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.
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The first author expresses his gratitude to the second and fourth authors, and the Department of Mathematics at the University of Hawai‘i at Mānoa for their hospitality during a visit in November 2019 where part of this work was carried out. The authors would also like to thank Robin Deeley for pointing out a mistake in an earlier version of this paper, and the anonymous referee for suggesting the approach to the comparison maps in Subsection 4.2 . Christian Bönicke was supported by the Alexander von Humboldt Foundation. Clément Dell'Aiera was partly supported by the US NSF (DMS 1564281). James Gabe was supported by the Carlsberg Foundation through an Internationalisation Fellowship, and by Australian Research Council grant DP180100595. Rufus Willet was partly supported by the US NSF (DMS 1564281 and DMS 1901522).
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