Automorphisms of categories of relatively free groups

AbbreviationFree group of a variety on a finite set#✓fully proved

Let \(W\) be a set of words in the free group \(F(\mathbb{N})\) on \(x_0, x_1, \ldots\), defining a variety \(\Theta\), and let \(X \subseteq \mathbb{N}\) be finite. The free group \(F_\Theta(X)\) of \(\Theta\) on \(X\) is the quotient of the free group \(F(X)\) on \(X\) by its verbal subgroup, the normal closure of the set of values \(\varphi(w)\) with \(w \in W\) and \(\varphi \colon F(\mathbb{N}) \to F(X)\) a homomorphism. This set of values is closed under conjugation, so the verbal subgroup is also the subgroup it generates.

abbrev relativelyFreeGroup (W : Set (FreeGroup ℕ)) (X : Finset ℕ) : Type :=
  FreeGroup X ⧸ Subgroup.normalClosure {g | ∃ w ∈ W, ∃ φ : FreeGroup ℕ →* FreeGroup X, φ w = g}

Used by 2

Used byPlotkin's question on outer automorphisms of the category of free groups of a variety · Plotkin's category of free groups of a variety

AbbreviationPlotkin's category of free groups of a variety#✓fully proved

Let \(W\) be a set of words in the free group on \(x_0, x_1, \ldots\), defining a variety \(\Theta\). The category \(\Theta^0\) has one object for each finite subset \(X\) of \(X^0 = \mathbb{N}\), namely the free group \(F_\Theta(X)\) of \(\Theta\) on \(X\), and its morphisms are all group homomorphisms between these groups.

abbrev ThetaZero (W : Set (FreeGroup ℕ)) : Type :=
  InducedCategory GrpCat fun X : Finset ℕ => GrpCat.of (relativelyFreeGroup W X)

Uses 1 · Used by 1

Statement usesFree group of a variety on a finite set

Used byPlotkin's question on outer automorphisms of the category of free groups of a variety

DefinitionPlotkin's question on outer automorphisms of the category of free groups of a variety#✓fully proved

The statement that there is a set \(W\) of words in the free group on \(x_0, x_1, \ldots\) such that, for the variety \(\Theta\) with laws \(W\), the category \(\Theta^0\) of the free groups of \(\Theta\) on the finite subsets of \(\mathbb{N}\) has an automorphism that is not inner: functors \(\Phi, \Psi \colon \Theta^0 \to \Theta^0\) whose composites \(\Psi \circ \Phi\) and \(\Phi \circ \Psi\) both equal the identity functor, with \(\Phi\) not naturally isomorphic to the identity functor. This is Kourovka Notebook Problem 20.72 (E. Plotkin).

def existsVarietyWithNonInnerCategoryAutomorphism : Prop :=
  ∃ (W : Set (FreeGroup ℕ)) (Φ Ψ : ThetaZero W ⥤ ThetaZero W),
    Φ ⋙ Ψ = 𝟭 _ ∧ Ψ ⋙ Φ = 𝟭 _ ∧ IsEmpty (Φ ≅ 𝟭 _)

Uses 2 · Used by 1

Statement usesPlotkin's category of free groups of a variety · Free group of a variety on a finite set

Used bySome variety of groups has an outer automorphism of its category of free groups

TheoremSome variety of groups has an outer automorphism of its category of free groups#●ready to prove

There is a set \(W\) of words in the free group on \(x_0, x_1, \ldots\) such that, for the variety \(\Theta\) with laws \(W\), the category \(\Theta^0\), whose objects are the free groups \(F_\Theta(X)\) of \(\Theta\) on the finite subsets \(X \subseteq \mathbb{N}\) and whose morphisms are all group homomorphisms, has an automorphism that is not inner: there are functors \(\Phi, \Psi \colon \Theta^0 \to \Theta^0\) whose composites \(\Psi \circ \Phi\) and \(\Phi \circ \Psi\) both equal the identity functor, and \(\Phi\) is not naturally isomorphic to the identity functor.

Barbosa Fernandes and Tsurkov (Theorem 9.1) show that for the variety of metabelian groups of exponent \(4\) and nilpotency class at most \(4\) (laws \(x^4 = 1\), \(((x_1, x_2), (x_3, x_4)) = 1\) and \(((((x_1, x_2), x_3), x_4), x_5) = 1\)) the group of automorphisms of \(\Theta^0\) modulo the inner ones has order \(2\). The 21st issue of the Kourovka Notebook marks 20.72 solved ("Yes, there is").

public theorem_wanted existsVarietyWithNonInnerCategoryAutomorphism_holds :
    existsVarietyWithNonInnerCategoryAutomorphism

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.72 (E. Plotkin); resolution: R. Barbosa Fernandes and A. Tsurkov, Automorphisms of the category of finitely generated free groups of the some subvariety of the variety of all groups, preprint, arXiv:1909.05955 (2019), Theorem 9.1: for the variety Θ defined by x⁴ = 1, ((x₁, x₂), (x₃, x₄)) = 1 and ((((x₁, x₂), x₃), x₄), x₅) = 1, the quotient A/Y of the automorphism group of Θ⁰ by the inner automorphisms has order 2; Kourovka Notebook (21st issue), Problem 20.72: Yes, there is, https://arxiv.org/abs/1909.05955.

Uses 1

Statement usesPlotkin's question on outer automorphisms of the category of free groups of a variety

3 fully proved1 ready to prove

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