关于CAT(0)立方复形嵌入树乘积的研究
度量几何
2026-04-23 v4 组合数学
摘要
我们证明了最大度为的二维CAT(0)立方复形的接触图可以用至多种颜色着色，其中为固定常数。这意味着（以及相关的中间图）可以等距嵌入至多棵树的笛卡尔积中，并且以为域的事件结构允许一个具有个标签的优美标号。另一方面，我们给出了一个5维CAT(0)立方复形的例子，其0-立方体具有一致有界度数，但无法嵌入有限棵树的笛卡尔积中。这否定了F. Haglund、G. Niblo、M. Sageev以及本文第一作者独立提出的一个问题。
引用
@article{arxiv.1107.0863,
title = {On embeddings of CAT(0) cube complexes into products of trees},
author = {Victor Chepoi and Mark F. Hagen},
journal= {arXiv preprint arXiv:1107.0863},
year = {2026}
}
备注
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