Finite asymptotic dimension for $\mathrm{CAT}(0)$ cube complexes

Abstract

We prove that the asymptotic dimension of a finite-dimensional $CAT\left(0\right)$ cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every $CAT\left(0\right)$ cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups.