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In this paper we prove that the asymptotic dimension of a finite-dimensionalCAT(0) cube complex is bounded above by the dimension. To achieve this we provea 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 anexample of the dimension theorem we obtain bounds on the asymptotic dimensionof small cancellation groups.

Finite asymptotic dimension for CAT(0) cube complexes