Finite asymptotic dimension for CAT(0) cube complexes

TL;DR

This paper proves that finite-dimensional CAT(0) cube complexes have asymptotic dimension bounded by their dimension, introduces a controlled colouring theorem, and applies these results to small cancellation groups.

Contribution

It establishes an upper bound on the asymptotic dimension of finite-dimensional CAT(0) cube complexes and introduces a controlled colouring theorem for these complexes.

Findings

Asymptotic dimension of finite-dimensional CAT(0) cube complexes is bounded by their dimension

Every CAT(0) cube complex is a contractive retraction of an infinite-dimensional cube

Bounds on asymptotic dimension of small cancellation groups

Abstract

In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) 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(0) 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.