arXiv 2011.017322020-11math.COmath.GRmath.MG

可被有效排序的空间： virtually free groups 与双曲性

Spaces that can be ordered effectively: virtually free groups and hyperbolicity

Anna Erschler, Ivan Mitrofanov

备注：: 33 pages. This is the first part of the paper, which appeared on arxiv in previous versions. The paper is split in two parts, and the second one appears now as "Assouad-Nagata dimension and gap for ordered metric spaces"

AI总结：

本文基于旅行商问题研究度量空间的渐近不变量，通过排序效率对群和空间进行分类，证明了 virtually free groups 可由 $4$ 点子集上的排序效率刻画，且所有 $δ$-hyperbolic spaces 在点数趋于无穷时可被极其高效地排序。

摘要：

我们研究度量空间的渐近不变量，这些不变量基于旅行商问题定义，我们的目标是根据群和空间在此背景下能被排序的程度对其进行分类。我们将 virtually free groups 刻画为那些允许在 $4$ 点子集上具有一定效率的排序的群。我们证明，对于子集点数趋于 $\infty$ 的情况，所有 $δ$-hyperbolic spaces 都能被极其高效地排序。

英文摘要：

We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on $4$-point subsets. We show that all $δ$-hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to $\infty$.

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。