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Understanding the lower central series of a group is, in general, a difficulttask. It is, however, a rewarding one: computing the lower central series andthe associated Lie algebras of a group or of some of its subgroups can lead toa deep understanding of the underlying structure of that group. Our goal hereis to showcase several techniques aimed at carrying out part of this task. Inparticular, we seek to answer the following question: when does the lowercentral series stop? We introduce a number of tools that we then apply tovarious groups related to braid groups: the braid groups themselves, surfacebraid groups, groups of virtual and welded braids, and partitioned versions ofall of these groups. The path from our general techniques to their applicationis far from being a straight one, and some astuteness and tenacity is requiredto deal with all of the cases encountered along the way. Nevertheless, wearrive 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. Inseveral cases, we even compute completely the lower central series. Someresults about the lower central series of Artin groups are also included.

When the lower central series stops: a comprehensive study for braid groups and their relatives