theorem GrowthInGroups.Lecture3.fact_4_1 {n : Type u_1} [Fintype n] [DecidableEq n] (S T : GL n ℂ) : ‖↑S * ↑T * ↑S⁻¹ * ↑T⁻¹ - 1‖ ≤ 2 * ‖(↑S)⁻¹‖ * ‖(↑T)⁻¹‖ * ‖↑S - 1‖ * ‖↑T - 1‖

theorem GrowthInGroups.Lecture3.lemma_4_2 {n : Type u_1} [Fintype n] [DecidableEq n] {C₀ : ℝ} (hC₀ : ↑(Fintype.card n) < C₀) (K : ℝ) : ∃ (δ : ℝ), ∀ (A : Finset (GL n ℂ)), IsApproximateSubgroup K ↑A → (∀ a ∈ A, ‖↑a‖ ≤ C₀) → ∃ γ ∈ A ^ 2, δ * ↑A.card ≤ ↑{x ∈ A ^ 4 | Commute γ x}.card

theorem GrowthInGroups.Lecture3.corollary_4_3 (K C₀ : ℝ) : ∃ C > 0, ∀ (A : Set (Matrix.SpecialLinearGroup (Fin 2) ℂ)), IsApproximateSubgroup K A → ∃ (Z : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℂ)) (_ : ∀ x ∈ Z, ∀ y ∈ Z, Commute x y) (F : Finset (Matrix.SpecialLinearGroup (Fin 2) ℂ)), F.card ≤ C ∧ A ⊆ ↑F * ↑Z

theorem GrowthInGroups.Lecture3.theorem_4_4 {K : ℝ} : ∃ C > 0, ∀ {G : Type u_2} [inst : Group G] [DecidableEq G] (A : Set G), IsApproximateSubgroup K A → ∃ (H : Subgroup G) (N : Subgroup ↥H) (_hD : N.Normal) (F : Finset G), Subgroup.upperCentralSeries (↥H ⧸ N) C = ⊤ ∧ Subtype.val '' ↑N ⊆ (A / A) ^ 4 ∧ A ⊆ ↑F * ↑H

The Breuillard-Green-Tao theorem.

theorem GrowthInGroups.Lecture3.theorem_4_5 {C : ℝ} {G : Type u_2} [Group G] [DecidableEq G] {S : Finset G} (hSsymm : S⁻¹ = S) (hSgen : ↑(Subgroup.closure ↑S) = Set.univ) {d : ℕ} (hS : ∀ (n : ℕ), ↑(S ^ n).card ≤ C * ↑n ^ d) : Group.IsVirtuallyNilpotent G