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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 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.

Publication Date


  • 2018

Citation


  • Farsi, C., Kumjian, A., Pask, D., & Sims, A. (2018). Ample groupoids: equivalence, homology, and Matui's HK conjecture. Retrieved from http://arxiv.org/abs/1808.07807v1

Web Of Science Accession Number


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.

Publication Date


  • 2018

Citation


  • Farsi, C., Kumjian, A., Pask, D., & Sims, A. (2018). Ample groupoids: equivalence, homology, and Matui's HK conjecture. Retrieved from http://arxiv.org/abs/1808.07807v1

Web Of Science Accession Number