Dynamic asymptotic dimension for actions of virtually cyclic groups

TL;DR

This paper proves that minimal free actions of infinite virtually cyclic groups on compact spaces have dynamic asymptotic dimension one, extending known results and generalizing to broader group actions with the marker property.

Contribution

It establishes the dynamic asymptotic dimension for virtually cyclic groups and proves the marker property for all free actions on finite dimensional compact spaces.

Findings

Dynamic asymptotic dimension is one for minimal free actions of virtually cyclic groups.

The marker property holds for all free actions of countable groups on finite dimensional compact spaces.

Extension of previous results from infinite cyclic to virtually cyclic groups.

Abstract

We show that the dynamic asymptotic dimension of a minimal free action of an infinite virtually cyclic group on a compact Hausdorff space is always one. This extends a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. Furthermore, the minimality assumption can be replaced by the marker property, and we prove the marker property for all free actions of countable groups on finite dimensional compact Hausdorff spaces, generalising a result of Szabo in the metrisable setting.