Growth rates of amenable groups

DefinitionGrowth rate of a \(k\)-element generating family (Kourovka 14.7)#✓fully proved

Let \(\Gamma\) be a group and \(S = (s_1, \dots, s_k)\) a family of \(k\) elements generating \(\Gamma\). For \(n \in \mathbb{N}\) let \(\beta_S(n)\) be the number of elements of \(\Gamma\) that are products of at most \(n\) elements of \(S \cup S^{-1}\). The growth rate of \(\Gamma\) with respect to \(S\) is the real number \(\omega(\Gamma, S) = \limsup_{n \to \infty} \beta_S(n)^{1/n}\).

noncomputable def growthRate {Γ : Type*} [Group Γ] {k : ℕ} (S : Group.Generators Γ (Fin k)) : ℝ :=
  Filter.limsup (fun n : ℕ => (Nat.card {g : Γ // S.wordLength g ≤ n} : ℝ) ^ ((1 : ℝ) / n))
    Filter.atTop

Used by 1

Used byKourovka Problem 15.4 (large growth rate implies non-amenability)

DefinitionKourovka Problem 15.4 (large growth rate implies non-amenability)#✓fully proved

The proposition, asked by P. de la Harpe, that there are \(k \ge 2\) and a real \(\varepsilon > 0\) such that every group \(\Gamma\) generated by a family \(S\) of \(k\) distinct elements whose growth rate satisfies \(\omega(\Gamma, S) \ge 2k - 1 - \varepsilon\) is non-amenable (amenability being the Følner condition).

def largeGrowthRateImpliesNonamenable : Prop :=
  ∃ k : ℕ, 2 ≤ k ∧ ∃ ε : ℝ, 0 < ε ∧ ∀ (Γ : Type) [Group Γ] (S : Group.Generators Γ (Fin k)),
    Function.Injective S.val → 2 * (k : ℝ) - 1 - ε ≤ growthRate S → ¬ IsAmenable Γ

Uses 2 · Used by 1

Statement usesAmenable group (Følner condition) · Growth rate of a \(k\)-element generating family (Kourovka 14.7)

Used byAmenable groups with growth rate close to \(2k - 1\) (Kourovka 15.4 refuted)

TheoremAmenable groups with growth rate close to \(2k - 1\) (Kourovka 15.4 refuted)#●ready to prove

For a group \(\Gamma\) generated by a family \(S\) of \(k\) elements and \(n \in \mathbb{N}\), let \(\beta_S(n)\) be the number of elements of \(\Gamma\) that are products of at most \(n\) elements of \(S \cup S^{-1}\), and let \(\omega(\Gamma, S) = \limsup_{n \to \infty} \beta_S(n)^{1/n}\) be the growth rate. For every \(k \ge 2\) and every real \(\varepsilon > 0\) there are an amenable group \(\Gamma\) (amenability being the Følner condition) and a family \(S\) of \(k\) distinct elements generating \(\Gamma\) such that \(\omega(\Gamma, S) \ge 2k - 1 - \varepsilon\). Equivalently, there are no \(k \ge 2\) and \(\varepsilon > 0\) such that every group generated by a family of \(k\) distinct elements with growth rate at least \(2k - 1 - \varepsilon\) is non-amenable.

Arzhantseva, Guba and Guyot showed that for every \(m \ge 2\), with \(F_m\) the free group of rank \(m\), \(R\) a normal subgroup of \(F_m\) and \(R'\) its commutator subgroup, the growth rate of \(F_m/R'\) tends to \(2m - 1\) as the shortest relation in \(R\) gets longer, so \(m\)-generated amenable groups (abelian-by-nilpotent ones, for instance) have growth rates arbitrarily close to \(2m - 1\); the Kourovka Archive answers 15.4 "No, it does not".

public theorem_wanted largeGrowthRateImpliesNonamenable_refuted :
    ¬ largeGrowthRateImpliesNonamenable

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 15.4 (P. de la Harpe); resolution: G. N. Arzhantseva, V. S. Guba and L. Guyot, Growth rates of amenable groups, J. Group Theory 8 (2005), 389-394, arXiv:math/0406013, abstract and main result (the growth rate of an m-generated amenable group, already in the abelian-by-nilpotent and metabelian-by-finite classes, can be arbitrarily close to 2m - 1); Kourovka Archive, Problem 15.4 (answer: No, it does not), https://arxiv.org/abs/math/0406013.

Uses 1

Statement usesKourovka Problem 15.4 (large growth rate implies non-amenability)

2 fully proved1 ready to prove

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