Free subgroups of one-relator groups

DefinitionKourovka Problem 20.41: one-relator groups with free infinite-index subgroups#✓fully proved

The proposition that every non-cyclic residually finite group \(G\) such that every subgroup of finite index in \(G\) (including \(G\) itself) is isomorphic to a one-relator group \(\langle x_0, \dots, x_{n-1} \mid r \rangle\) for some natural number \(n\) and some element \(r\) of the free group on \(x_0, \dots, x_{n-1}\), and every subgroup of infinite index in \(G\) is free, is free, or is isomorphic to \(\langle a_0, b_0, \dots, a_{g-1}, b_{g-1} \mid [a_0, b_0] \cdots [a_{g-1}, b_{g-1}] \rangle\) for some \(g \ge 0\), where \([a, b] = a b a^{-1} b^{-1}\), or is isomorphic to \(\langle c_0, \dots, c_{k-1} \mid c_0^2 \cdots c_{k-1}^2 \rangle\) for some \(k \ge 1\). The problem is due to L. Ciobanu, B. Fine and G. Rosenberger.

These presentations are the fundamental groups of the closed orientable surface of genus \(g\) and the closed non-orientable surface of genus \(k\); compact surfaces with boundary have free fundamental groups.

def oneRelatorInfiniteIndexFreeIsFreeOrSurface : Prop :=
  ∀ (G : Type) [Group G], ¬ IsCyclic G → Group.ResiduallyFinite G →
    (∀ H : Subgroup G, H.FiniteIndex →
      ∃ (n : ℕ) (r : FreeGroup (Fin n)), Nonempty (H ≃* PresentedGroup {r})) →
    (∀ H : Subgroup G, H.index = 0 → IsFreeGroup H) →
    IsFreeGroup G ∨
      (∃ g : ℕ, Nonempty (G ≃* PresentedGroup
        {((List.finRange g).map fun i => FreeGroup.of (i, false) * FreeGroup.of (i, true) *
          (FreeGroup.of (i, false))⁻¹ * (FreeGroup.of (i, true))⁻¹).prod})) ∨
      ∃ k : ℕ, 1 ≤ k ∧ Nonempty (G ≃* PresentedGroup
        {((List.finRange k).map fun i => FreeGroup.of i ^ 2).prod})

Used by 1

Used byKourovka Problem 20.41 has a positive answer

TheoremKourovka Problem 20.41 has a positive answer#●ready to prove

Let \(G\) be a non-cyclic residually finite group such that every subgroup of finite index in \(G\) (including \(G\) itself) is isomorphic to a one-relator group \(\langle x_0, \dots, x_{n-1} \mid r \rangle\) for some natural number \(n\) and some element \(r\) of the free group on \(x_0, \dots, x_{n-1}\), and every subgroup of infinite index in \(G\) is free. Then \(G\) is free, or \(G\) is isomorphic to \(\langle a_0, b_0, \dots, a_{g-1}, b_{g-1} \mid [a_0, b_0] \cdots [a_{g-1}, b_{g-1}] \rangle\) for some \(g \ge 0\), where \([a, b] = a b a^{-1} b^{-1}\), or \(G\) is isomorphic to \(\langle c_0, \dots, c_{k-1} \mid c_0^2 \cdots c_{k-1}^2 \rangle\) for some \(k \ge 1\).

These presentations are the fundamental groups of the closed orientable surface of genus \(g\) and the closed non-orientable surface of genus \(k\); compact surfaces with boundary have free fundamental groups.

public theorem_wanted oneRelatorInfiniteIndexFreeIsFreeOrSurface_holds :
    oneRelatorInfiniteIndexFreeIsFreeOrSurface

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.41 (L. Ciobanu, B. Fine, G. Rosenberger); resolution: H. Wilton, Surface groups among cubulated hyperbolic and one-relator groups, preprint, arXiv:2406.02121 (2024, v3 2026), Theorem D: an infinite one-relator group that is neither free nor a surface group has a subgroup of infinite index that is not free (a non-cyclic one-relator group has at least two generators and is infinite); Kourovka Notebook (21st issue), Problem 20.41: Yes, it is true, https://arxiv.org/abs/2406.02121.

Uses 1

Statement usesKourovka Problem 20.41: one-relator groups with free infinite-index subgroups

DefinitionOne-relator groups with finitely many generators#✓fully proved

A group \(G\) is a one-relator group if, for some \(n \ge 0\) and some element \(r\) of the free group on \(x_1, \dots, x_n\), \(G\) is isomorphic to the group \(\langle x_1, \dots, x_n \mid r \rangle\) with generators \(x_1, \dots, x_n\) and the single relator \(r\).

def IsOneRelatorGroup (G : Type) [Group G] : Prop :=
  ∃ (n : ℕ) (r : FreeGroup (Fin n)), Nonempty (G ≃* PresentedGroup ({r} : Set (FreeGroup (Fin n))))

Used by 1

Used byKourovka Problem 20.22 (statement)

DefinitionOrientable surface relator#✓fully proved

For \(g \ge 0\), the orientable surface relator of genus \(g\) is the element \([a_1, b_1] [a_2, b_2] \cdots [a_g, b_g]\) of the free group on the \(2g\) generators \(a_1, \dots, a_g, b_1, \dots, b_g\), where \([a, b] = a b a^{-1} b^{-1}\); for \(g = 0\) it is the identity.

def orientableSurfaceRelator (g : ℕ) : FreeGroup (Fin g ⊕ Fin g) :=
  (List.ofFn fun i : Fin g => FreeGroup.of (Sum.inl i) * FreeGroup.of (Sum.inr i) *
    (FreeGroup.of (Sum.inl i))⁻¹ * (FreeGroup.of (Sum.inr i))⁻¹).prod

Used by 1

Used bySurface groups

DefinitionNon-orientable surface relator#✓fully proved

For \(k \ge 0\), the non-orientable surface relator of genus \(k\) is the element \(a_1^2 a_2^2 \cdots a_k^2\) of the free group on the \(k\) generators \(a_1, \dots, a_k\); for \(k = 0\) it is the identity.

def nonorientableSurfaceRelator (k : ℕ) : FreeGroup (Fin k) :=
  (List.ofFn fun i : Fin k => FreeGroup.of i * FreeGroup.of i).prod

Used by 1

Used bySurface groups

DefinitionSurface groups#✓fully proved

A group \(G\) is a surface group if either \(G\) is isomorphic to \(\langle a_1, b_1, \dots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle\) for some \(g \ge 1\), where \([a, b] = a b a^{-1} b^{-1}\), or \(G\) is isomorphic to \(\langle a_1, \dots, a_k \mid a_1^2 \cdots a_k^2 \rangle\) for some \(k \ge 2\).

These are the standard one-relator presentations of the fundamental groups of the closed orientable surfaces of genus \(g \ge 1\) and the closed non-orientable surfaces of genus \(k \ge 2\), that is, of the closed aspherical surfaces.

def IsSurfaceGroup (G : Type) [Group G] : Prop :=
  (∃ g : ℕ, 1 ≤ g ∧ Nonempty (G ≃* PresentedGroup {orientableSurfaceRelator g})) ∨
    ∃ k : ℕ, 2 ≤ k ∧ Nonempty (G ≃* PresentedGroup {nonorientableSurfaceRelator k})

Uses 2 · Used by 1

Statement usesNon-orientable surface relator · Orientable surface relator

Used byKourovka Problem 20.22 (statement)

DefinitionKourovka Problem 20.22 (statement)#✓fully proved

The statement that every group \(G\) with the following properties is a surface group: \(G\) is isomorphic to \(\langle x_1, \dots, x_n \mid r \rangle\) for some \(n \ge 0\) and some element \(r\) of the free group on \(x_1, \dots, x_n\); \(G\) is neither free nor cyclic; and every subgroup of infinite index in \(G\) is free. Here \(G\) is a surface group if it is isomorphic to \(\langle a_1, b_1, \dots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle\) for some \(g \ge 1\), where \([a, b] = a b a^{-1} b^{-1}\), or to \(\langle a_1, \dots, a_k \mid a_1^2 \cdots a_k^2 \rangle\) for some \(k \ge 2\).

def oneRelatorInfiniteIndexFreeIsSurfaceGroup : Prop :=
  ∀ (G : Type) [Group G], IsOneRelatorGroup G → ¬ IsFreeGroup G → ¬ IsCyclic G →
    (∀ H : Subgroup G, H.index = 0 → IsFreeGroup H) → IsSurfaceGroup G

Uses 2 · Used by 1

Statement usesOne-relator groups with finitely many generators · Surface groups

Used byWilton's theorem (Kourovka Problem 20.22)

TheoremWilton's theorem (Kourovka Problem 20.22)#●ready to prove

Let \(G\) be a group isomorphic to \(\langle x_1, \dots, x_n \mid r \rangle\) for some \(n \ge 0\) and some element \(r\) of the free group on \(x_1, \dots, x_n\). If \(G\) is neither free nor cyclic and every subgroup of infinite index in \(G\) is free, then \(G\) is a surface group: \(G\) is isomorphic to \(\langle a_1, b_1, \dots, a_g, b_g \mid [a_1, b_1] \cdots [a_g, b_g] \rangle\) for some \(g \ge 1\), where \([a, b] = a b a^{-1} b^{-1}\), or to \(\langle a_1, \dots, a_k \mid a_1^2 \cdots a_k^2 \rangle\) for some \(k \ge 2\).

public theorem_wanted oneRelatorInfiniteIndexFreeIsSurfaceGroup_holds :
    oneRelatorInfiniteIndexFreeIsSurfaceGroup

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.22 (B. Fine, G. Rosenberger, L. Wienke); resolution: H. Wilton, Surface groups among cubulated hyperbolic and one-relator groups, preprint, arXiv:2406.02121 (2024), Theorem D: an infinite one-relator group that is neither free nor a surface group has a subgroup of infinite index that is not free; the Kourovka Notebook (21st issue) marks 20.22 solved, https://arxiv.org/abs/2406.02121.

Uses 1

Statement usesKourovka Problem 20.22 (statement)

6 fully proved2 ready to prove

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