The type semigroup, comparison and almost finiteness for ample groupoids

Ara, P., Bonicke, C. , Bosa, J. and Li, K. (2023) The type semigroup, comparison and almost finiteness for ample groupoids. Ergodic Theory and Dynamical Systems, 43(2), pp. 361-400. (doi: 10.1017/etds.2021.115)

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We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Bonicke, Dr Christian and Bosa, Dr Joan
Authors: Ara, P., Bonicke, C., Bosa, J., and Li, K.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Ergodic Theory and Dynamical Systems
Publisher:Cambridge University Press
ISSN (Online):1469-4417
Published Online:27 October 2021
Copyright Holders:Copyright © 2021 The Authors
First Published:First published in Ergodic Theory and Dynamical Systems 43(2): 361-400
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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