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import FormalConjecturesUtil
namespace GapConjecture
open Filter GromovPolynomialGrowth
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least $e^{\sqrt n}$ in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.
@[category research open, AMS 20]
theorem gap_conjecture :
∀ (G : Type) [Group G] (S : Set G), S.Finite → Subgroup.closure S = ⊤ →
HasSuperPolynomialGrowth G →
∃ C : ℕ, 0 < C ∧
∀ᶠ n : ℕ in atTop, Real.exp (Real.sqrt (n : ℝ)) ≤
(GrowthFunction S (C * n) : ℝ) := ⊢ ∀ (G : Type) [inst : Group G] (S : Set G),
S.Finite →
Subgroup.closure S = ⊤ →
HasSuperPolynomialGrowth G → ∃ C, 0 < C ∧ ∀ᶠ (n : ℕ) in atTop, Real.exp √↑n ≤ ↑(GrowthFunction S (C * n))
All goals completed! 🐙
end GapConjecture