When the lower central series stops: a comprehensive study for braid groups and their relatives
with Jacques Darné and Arthur Soulié

To appear in the Memoirs of the American Mathematical Society
arXiv: abs — pdf v3 — v2 — v1

Abstract

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.

GAP code

We made some conjectures in the article about the few remaining unknown (or known-only-up-to-an-ambiguity) cases. These were based on experimental calculations of lower central series of certain finitely-presented groups using GAP and the NQ package. The code used for these is below (and also available here):

For the (1,m)-th partitioned braid groups on the projective plane – B1mP.g

For the (2,m)-th partitioned braid groups on the projective plane – B2mP.g

For the (2,m)-th partitioned braid groups on the sphere – B2mS.g

A looped version of the code for the (2,m)-th partitioned braid groups on the projective plane is here – B2mP-for-small-m.g – and verifies that the lower central series has length at least 100 for all values of m up to 1024. Another looped version is here – B2mP-for-powers-of-2.g – and verifies the same for all values of m that are powers of 2, up to 2^{23}.