Dynamic asymptotic dimension and Matui’s HK conjecture

Bönicke, C. , Dell'Aiera, C., Gabe, J. and Willett, R. (2023) Dynamic asymptotic dimension and Matui’s HK conjecture. Proceedings of the London Mathematical Society, 126(4), pp. 1182-1253. (doi: 10.1112/plms.12510)

Abstract

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 C*-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 C*-algebras in degrees zero and one, and investigate their properties.

• Item Type: Articles
• Additional Information: 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).
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bonicke, Dr Christian
• Authors: Bönicke, C., Dell'Aiera, C., Gabe, J., and Willett, R.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Proceedings of the London Mathematical Society
• Publisher: Wiley
• ISSN: 0024-6115
• ISSN (Online): 1460-244X
• Published Online: 15 January 2023
• Copyright Holders: Copyright © 2023 The Authors
• First Published: First published in Proceedings of the London Mathematical Society 126(4):1182-1253
• Publisher Policy: Reproduced under a Creative Commons License