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Corollary 1.8.4 (Chow): the centre of BnB_n is generated by (σ1⋯σn−1)n(\sigma_1\cdots\sigma_{n-1})^n

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BraidsLinksMCG.cor_1_8_4_center_eq

by Gabewhigham · Sep 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-topologybraid-groupsgroup-theory

Corollary 1.8.4 (Chow). For n≥3n \ge 3 the centre of the braid group BnB_n is the cyclic subgroup generated by the full twist

Δ2  =  (σ1σ2⋯σn−1)n.\Delta^2 \;=\; (\sigma_1\sigma_2\cdots\sigma_{n-1})^n .

That is, an element of BnB_n commutes with every braid if and only if it is an integral power of (σ1⋯σn−1)n(\sigma_1\cdots\sigma_{n-1})^n. Geometrically the generator is the braid in which the whole bundle of nn strands is given one full rotation.

This is the equality-of-subgroups half of Corollary 1.8.4; the complementary assertion, that the generator has infinite order, so that the centre is infinite cyclic, follows from the exponent-sum homomorphism.

The result is the reason the centre plays no role in the conjugacy and word problems for BnB_n: quotienting by it loses only the number of full twists. The hypothesis n≥3n \ge 3 is necessary, since B2B_2 is infinite cyclic and equal to its own centre, whereas (σ1)2(\sigma_1)^2 generates a proper subgroup of it.

Formalization Note In the formal statement the generator is written as the nn-th power of the ordered product of all the Artin generators, and the subgroup it generates is the set of its integer powers.

Preamble
import Definitions.Def_BraidsLinksMCG_ArtinBraidGroup
import Definitions.Def_BraidsLinksMCG_ArtinEndo
Formal statement
namespace BraidsLinksMCG

theorem cor_1_8_4_center_eq (n : ℕ) (hn : 3 ≤ n) :
    Subgroup.center (ArtinBraidGroup n) = Subgroup.zpowers (sigmaProd n ^ n) := by sorry

end BraidsLinksMCG
Source
Joan S. Birman, *Braids, Links, and Mapping Class Groups*, Annals of Mathematics Studies 82, Princeton University Press, 1974, Chapter 1, Corollary 1.8.4, p. 28 (due to W.-L. Chow, 1948)

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