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The type semigroup, comparison, and almost finiteness for ample groupoids

Part of: Special aspects of infinite or finite groups Ordered structures Topological dynamics Topological and differentiable algebraic systems

Published online by Cambridge University Press:  27 October 2021

PERE ARA ,
CHRISTIAN BÖNICKE Open the ORCID record for CHRISTIAN BÖNICKE [Opens in a new window] ,
JOAN BOSA  and
KANG LI
PERE ARA
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 01893 Cerdanyola del Vallès (Barcelona), Spain Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra, 01893 Cerdanyola del Vallès (Barcelona), Spain (e-mail: para@mat.uab.cat)
CHRISTIAN BÖNICKE*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow, G12 8QQ, UK
JOAN BOSA
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 01893 Bellaterra (Barcelona), Spain (e-mail: jbosa@mat.uab.cat)
KANG LI
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland (e-mail: kli@impan.pl)
*
e-mail: christian.bonicke@glasgow.ac.uk
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Abstract

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.

Keywords

almost finite groupoidsamenabilitydynamical comparisontype semigroup

MSC classification

Primary: 22A22: Topological groupoids (including differentiable and Lie groupoids)
Secondary: 06F05: Ordered semigroups and monoids 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 20F65: Geometric group theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 361 - 400
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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