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

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.

arXiv comment34 pages

References in corpus (3)

• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
• Renault's Equivalence Theorem for Reduced Groupoid C*-algebras
• Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

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• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
• On groupoids and $C^{\ast }$-algebras from self-similar actions
• The homology of the groupoid of the self-similar dihedral group
• Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
• Dynamic asymptotic dimension and Matui's HK conjecture