Braid groups and Burnside groups

Ethan Dlugie

group theory

Braid groups and Burnside groups

arXiv:2607.13316

summary

The paper examines a special quotient from the 4‑strand braid group to the 3‑strand braid group, extends this to truncated braid groups with torsion relations, and describes the kernel as a primitive Burnside group, giving finiteness results and a geometric interpretation using Lefschetz fibrations.

Abstract

There exists an exceptional quotient of braid groups $Br_{4} ↠ Br_{3}$ that is related to many interesting constructions in algebra, topology, and geometry. This quotient map also descends to a quotient of "truncated" braid groups $Br_{4} (d) ↠ Br_{3} (d)$ , which have an added torsion relation on their half twist generators. In this article, we find a presentation for the kernel of this truncated quotient map that takes the form of what we deem a "primitive" Burnside group. We give a few finiteness results on these primitive Burnside groups. Our methods are purely group theoretic, but we comment on an interpretation involving Lefschetz fibrations at the end.

10 pages, 2 figures

Topics & keywords

#braid groups #burnside groups #truncated braid groups #group presentations #geometric topologybraid group quotientprimitive Burnside groupkernel presentationtorsion relationLefschetz fibration