On the dynamic asymptotic dimension of \'etale groupoids

TL;DR

This paper studies the dynamic asymptotic dimension of étale groupoids, establishing key properties, invariance under Morita equivalence, and connections with coarse geometry, advancing understanding of their large-scale structure.

Contribution

It provides new permanence properties, invariance results, and compares dynamic asymptotic dimension with coarse space asymptotic dimension for étale groupoids.

Findings

Permanence properties for products and unions of groupoids

Invariance of dynamic asymptotic dimension under Morita equivalence

Comparison between asymptotic dimensions of groupoids and their coarse spaces

Abstract

We investigate the dynamic asymptotic dimension for \'etale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also establish invariance of the dynamic asymptotic dimension under Morita equivalence. In the second part of the article, we consider a canonical coarse structure on an \'etale groupoid and compare the asymptotic dimension of the resulting coarse space with the dynamic asymptotic dimension of the underlying groupoid.