当下中心列停止时：辫群及其相关群的全面研究
When the lower central series stops: a comprehensive study for braid groups and their relatives
AI总结：
本文研究群的下中心列何时停止的问题，引入多种工具并应用于辫群及其相关群，除射影平面上的一族分割辫群外均给出答案，部分情形完整计算了下中心列。
摘要：
理解一个群的下中心列通常是一项困难的任务。然而，这是一项有回报的工作：计算一个群或其某些子群的下中心列以及相关的李代数，可以深刻理解该群的底层结构。我们的目标是展示几种旨在执行此任务部分工作的技术。具体而言，我们试图回答以下问题：下中心列何时停止？我们引入了若干工具，随后将其应用于与辫群相关的各种群：辫群本身、曲面辫群、虚拟辫群与焊接辫群，以及所有这些群的分割版本。从通用技术到其应用的路径绝非一条直线，需要一定的机敏与坚韧才能处理沿途遇到的所有情况。尽管如此，除射影平面上的一族分割辫群外，我们对每一个群都给出了问题的答案。在若干情形中，我们甚至完整计算了下中心列。此外，还包含了关于Artin群下中心列的一些结果。
英文摘要：
Understanding the lower central series of a group is, in general, a difficult task. It is, however, a rewarding one: computing the lower central series and the associated Lie algebras of a group or of some of its subgroups can lead to a deep understanding of the underlying structure of that group. Our goal here is to showcase several techniques aimed at carrying out part of this task. In particular, we seek to answer the following question: when does the lower central series stop? We introduce a number of tools that we then apply to various groups related to braid groups: the braid groups themselves, surface braid groups, groups of virtual and welded braids, and partitioned versions of all of these groups. The path from our general techniques to their application is far from being a straight one, and some astuteness and tenacity is required to deal with all of the cases encountered along the way. Nevertheless, we arrive at an answer to our question for each and every one of these groups, save for one family of partitioned braid groups on the projective plane. In several cases, we even compute completely the lower central series. Some results about the lower central series of Artin groups are also included.
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。
