Dynamic asymptotic dimension and Matui's HK conjecture

Christian Bönicke*, Clément Dell'Aiera, James Gabe, Rufus Willett

*Corresponding author for this work

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We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side-effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K-theory of the (Formula presented.) -algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK-conjecture. We also construct explicit maps from the groupoid homology groups to the K-theory groups of their (Formula presented.) -algebras in degrees zero and one, and investigate their properties.

Original languageEnglish
JournalProceedings of the London Mathematical Society
Issue number4
Pages (from-to)1182-1253
Publication statusPublished - 15. Jan 2023


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