A characterization for asymptotic dimension growth

TL;DR

This paper introduces a new characterization for asymptotic dimension growth, applies it to CAT(0) cube complexes and geodesic coarse median spaces, and establishes finite or subexponential growth results, strengthening existing theorems.

Contribution

It provides a novel characterization for asymptotic dimension growth and applies it to specific geometric spaces, offering alternative proofs and new growth bounds.

Findings

Finite asymptotic dimension for CAT(0) cube complexes

Subexponential asymptotic dimension growth for geodesic coarse median spaces

Strengthens previous results on asymptotic dimension growth

Abstract

We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of J. Spakula and N. Wright.