Pure braid groups

DefinitionPure braid group#✓fully proved

For \(n \in \mathbb{N}\), the pure (coloured) braid group \(P_{n+1}\) on \(n + 1\) strands is the normal closure, in the braid group \(B_{n+1}\), of the squares \(\sigma_i^2\) of the standard generators \(\sigma_1, \dots, \sigma_n\) of \(B_{n+1}\), where \(B_{n+1}\) is the Artin group of the Coxeter matrix of type \(A_n\).

This is the kernel of the map \(B_{n+1} \to S_{n+1}\), since adding the relations \(\sigma_i^2 = 1\) to the braid relations presents the Coxeter group of type \(A_n\), the symmetric group \(S_{n+1}\).

def pureBraidGroup (n : ℕ) : Subgroup (CoxeterMatrix.A n).ArtinGroup :=
  Subgroup.normalClosure (Set.range fun i => CoxeterMatrix.ArtinGroup.of (CoxeterMatrix.A n) i ^ 2)

Uses 3 · Used by 1

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

Used byKourovka 11.54: pure braids conjugate to their inverses

DefinitionKourovka 11.54: pure braids conjugate to their inverses#✓fully proved

The proposition that for every \(n \in \mathbb{N}\), every pure braid \(\beta\) on \(n + 1\) strands that is conjugate to its inverse inside the pure braid group, that is, \(\gamma \beta \gamma^{-1} = \beta^{-1}\) for some pure braid \(\gamma\) on \(n + 1\) strands, is the identity braid.

This is Kourovka Problem 11.54 (G. S. Makanin).

def pureBraidConjInvTrivial : Prop :=
  ∀ n : ℕ, ∀ β ∈ pureBraidGroup n, (∃ γ ∈ pureBraidGroup n, γ * β * γ⁻¹ = β⁻¹) → β = 1

Uses 3 · Used by 1

Statement usesArtin group · Artin relators of a Coxeter matrix · Pure braid group

Used byKourovka 11.54: only the trivial pure braid is conjugate to its inverse

TheoremKourovka 11.54: only the trivial pure braid is conjugate to its inverse#●ready to prove

For every \(n \in \mathbb{N}\) and every pure braid \(\beta\) on \(n + 1\) strands, if \(\gamma \beta \gamma^{-1} = \beta^{-1}\) for some pure braid \(\gamma\) on \(n + 1\) strands, then \(\beta = 1\).

The Kourovka Archive answers 11.54 affirmatively (V. A. Roman'kov): the pure braid group embeds in the IA-automorphism group of a free group, which is residually torsion-free nilpotent, and in a torsion-free nilpotent group an element conjugate to its inverse is trivial.

public theorem_wanted pureBraidConjInvTrivial_holds : pureBraidConjInvTrivial

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 11.54 (G. S. Makanin); resolution: V. A. Roman'kov, letter of 3 October 1990; see also V. G. Bardakov, Russ. Acad. Sci. Sb. Math. 76, no. 1 (1993), 123-153, Kourovka Archive, Problems 11.54 and 11.55: both affirmative, since the coloured braid group embeds in the group of automorphisms of F_n acting trivially modulo F_n', which is residually torsion-free nilpotent, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka 11.54: pure braids conjugate to their inverses

DefinitionPure braid group#✓fully proved

For \(n \in \mathbb{N}\), the pure (coloured) braid group \(P_{n+1}\) on \(n + 1\) strands is the normal closure, in the braid group \(B_{n+1}\), of the squares \(\sigma_i^2\) of the standard generators \(\sigma_1, \dots, \sigma_n\) of \(B_{n+1}\), where \(B_{n+1}\) is the Artin group of the Coxeter matrix of type \(A_n\).

This is the kernel of the map \(B_{n+1} \to S_{n+1}\), since adding the relations \(\sigma_i^2 = 1\) to the braid relations presents the Coxeter group of type \(A_n\), the symmetric group \(S_{n+1}\).

def pureBraidGroup (n : ℕ) : Subgroup (CoxeterMatrix.A n).ArtinGroup :=
  Subgroup.normalClosure (Set.range fun i => CoxeterMatrix.ArtinGroup.of (CoxeterMatrix.A n) i ^ 2)

Uses 3 · Used by 1

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

Used byKourovka 11.55: unique roots in pure braid groups

DefinitionKourovka 11.55: unique roots in pure braid groups#✓fully proved

The proposition that for every \(n \in \mathbb{N}\) and every integer \(k \ge 1\), if pure braids \(\alpha\) and \(\beta\) on \(n + 1\) strands satisfy \(\alpha^k = \beta^k\), then \(\alpha = \beta\).

This is Kourovka Problem 11.55 (G. S. Makanin): extraction of roots in the group of coloured braids is unique.

def pureBraidUniqueRoots : Prop :=
  ∀ n k : ℕ, 0 < k → ∀ α ∈ pureBraidGroup n, ∀ β ∈ pureBraidGroup n, α ^ k = β ^ k → α = β

Uses 3 · Used by 1

Statement usesArtin group · Artin relators of a Coxeter matrix · Pure braid group

Used byKourovka 11.55: roots in pure braid groups are unique

TheoremKourovka 11.55: roots in pure braid groups are unique#●ready to prove

For every \(n \in \mathbb{N}\) and every integer \(k \ge 1\), if pure braids \(\alpha\) and \(\beta\) on \(n + 1\) strands satisfy \(\alpha^k = \beta^k\), then \(\alpha = \beta\).

The Kourovka Archive answers 11.55 affirmatively (V. A. Roman'kov): the pure braid group embeds in the IA-automorphism group of a free group, which is residually torsion-free nilpotent, and torsion-free nilpotent groups have unique roots.

public theorem_wanted pureBraidUniqueRoots_holds : pureBraidUniqueRoots

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 11.55 (G. S. Makanin); resolution: V. A. Roman'kov, letter of 3 October 1990; see also V. G. Bardakov, Russ. Acad. Sci. Sb. Math. 76, no. 1 (1993), 123-153, Kourovka Archive, Problems 11.54 and 11.55: both affirmative, since the coloured braid group embeds in the group of automorphisms of F_n acting trivially modulo F_n', which is residually torsion-free nilpotent, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka 11.55: unique roots in pure braid groups

4 fully proved2 ready to prove

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