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

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 a...

Authors: Farsi, Carla
Kumjian, Alex
Pask, David
Sims, Aidan
Document types:Article
Media types:Text
Publication date:2019
Date of publication on miami:24.10.2019
Modification date:28.10.2019
Source:Münster Journal of Mathematics, 12 (2019), S. 411-451
Publisher: Mathematisches Institut (Universität Münster)
Edition statement:[Electronic ed.]
DDC Subject:510: Mathematik
License:InC 1.0
Language:English
Format:PDF document
URN:urn:nbn:de:hbz:6-53149724313
Permalink:http://nbn-resolving.de/urn:nbn:de:hbz:6-53149724313
Other Identifiers:DOI: 10.17879/53149724091
Digital documents:mjm_2019_12_411-451.pdf

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 pairwise-commuting 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 specialize 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.