论文标题
当下部中央系列停止时：对辫子及其亲戚的全面研究
When the lower central series stops: a comprehensive study for braid groups and their relatives
论文作者
论文摘要
一般而言，了解小组的较低中央系列是一项艰巨的任务。但是，这是一个有意义的一个：计算一个群体或其某些子组的相关代数的较低中央序列和相关的谎言代数可能会深入了解该组的基础结构。我们的目标是展示旨在执行部分任务的几种技术。特别是，我们试图回答以下问题：下部中央系列何时停止？我们介绍了许多工具，然后将其应用于与辫子组相关的各个组：辫子组本身，表面辫子组，虚拟和焊接辫子组以及所有这些组的分区版本。从我们的一般技术到其应用的道路远非直接的技术，并且需要一些敏锐和韧性来处理沿途遇到的所有情况。尽管如此，我们为这些小组中的每个小组提供了一个答案，除了在投影飞机上划分的辫子小组。在某些情况下，我们甚至完全计算了下部中央系列。还包括有关下部中央系列Artin组的一些结果。
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.
