Braid groups and Burnside groups

Abstract

There exists an exceptional quotient of braid groups $\text{Br}_4 \twoheadrightarrow \text{Br}_3$ that is related to many interesting constructions in algebra, topology, and geometry. This quotient map also descends to a quotient of "truncated" braid groups $\text{Br}_4(d) \twoheadrightarrow \text{Br}_3(d)$, 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.

Disclosure

“at led to the semidi- rect product description of Proposition 2.3. Thanks to Seraphina Lee, Faye Jackson, and Trent Lucas for discussions about the Lefschetz fibration in- terpretation of these results (as discussed in Section 3). Google’s Gemini assisted drafting Lemma 2.5. 2 Proofs The main tool for proving Theorem 1.2 is a semidirect product decomposition of Br4 coming from a section of ψ. First, we define the relevant action. Write F2 = ⟨x, y⟩ for the free group of rank 2”

PDF page 3

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:53:25.455962Z

Structural counts