Normally independent sets in free groups

DefinitionNormally independent sets in the free group of rank two#✓fully proved

A subset \(E\) of the free group \(F_2\) on two generators is normally independent when no element \(w \in E\) lies in the normal closure of \(E \setminus \{w\}\), the smallest normal subgroup of \(F_2\) containing \(E \setminus \{w\}\).

def IsNormallyIndependent (E : Set (FreeGroup (Fin 2))) : Prop :=
  ∀ w ∈ E, w ∉ Subgroup.normalClosure (E \ {w})

Used by 1

Used byGrigorchuk's problem on massive independent sets in the free group of rank two

DefinitionGrigorchuk's problem on massive independent sets in the free group of rank two#✓fully proved

The statement that some subset \(E\) of the free group \(F_2\) on two generators is normally independent (no \(w \in E\) lies in the normal closure of \(E \setminus \{w\}\)) and massive: for every real number \(c\) with \(0 < c < 3\) and every natural number \(N\) there is \(n \ge N\) such that the set \(E_n\) of elements of \(E\) of reduced word length \(n\) has at least \(c^n\) elements.

Since \(F_2\) has \(4 \cdot 3^{n-1}\) reduced words of length \(n \ge 1\), \(\limsup |E_n|^{1/n} \le 3\) always, so massiveness says \(\limsup |E_n|^{1/n} = 3\), as in Kourovka Notebook Problem 10.13 (R. I. Grigorchuk).

def existsMassiveNormallyIndependentSet : Prop :=
  ∃ E : Set (FreeGroup (Fin 2)), IsNormallyIndependent E ∧
    ∀ c : ℝ, 0 < c → c < 3 → ∀ N : ℕ, ∃ n ≥ N, c ^ n ≤ ({w ∈ E | w.norm = n}.ncard : ℝ)

Uses 1 · Used by 1

Statement usesNormally independent sets in the free group of rank two

Used byNo massive normally independent set in the free group of rank two

TheoremNo massive normally independent set in the free group of rank two#●ready to prove

There is no subset \(E\) of the free group \(F_2\) on two generators such that no element \(w \in E\) lies in the normal closure of \(E \setminus \{w\}\) and, for every real number \(c\) with \(0 < c < 3\) and every natural number \(N\), there is \(n \ge N\) for which at least \(c^n\) elements of \(E\) have reduced word length \(n\).

Koch-Hyde and Olive prove that a subset \(S\) of the ball \(B_n\) of the free group \(F_r\) in which no \(x \in S\) lies in the subgroup generated by \(S \setminus \{x\}\) has \(|S| = O((2r - 1)^{n/2})\). Normal independence implies this independence, so for every such \(E \subseteq F_2\) the set \(E_n\) of its elements of length \(n\) has \(|E_n| = O(3^{n/2})\), and the growth rate is at most \(\sqrt 3 < 3\). The Kourovka Notebook (21st issue) marks Problem 10.13 solved, "No, there is no such set".

public theorem_wanted existsMassiveNormallyIndependentSet_refuted :
    ¬ existsMassiveNormallyIndependentSet

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 10.13 (R. I. Grigorchuk); resolution: L. Koch-Hyde and É. Olive, Two problems about bases and independent sets in free groups, arXiv:2609.00382 (2026), Theorem 1.7 (an independent set S ⊆ B_n ⊆ F_r has |S| = O((2r - 1)^{n/2})) and the Solution to Problem 1.5; the Kourovka Notebook (21st issue) marks 10.13 solved, https://arxiv.org/abs/2609.00382.

Uses 1

Statement usesGrigorchuk's problem on massive independent sets in the free group of rank two

2 fully proved1 ready to prove

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