Endomorphisms of braid groups

DefinitionArtin's braid relators#✓fully proved

Artin's relators for the braid group on n strands, in the free group on the generators σ₀, …, σ_{n-2} (indexed by Fin (n - 1)): σᵢ σⱼ = σⱼ σᵢ when j ≥ i + 2, and σᵢ σᵢ₊₁ σᵢ = σᵢ₊₁ σᵢ σᵢ₊₁.

def braidRelators (n : ℕ) : Set (FreeGroup (Fin (n - 1))) :=
  {r | ∃ i j : Fin (n - 1),
    (i.val + 2 ≤ j.val ∧ r = of i * of j * (of i)⁻¹ * (of j)⁻¹) ∨
    (i.val + 1 = j.val ∧ r = of i * of j * of i * (of j * of i * of j)⁻¹)}

Used by 4

Used byKourovka Problem 14.102(b): endomorphisms of the braid commutator subgroup · Braid group on n strands · Kourovka Problem 14.102(a): non-injective braid endomorphisms with non-cyclic image · Kourovka Problem 14.102(c): torsion-free quotients of braid groups

AbbreviationBraid group on n strands#✓fully proved

The braid group Bₙ on n strands, by Artin's presentation.

abbrev BraidGroup (n : ℕ) : Type := PresentedGroup (braidRelators n)

def braidRelators (n : ℕ) : Set (FreeGroup (Fin (n - 1))) :=
  {r | ∃ i j : Fin (n - 1),
    (i.val + 2 ≤ j.val ∧ r = of i * of j * (of i)⁻¹ * (of j)⁻¹) ∨
    (i.val + 1 = j.val ∧ r = of i * of j * of i * (of j * of i * of j)⁻¹)}

WantedExt/GroupTheory/BraidGroupEndomorphismsKourovka14Problem102Wanted.lean:45

Uses 1 · Used by 3

Statement usesArtin's braid relators

Used byKourovka Problem 14.102(b): endomorphisms of the braid commutator subgroup · Kourovka Problem 14.102(a): non-injective braid endomorphisms with non-cyclic image · Kourovka Problem 14.102(c): torsion-free quotients of braid groups

DefinitionKourovka Problem 14.102(a): non-injective braid endomorphisms with non-cyclic image#✓fully proved

The statement that for some \(n > 4\) the braid group \(B_n\) on \(n\) strands has a non-injective endomorphism whose image is not cyclic (in particular not trivial). Posed by V. Lin and V. Shpilrain.

def braidNonInjectiveNoncyclicEndomorphism : Prop :=
  ∃ n : ℕ, 4 < n ∧ ∃ φ : BraidGroup n →* BraidGroup n, ¬ Function.Injective φ ∧
    ¬ IsCyclic φ.range

Uses 2 · Used by 1

Statement usesBraid group on n strands · Artin's braid relators

Used byNon-injective endomorphisms of braid groups on five or more strands have cyclic image

DefinitionKourovka Problem 14.102(b): endomorphisms of the braid commutator subgroup#✓fully proved

The statement that for every \(n > 4\), every endomorphism of the commutator subgroup \([B_n, B_n]\) of the braid group \(B_n\) on \(n\) strands other than the trivial endomorphism is bijective, that is, an automorphism. Posed by V. Lin and V. Shpilrain.

def braidCommutatorEndomorphismAutomorphism : Prop :=
  ∀ n : ℕ, 4 < n → ∀ φ : commutator (BraidGroup n) →* commutator (BraidGroup n), φ ≠ 1 →
    Function.Bijective φ

Uses 2 · Used by 1

Statement usesBraid group on n strands · Artin's braid relators

Used byNontrivial endomorphisms of the braid commutator subgroup on five or more strands are automorphisms

DefinitionKourovka Problem 14.102(c): torsion-free quotients of braid groups#✓fully proved

The statement that for every \(n > 4\), the braid group \(B_n\) on \(n\) strands has a nontrivial normal subgroup \(N\) such that the quotient \(B_n / N\) is non-abelian and torsion-free (no element other than the identity has finite order). Posed by V. Lin and V. Shpilrain.

def braidTorsionFreeNonabelianProperQuotient : Prop :=
  ∀ n : ℕ, 4 < n → ∃ (N : Subgroup (BraidGroup n)) (_ : N.Normal), N ≠ ⊥ ∧
    (∃ x y : BraidGroup n ⧸ N, x * y ≠ y * x) ∧
    ∀ x : BraidGroup n ⧸ N, x ≠ 1 → ¬ IsOfFinOrder x

Uses 2 · Used by 1

Statement usesBraid group on n strands · Artin's braid relators

Used byBraid groups on five or more strands have proper non-abelian torsion-free quotients

TheoremNon-injective endomorphisms of braid groups on five or more strands have cyclic image#●ready to prove

The answer to Kourovka Problem 14.102(a) is negative: there is no \(n > 4\) for which the braid group \(B_n\) on \(n\) strands has a non-injective endomorphism whose image is not cyclic. Equivalently, for every \(n \ge 5\), every non-injective endomorphism of \(B_n\) has cyclic image.

This was proved by Castel for \(n \ge 6\) and by Chen, Kordek and Margalit for \(n = 5\); the Kourovka Notebook (21st issue) answers 14.102(a) "No".

public theorem_wanted braidNonInjectiveNoncyclicEndomorphism_refuted :
    ¬ braidNonInjectiveNoncyclicEndomorphism

def braidNonInjectiveNoncyclicEndomorphism : Prop :=
  ∃ n : ℕ, 4 < n ∧ ∃ φ : BraidGroup n →* BraidGroup n, ¬ Function.Injective φ ∧
    ¬ IsCyclic φ.range

WantedExt/GroupTheory/BraidGroupEndomorphismsKourovka14Problem102Wanted.lean:55

Sources

E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.102 (V. Lin; V. Shpilrain); part (c) in the Archive of Solved Problems; resolution: (a) F. Castel, Geometric representations of the braid groups, Astérisque 378 (2016) (n ≥ 6), and L. Chen, K. Kordek and D. Margalit, Homomorphisms between braid groups, arXiv:1910.00712 (n = 5); (b) S. Orevkov, Ann. Fac. Sci. Toulouse Math. 33 (2024), no. 1, 105-121, extending K. Kordek and D. Margalit, Bull. London Math. Soc. 54 (2022), 95-111 (n ≥ 7); (c) P. Linnell and T. Schick, J. Amer. Math. Soc. 20 (2007), no. 4, 1003-1051, the Kourovka Notebook (21st issue) marks 14.102(a), (b) solved (for n ≥ 5 every non-injective endomorphism of B_n has cyclic image; every non-trivial endomorphism of [B_n, B_n] is an automorphism); Kourovka Archive, Problem 14.102(c) (Yes), https://arxiv.org/abs/1910.00712.

Uses 1

Statement usesKourovka Problem 14.102(a): non-injective braid endomorphisms with non-cyclic image

TheoremNontrivial endomorphisms of the braid commutator subgroup on five or more strands are automorphisms#●ready to prove

Nontrivial endomorphisms of the braid commutator subgroup on five or more strands are automorphisms. For every \(n > 4\), every endomorphism of the commutator subgroup \([B_n, B_n]\) of the braid group \(B_n\) on \(n\) strands other than the trivial endomorphism is bijective, that is, an automorphism.

This was proved by Kordek and Margalit for \(n \ge 7\) and extended by Orevkov to all \(n \ge 5\); the Kourovka Notebook (21st issue) answers 14.102(b) "Yes".

public theorem_wanted braidCommutatorEndomorphismAutomorphism_holds :
    braidCommutatorEndomorphismAutomorphism

def braidCommutatorEndomorphismAutomorphism : Prop :=
  ∀ n : ℕ, 4 < n → ∀ φ : commutator (BraidGroup n) →* commutator (BraidGroup n), φ ≠ 1 →
    Function.Bijective φ

WantedExt/GroupTheory/BraidGroupEndomorphismsKourovka14Problem102Wanted.lean:61

Sources

E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.102 (V. Lin; V. Shpilrain); part (c) in the Archive of Solved Problems; resolution: (a) F. Castel, Geometric representations of the braid groups, Astérisque 378 (2016) (n ≥ 6), and L. Chen, K. Kordek and D. Margalit, Homomorphisms between braid groups, arXiv:1910.00712 (n = 5); (b) S. Orevkov, Ann. Fac. Sci. Toulouse Math. 33 (2024), no. 1, 105-121, extending K. Kordek and D. Margalit, Bull. London Math. Soc. 54 (2022), 95-111 (n ≥ 7); (c) P. Linnell and T. Schick, J. Amer. Math. Soc. 20 (2007), no. 4, 1003-1051, the Kourovka Notebook (21st issue) marks 14.102(a), (b) solved (for n ≥ 5 every non-injective endomorphism of B_n has cyclic image; every non-trivial endomorphism of [B_n, B_n] is an automorphism); Kourovka Archive, Problem 14.102(c) (Yes), https://arxiv.org/abs/1910.00712.

Uses 1

Statement usesKourovka Problem 14.102(b): endomorphisms of the braid commutator subgroup

TheoremBraid groups on five or more strands have proper non-abelian torsion-free quotients#●ready to prove

For every \(n > 4\), the braid group \(B_n\) on \(n\) strands has a nontrivial normal subgroup \(N\) such that the quotient \(B_n / N\) is non-abelian and torsion-free (no element other than the identity has finite order).

The Kourovka Archive answers 14.102(c) "Yes, it does", citing Linnell and Schick (every braid group is residually torsion-free nilpotent-by-finite); Humphries had constructed such quotients for \(n < 7\).

public theorem_wanted braidTorsionFreeNonabelianProperQuotient_holds :
    braidTorsionFreeNonabelianProperQuotient

def braidTorsionFreeNonabelianProperQuotient : Prop :=
  ∀ n : ℕ, 4 < n → ∃ (N : Subgroup (BraidGroup n)) (_ : N.Normal), N ≠ ⊥ ∧
    (∃ x y : BraidGroup n ⧸ N, x * y ≠ y * x) ∧
    ∀ x : BraidGroup n ⧸ N, x ≠ 1 → ¬ IsOfFinOrder x

WantedExt/GroupTheory/BraidGroupEndomorphismsKourovka14Problem102Wanted.lean:67

Sources

E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.102 (V. Lin; V. Shpilrain); part (c) in the Archive of Solved Problems; resolution: (a) F. Castel, Geometric representations of the braid groups, Astérisque 378 (2016) (n ≥ 6), and L. Chen, K. Kordek and D. Margalit, Homomorphisms between braid groups, arXiv:1910.00712 (n = 5); (b) S. Orevkov, Ann. Fac. Sci. Toulouse Math. 33 (2024), no. 1, 105-121, extending K. Kordek and D. Margalit, Bull. London Math. Soc. 54 (2022), 95-111 (n ≥ 7); (c) P. Linnell and T. Schick, J. Amer. Math. Soc. 20 (2007), no. 4, 1003-1051, the Kourovka Notebook (21st issue) marks 14.102(a), (b) solved (for n ≥ 5 every non-injective endomorphism of B_n has cyclic image; every non-trivial endomorphism of [B_n, B_n] is an automorphism); Kourovka Archive, Problem 14.102(c) (Yes), https://arxiv.org/abs/1910.00712.

Uses 1

Statement usesKourovka Problem 14.102(c): torsion-free quotients of braid groups

5 fully proved3 ready to prove

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