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

Journal Article


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 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.

Publication Date


  • 2019

Web Of Science Accession Number


Number Of Pages


  • 41

Start Page


  • 411

End Page


  • 451

Volume


  • 12

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 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.

Publication Date


  • 2019

Web Of Science Accession Number


Number Of Pages


  • 41

Start Page


  • 411

End Page


  • 451

Volume


  • 12