A characterization for asymptotic dimension growth

Abstract

We give a characterization for asymptotic dimension growth. We apply it to $CAT\left(0\right)$ cube complexes of finite dimension, giving an alternative proof of 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 S̆pakula and Wright.