On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

Authors:
Victor Chepoi;Mark F. Hagen
Affiliations:
Laboratoire dInformatique Fondamentale, Université dAix-Marseille, Faculté des Sciences de Luminy, F-13288 Marseille Cedex 9, France;Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2013

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Abstract

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree @D can be coloured with at most @e(@D)=M@D^2^6 colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most @e(@D) trees, and that the event structure whose domain is X admits a nice labelling with @e(@D) labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.