conjecture

TL;DR

This paper explores the homology of ample Hausdorff groupoids, establishing equivalences that preserve homology, computing specific examples, and verifying Matui's HK conjecture for certain classes of groupoids and k-graphs.

Contribution

It demonstrates the equivalence of various notions of groupoid equivalence, computes homology for specific groupoids, and verifies Matui's HK conjecture in new cases involving k-graphs.

Findings

Groupoid equivalences preserve homology.

Matui's HK conjecture holds for certain k-graph groupoids.

Homology of k-graph groupoids can be computed via adjacency matrices.

Abstract

We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu{Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain complex developed by Evans. We show that Matui's HK conjecture holds for the groupoids of single vertex k-graphs which satisfy a mild joint-coprimality condition. We also prove that there is a natural homomorphism from the categorical homology of a k-graph to the homology of its groupoid.