Relatively convex subgroups

DefinitionRight-invariant linear order#✓fully proved

A linear order \(\le\) on a group \(G\) is right-invariant (a right order) when \(a \le b\) implies \(ac \le bc\) for all \(a, b, c \in G\).

def IsRightInvariantOrder {G : Type} [Group G] (o : LinearOrder G) : Prop :=
  ∀ a b : G, o.le a b → ∀ c : G, o.le (a * c) (b * c)

Used by 1

Used byRight-relatively convex subgroup

DefinitionRight-relatively convex subgroup#✓fully proved

A subgroup \(H\) of a group \(G\) is right-relatively convex when there is a right-invariant linear order \(\le\) on \(G\) for which \(H\) is convex: whenever \(h_1 \le g \le h_2\) with \(h_1, h_2 \in H\) and \(g \in G\), also \(g \in H\).

def IsRightRelativelyConvex {G : Type} [Group G] (H : Subgroup G) : Prop :=
  ∃ o : LinearOrder G, IsRightInvariantOrder o ∧
    ∀ h₁ ∈ H, ∀ h₂ ∈ H, ∀ g : G, o.le h₁ g → o.le g h₂ → g ∈ H

Uses 1 · Used by 1

Statement usesRight-invariant linear order

Used bySublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case)

DefinitionSublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case)#✓fully proved

Sublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case). The statement that for every non-abelian right-orderable group \(G\) and all right-relatively convex subgroups \(H\) and \(K\) of \(G\), both the intersection \(H \cap K\) and the join \(H \vee K\) (the subgroup generated by \(H\) and \(K\)) are right-relatively convex; that is, the right-relatively convex subgroups of \(G\) form a sublattice of the lattice of subgroups of \(G\).

The question is restricted to non-abelian groups, as the proposers meant.

def rightRelativelyConvexSublattice : Prop :=
  ∀ (G : Type) [Group G], (∃ a b : G, a * b ≠ b * a) → IsRightOrderable G → ∀ H K : Subgroup G,
    IsRightRelativelyConvex H → IsRightRelativelyConvex K →
      IsRightRelativelyConvex (H ⊓ K) ∧ IsRightRelativelyConvex (H ⊔ K)

Uses 2 · Used by 1

Statement usesRight-orderable groups · Right-relatively convex subgroup

Used byNegative answer to Kourovka Problem 21.147 for non-abelian groups

TheoremNegative answer to Kourovka Problem 21.147 for non-abelian groups#●ready to prove

It is not the case that for every non-abelian right-orderable group \(G\) and all right-relatively convex subgroups \(H\) and \(K\) of \(G\), both the intersection \(H \cap K\) and the join \(H \vee K\) (the subgroup generated by \(H\) and \(K\)) are right-relatively convex.

In the integer Heisenberg group, which is non-abelian, the cyclic subgroups \(\langle (1, 0, 1) \rangle\) and \(\langle (1, 0, -1) \rangle\) are convex for suitable right orders, but the subgroup \(J\) they generate contains \(2e\) and not \(e = (1, 0, 0)\), and every right order puts \(e\) between \(0\) and \(2e\), so \(J\) is convex for no right order (van Doorn, Judin, Monticone and Morrison, Theorem 8.1; Monticone's note). The 21st issue of the Kourovka Notebook marks 21.147 solved ("No, not always"; AI solution confirmed by A. V. Zenkov).

public theorem_wanted rightRelativelyConvexSublattice_refuted : ¬ rightRelativelyConvexSublattice

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 21.147 (V. M. Kopytov, N. Ya. Medvedev; A. V. Zenkov); resolution: P. Monticone, Kourovka Notebook Problem 21.147, solution note (14 May 2026), Kourovka Notebook site; W. van Doorn, E. Judin, P. Monticone and D. Morrison, On some problems from the Kourovka Notebook, preprint, arXiv:2607.17477 (2026), van Doorn-Judin-Monticone-Morrison, Theorem 8.1: in the integer Heisenberg group the subgroups ⟨(1, 0, 1)⟩ and ⟨(1, 0, -1)⟩ are right-relatively convex, while their join contains 2e but not e = (1, 0, 0) and so is not convex for any right order; an AI solution (Aristotle) confirmed by A. V. Zenkov, Kourovka Notebook (21st issue), Problem 21.147: No, not always, https://arxiv.org/abs/2607.17477.

Uses 1

Statement usesSublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case)

3 fully proved1 ready to prove

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