• Corpus ID: 119141736

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

@article{Farsi2018AmpleGE,
  title={Ample groupoids: equivalence, homology, and Matui's HK conjecture},
  author={Carla Farsi and Alex Kumjian and David Pask and Aidan Sims},
  journal={arXiv: Operator Algebras},
  year={2018}
}

• Carla Farsi, A. Kumjian, +1 author A. Sims
• Published 23 August 2018
• Mathematics
• arXiv: Operator Algebras

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.