Abstract
We prove that the asymptotic dimension of a finite-dimensional 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 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.
Citation
Nick Wright. "Finite asymptotic dimension for $\mathrm{CAT}(0)$ cube complexes." Geom. Topol. 16 (1) 527 - 554, 2012. https://doi.org/10.2140/gt.2012.16.527
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