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Dynamic asymptotic dimension for actions of virtually cyclic groups

Published online by Cambridge University Press:  04 May 2021

Massoud Amini
Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran14115-134, Iran (
Kang Li
Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, Warsaw00-656, Poland (
Damian Sawicki
Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn53111, Germany (
Ali Shakibazadeh
Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran14115-134, Iran (


We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$-algebras arising from minimal free $\mathbb {Z}$-actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Research Article
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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