Roots in braid groups

DefinitionArtin's braid relators#✓fully proved

For a natural number \(n\), the set of Artin's relators for the braid group on \(n\) strands, as elements of the free group on the \(n - 1\) generators \(\sigma_0, \dots, \sigma_{n-2}\) (no generators when \(n \le 1\)): the commutators \(\sigma_i \sigma_j \sigma_i^{-1} \sigma_j^{-1}\) for \(0 \le i\) and \(i + 2 \le j \le n - 2\), and the words \(\sigma_i \sigma_{i+1} \sigma_i (\sigma_{i+1} \sigma_i \sigma_{i+1})^{-1}\) for \(0 \le i \le n - 3\).

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 2

Used byThe braid group \(B_n\) by Artin's presentation · Makanin's root-uniqueness question for braid groups

AbbreviationThe braid group \(B_n\) by Artin's presentation#✓fully proved

For a natural number \(n\), the braid group \(B_n\) on \(n\) strands is the group generated by \(\sigma_0, \dots, \sigma_{n-2}\) subject to the relations \(\sigma_i \sigma_j = \sigma_j \sigma_i\) for \(j \ge i + 2\) and \(\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}\) for \(0 \le i \le n - 3\); it is the trivial group when \(n \le 1\).

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

Uses 1 · Used by 1

Statement usesArtin's braid relators

Used byMakanin's root-uniqueness question for braid groups

DefinitionMakanin's root-uniqueness question for braid groups#✓fully proved

The statement that for every natural number \(n\) and every integer \(k \ge 1\), any two elements \(\alpha, \beta\) of the braid group \(B_n\) on \(n\) strands with \(\alpha^k = \beta^k\) are conjugate in \(B_n\). This is Kourovka Problem 10.23 (G. S. Makanin).

def braidRootsUniqueUpToConjugacy : Prop :=
  ∀ (n k : ℕ), 0 < k → ∀ α β : BraidGroup n, α ^ k = β ^ k → IsConj α β

Uses 2 · Used by 1

Statement usesThe braid group \(B_n\) by Artin's presentation · Artin's braid relators

Used byRoots in braid groups are unique up to conjugacy (González-Meneses)

TheoremRoots in braid groups are unique up to conjugacy (González-Meneses)#●ready to prove

For every natural number \(n\) and every integer \(k \ge 1\), if two elements \(\alpha, \beta\) of the braid group \(B_n\) on \(n\) strands satisfy \(\alpha^k = \beta^k\), then \(\alpha\) and \(\beta\) are conjugate in \(B_n\). This answers Kourovka Problem 10.23 (G. S. Makanin) affirmatively.

public theorem_wanted braidRootsUniqueUpToConjugacy_holds : braidRootsUniqueUpToConjugacy

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 10.23 (G. S. Makanin); resolution: J. González-Meneses, The nth root of a braid is unique up to conjugacy, Algebr. Geom. Topol. 3 (2003), 1103-1118, main theorem; Kourovka Archive, Problem 10.23 ("Yes, it is true"), https://doi.org/10.2140/agt.2003.3.1103.

Uses 1

Statement usesMakanin's root-uniqueness question for braid groups

3 fully proved1 ready to prove

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