Topological Rigidity of the Dynamic Asymptotic Dimension

TL;DR

This paper establishes a fundamental link between the dynamic asymptotic dimension of a group action and the group's asymptotic dimension, revealing a rigidity phenomenon in topological dynamics.

Contribution

It proves that for free actions of countable groups on finite-dimensional compact spaces, the dynamic asymptotic dimension is either infinite or equal to the group's asymptotic dimension, demonstrating a rigidity property.

Findings

Dynamic asymptotic dimension equals the group's asymptotic dimension for finite-dimensional spaces.

The dynamic asymptotic dimension is either infinite or matches the group's asymptotic dimension.

The result applies to free actions of countable groups on compact metric spaces.

Abstract

We show for a free action of a countable group $\Gamma$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$.