Generic properties of group presentations

DefinitionPresentations counted by Arzhantseva and Olshanskii#✓fully proved

For natural numbers \(m\), \(n\) and \(d\), this is the set of \(n\)-tuples \((r_1, \dots, r_n)\) of elements of the free group \(F_m\) on generators \(x_1, \dots, x_m\) such that the reduced word of each \(r_i\) is cyclically reduced (the empty word counts as cyclically reduced) and the reduced word lengths satisfy \(|r_1| + \dots + |r_n| = d\). Each tuple stands for the presentation \(\langle x_1, \dots, x_m \mid r_1, \dots, r_n \rangle\).

def presentations (m n d : ℕ) : Set (Fin n → FreeGroup (Fin m)) :=
  {r | (∀ i, FreeGroup.IsCyclicallyReduced (r i).toWord) ∧ ∑ i, (r i).norm = d}

Used by 1

Used byOlshanskii's genericity problem (Kourovka 11.75)

DefinitionGroups whose subgroups with fewer than \(m\) generators are free#✓fully proved

For a natural number \(m\), a group \(G\) has this property if, for every finite set \(S \subseteq G\) with fewer than \(m\) elements, the subgroup of \(G\) generated by \(S\) is a free group.

def SubgroupsWithFewerGeneratorsFree (m : ℕ) (G : Type) [Group G] : Prop :=
  ∀ s : Finset G, s.card < m → IsFreeGroup (Subgroup.closure (s : Set G))

Used by 1

Used byOlshanskii's genericity problem (Kourovka 11.75)

DefinitionOlshanskii's genericity problem (Kourovka 11.75)#✓fully proved

This is the proposition, in the form proved by Arzhantseva and Olshanskii, that the following holds for all natural numbers \(m \ge 2\) and \(n \ge 1\). Let \(F_m\) be the free group on \(x_1, \dots, x_m\). For each \(d\), let \(N_d\) be the number of \(n\)-tuples \((r_1, \dots, r_n)\) of elements of \(F_m\) whose reduced words are cyclically reduced and have total length \(|r_1| + \dots + |r_n| = d\), and let \(N_d^f\) be the number of those tuples for which every subgroup of \(\langle x_1, \dots, x_m \mid r_1, \dots, r_n \rangle\) generated by fewer than \(m\) elements is free. Then \(N_d^f / N_d \to 1\) as \(d \to \infty\).

def fewerGeneratorSubgroupsFreeGeneric : Prop :=
  ∀ m n : ℕ, 2 ≤ m → 1 ≤ n →
    Filter.Tendsto
      (fun d => (({r ∈ presentations m n d |
          SubgroupsWithFewerGeneratorsFree m (PresentedGroup (Set.range r))} : Set _).ncard : ℝ) /
        (presentations m n d).ncard)
      Filter.atTop (nhds 1)

Uses 2 · Used by 1

Statement usesGroups whose subgroups with fewer than \(m\) generators are free · Presentations counted by Arzhantseva and Olshanskii

Used byGenerically, subgroups with fewer generators are free (Kourovka 11.75)

TheoremGenerically, subgroups with fewer generators are free (Kourovka 11.75)#●ready to prove

Let \(m \ge 2\) and \(n \ge 1\) be natural numbers and let \(F_m\) be the free group on \(x_1, \dots, x_m\). For each \(d\), let \(N_d\) be the number of \(n\)-tuples \((r_1, \dots, r_n)\) of elements of \(F_m\) whose reduced words are cyclically reduced and have total length \(|r_1| + \dots + |r_n| = d\), and let \(N_d^f\) be the number of those tuples for which every subgroup of \(\langle x_1, \dots, x_m \mid r_1, \dots, r_n \rangle\) generated by fewer than \(m\) elements is free. Then \(N_d^f / N_d \to 1\) as \(d \to \infty\).

Resolved true: Arzhantseva and Olshanskii prove this; the Kourovka Archive records 11.75 as proved by this paper.

public theorem_wanted fewerGeneratorSubgroupsFreeGeneric_holds : fewerGeneratorSubgroupsFreeGeneric

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 11.75 (A. Yu. Olshanskii); resolution: G. N. Arzhantseva and A. Yu. Olshanskii, The class of groups all of whose subgroups with lesser number of generators are free is generic, Math. Notes 59 (1996), no. 4, 350-355 (Mat. Zametki 59 (1996), 489-496), Kourovka Archive, Problem 11.75: This is proved (Arzhantseva-Olshanskii); paper, abstract ("this solves Problem 11.75 from the Kourovka Notebook") and main theorem as summarized in zbMATH Zbl 0877.20021 (lim N_d^f / N_d = 1 over presentations with cyclically reduced relators of total length d), https://doi.org/10.1007/BF02308683.

Uses 1

Statement usesOlshanskii's genericity problem (Kourovka 11.75)

3 fully proved1 ready to prove

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