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arXiv · 1107.0863

On embeddings of CAT(0) cube complexes into products of trees

Victor Chepoi·Mark F. Hagen
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Abstract

We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $\Delta$ can be coloured with at most $\epsilon(\Delta)=M\Delta^{26}$ colours, for a fixed constant $M$. This implies that ${\bf X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $\epsilon(\Delta)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $\epsilon(\Delta)$ 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.

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Victor Chepoi, Mark F. Hagen. 2011-07-05. On embeddings of CAT(0) cube complexes into products of trees. https://doi.org/10.1016/j.jctb.2013.04.003

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