Conjugators in braid groups

DefinitionThe braid represented by a braid word#✓fully proved

For \(n \in \mathbb{N}\), let \(B_{n+1}\) be the braid group on \(n + 1\) strands, realized as the Artin group of the Coxeter matrix of type \(A_n\), with standard generators \(\sigma_1, \dots, \sigma_n\). A braid word is an element of the free group on \(\sigma_1, \dots, \sigma_n\); the braid it represents is its image under the group homomorphism from this free group to \(B_{n+1}\) that sends each \(\sigma_i\) to the generator \(\sigma_i\) of \(B_{n+1}\).

def braidOf (n : ℕ) : FreeGroup (Fin n) →* (CoxeterMatrix.A n).ArtinGroup :=
  FreeGroup.lift (CoxeterMatrix.ArtinGroup.of (CoxeterMatrix.A n))

Uses 3 · Used by 1

Statement usesArtin group · Generators of the Artin group · Artin relators of a Coxeter matrix

Used byPolynomially short conjugators in braid groups

DefinitionPolynomially short conjugators in braid groups#✓fully proved

The proposition (the stronger question of Kourovka Problem 19.81, V. Shpilrain) that for every \(n \in \mathbb{N}\) there are natural numbers \(C\) and \(d\) with the following property. Let \(B_{n+1}\) be the braid group on \(n + 1\) strands, realized as the Artin group of the Coxeter matrix of type \(A_n\), with standard generators \(\sigma_1, \dots, \sigma_n\); a braid word is an element of the free group on \(\sigma_1, \dots, \sigma_n\), and its length \(|u|\) is the length of its freely reduced form. For all braid words \(u\) and \(v\) whose braids are conjugate in \(B_{n+1}\) there is a braid word \(w\) with \(|w| \le C (\max(|u|, |v|) + 1)^d\) such that \(w u w^{-1}\) and \(v\) represent the same braid.

def braidPolynomialConjugatorLength : Prop :=
  ∀ n : ℕ, ∃ C d : ℕ, ∀ u v : FreeGroup (Fin n), IsConj (braidOf n u) (braidOf n v) →
    ∃ w : FreeGroup (Fin n), w.norm ≤ C * (max u.norm v.norm + 1) ^ d ∧
      braidOf n (w * u * w⁻¹) = braidOf n v

Uses 3 · Used by 1

Statement usesArtin group · Artin relators of a Coxeter matrix · The braid represented by a braid word

Used byBraid groups have polynomially short conjugators

TheoremBraid groups have polynomially short conjugators#●ready to prove

For \(n \in \mathbb{N}\), let \(B_{n+1}\) be the braid group on \(n + 1\) strands, realized as the Artin group of the Coxeter matrix of type \(A_n\), with standard generators \(\sigma_1, \dots, \sigma_n\). A braid word is an element of the free group on \(\sigma_1, \dots, \sigma_n\), and its length \(|u|\) is the length of its freely reduced form. For every \(n\) there are natural numbers \(C\) and \(d\) such that for all braid words \(u\) and \(v\) whose braids are conjugate in \(B_{n+1}\) there is a braid word \(w\) with \(|w| \le C (\max(|u|, |v|) + 1)^d\) such that \(w u w^{-1}\) and \(v\) represent the same braid.

This answers the stronger question of Kourovka Problem 19.81 positively: the Kourovka Archive answers "Yes, there is such a conjugator, even for any mapping class groups", citing Tao's linearly bounded conjugator property for mapping class groups (Theorem B: a conjugator of length at most \(K(|f| + |g|)\)).

public theorem_wanted braidPolynomialConjugatorLength_holds : braidPolynomialConjugatorLength

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 19.81 (V. Shpilrain); resolution: J. Tao, Linearly bounded conjugator property for mapping class groups, Geom. Funct. Anal. 23 (2013), no. 1, 415-466 (arXiv:1106.2341), Theorem B: for a finite generating set of the mapping class group of a surface of finite type there is K such that conjugate f, g have a conjugator ω with |ω| ≤ K(|f| + |g|); Kourovka Archive, Problem 19.81: Yes, there is such a conjugator, even for any mapping class groups, https://arxiv.org/abs/1106.2341.

Uses 1

Statement usesPolynomially short conjugators in braid groups

2 fully proved1 ready to prove

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