Finite asymptotic dimension for CAT(0) cube complexes

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.

References in corpus (2)

• Cubulating spaces with walls
• On asymptotic dimension of amalgamated products and right-angled Coxeter groups

Cited by in corpus (6)

• Finiteness properties of cubulated groups
• Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups
• Asymptotic Dimension of Graphs of Groups and One Relator Groups
• A characterization for asymptotic dimension growth
• Coarse cubical rigidity
• $L^p$ coarse Baum-Connes conjecture via $C_0$ coarse geometry