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

TL;DR

This paper demonstrates that 2-dimensional CAT(0) cube complexes with bounded degree can be embedded into products of trees with a bounded number of factors, but provides a counterexample in higher dimensions.

Contribution

It establishes bounds on embeddings of 2D CAT(0) cube complexes into products of trees and presents a higher-dimensional counterexample.

Findings

Contact graph of 2D CAT(0) cube complex is colorable with at most MΔ^{26} colors.

Such complexes embed isometrically into a product of at most MΔ^{26} trees.

Counterexample shows some 5D CAT(0) complexes cannot embed into finite products of trees.

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.