Dynamic asymptotic dimension for actions of virtually cyclic groups

Abstract

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.