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

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

2011-07-05

arxiv.org/abs/1107.0863v2

Mathematics

Combinatorics

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Scientific paper

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 $\epsilon(\Delta)=M\Delta^{15}$ 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 M. Sageev and the first author of this paper.

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