#### Volume 16, issue 1 (2012)

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

### Nick Wright

Geometry & Topology 16 (2012) 527–554
##### 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.

##### Keywords
asymptotic dimension, $\mathrm{CAT}(0)$ cube complex, small cancellation group
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F69, 54F45