#### 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
##### Publication
Received: 12 July 2010
Revised: 12 January 2012
Accepted: 23 September 2011
Published: 8 April 2012
Proposed: Martin Bridson
Seconded: Dmitri Burago, Steve Ferry
##### Authors
 Nick Wright Mathematics University of Southampton University Road Southampton SO17 1BJ UK