Gap conjecture
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the ...
Mathematical statement
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.
Statement source: Wikipedia statement material
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Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
gap_conjecture
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) := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References