关于CAT(0)立方复形嵌入到树积中的研究
On embeddings of CAT(0) cube complexes into products of trees
AI总结：
本文证明了二维CAT(0)立方复形的最大度数Δ的接触图可被最多MΔ^26种颜色着色，从而证明该复形可嵌入到有限个树的积中，同时给出了无法嵌入到有限树积中的五维立方复形的例子。
摘要：
我们证明了二维CAT(0)立方复形X的最大度数Δ的接触图可以用至多ε(Δ)=MΔ^{26}种颜色着色，其中M是一个固定常数。这表明X（及其关联的中位图）可以等距嵌入到最多ε(Δ)个树的积中，并且该事件结构在其域X上可以有良好的标签。另一方面，我们给出了一个五维CAT(0)立方复形的例子，其0立方的度数有界，但无法嵌入到有限个树的积中。这回答了F.Haglund、G.Niblo、M.Sageev和本文第一作者之前提出的问题。
英文摘要：
We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $Δ$ can be coloured with at most $ε(Δ)=MΔ^{26}$ 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 $ε(Δ)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $ε(Δ)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
