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

Abstract

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

Previous version had an error in Lemma 12, affecting Theorem 1. Current version has appendix correcting Theorem 1 under additional hypothesis: no vertex has a 5-cycle in its link or, equivalently, the crossing graph has no 5-cycle. (4-cycles, and cycles larger than 5, are allowed.) Theorem 2, is unchanged. Appendix appears as journal correction: https://doi.org/10.1016/j.jctb.2026.04.001