Growth in Groups - Lecture 1

This file contains a Lean formalisation of the statements and proofs given in lecture 1 of the ETH course Growth in Groups lectured by Simon Machado in autumn/winter 2024.

References

Lecture notes by Simon Machado

theorem GrowthInGroups.Lecture1.fact_1_1_1 {G : Type u_1} [Group G] [DecidableEq G] {X : Finset G} [Infinite G] (hX₁ : 1 ∈ X) (hXgen : Subgroup.closure ↑X = ⊤) (n : ℕ) : n + 1 ≤ (X ^ n).card

The growth of a set generating an infinite group is at least linear.

theorem GrowthInGroups.Lecture1.fact_1_1_2 {G : Type u_1} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ} : (X ^ n).card ≤ X.card ^ n

The growth of a set is at most exponential.

def GrowthInGroups.Lecture1.HasPolynomialGrowth (G : Type u_1) [Group G] [DecidableEq G] : Prop

A group has polynomial growth if any (equivalently, all) of its finite symmetric sets has polynomial growth.

Equations

• GrowthInGroups.Lecture1.HasPolynomialGrowth G = ∃ (X : Finset G), X⁻¹ = X ∧ Subgroup.closure ↑X = ⊤ ∧ ∃ (d : ℕ), ∀ n ≥ 2, (X ^ n).card ≤ n ^ d

theorem GrowthInGroups.Lecture1.theorem_1_2 {G : Type u_1} [Group G] [DecidableEq G] [Group.FG G] : HasPolynomialGrowth G ↔ Group.IsVirtuallyNilpotent G

Gromov's theorem.

A group has polynomial growth iff it's virtually nilpotent.

theorem GrowthInGroups.Lecture1.fact_1_3 {G : Type u_1} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ} [Fintype G] (hn : X ^ n = Finset.univ) : Real.log ↑(Fintype.card G) / Real.log ↑X.card ≤ ↑n

theorem GrowthInGroups.Lecture1.conjecture_1_4 : ∃ Cᵤ ≥ 0, ∃ dᵤ ≥ 0, ∀ {G : Type u_2} [inst : Group G] [IsSimpleGroup G] [inst_2 : Fintype G] [inst_3 : DecidableEq G] (X : Finset G), 1 ∈ X → X⁻¹ = X → Subgroup.closure ↑X = ⊤ → ∃ (m : ℕ), ↑m ≤ Cᵤ * Real.log ↑(Fintype.card G) ^ dᵤ ∧ X ^ m = Finset.univ

Babai's conjecture.

For all finite sets X generating a simple group G, there exists a universal polynomial (in log |G|) upper bound to the number of steps X takes to generate G.

theorem GrowthInGroups.Lecture1.proposition_1_7 : ∃ ε > 0, ∀ (X : Finset (Matrix.SpecialLinearGroup (Fin 2) ℝ)), (X ^ 2).card ≤ 1000 * X.card → (∀ M ∈ X, ∀ (i j : Fin 2), |↑M i j| ≤ ε) → ∃ (A : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℝ)), IsMulCommutative ↥A ∧ ∃ (a : Fin 10000000 → Matrix.SpecialLinearGroup (Fin 2) ℝ), ↑X ⊆ ⋃ (i : Fin 10000000), a i • ↑A

An auxiliary lemma used in the proof of the collar theorem.

theorem GrowthInGroups.Lecture1.theorem_1_8 {K : ℝ} : ∃ C > 0, ∀ {G : Type u_2} [inst : Group G] [inst_1 : DecidableEq G] (A : Finset G), ↑(A.mulConst A) ≤ K → ∃ (N : Subgroup G) (D : Subgroup ↥N) (_hD : D.Normal), Subgroup.upperCentralSeries (↥N ⧸ D) C = ⊤ ∧ Subtype.val '' ↑D ⊆ (↑A / ↑A) ^ 4 ∧ ∃ (a : Fin C → G), ↑A ⊆ ⋃ (i : Fin C), a i • ↑N

The Breuillard-Green-Tao theorem.

theorem GrowthInGroups.Lecture1.theorem_1_9 {n : ℕ} : ∃ δ > 0, ∃ ε > 0, ∀ (k : Type u_2) [inst : Field k] [inst_1 : Fintype k] [inst_2 : DecidableEq k] (A : Finset (Matrix.SpecialLinearGroup (Fin n) k)), Subgroup.closure ↑A = ⊤ → A.card ^ (1 + δ) ≤ (A ^ 3).card ∨ Fintype.card (Matrix.SpecialLinearGroup (Fin n) k) ^ (1 - ε) ≤ A.card

The product theorem, due Breuillard-Green-Tao and Pyber-Szabo.

A set in SLₙ(k) either has big tripling or is very big. In other words, there is no small tripling, except in trivial situations.

theorem GrowthInGroups.Lecture1.fact_1_10 {G : Type u_1} [Group G] [DecidableEq G] {A : Finset G} (hA : (A * A).card ≤ A.card) : ∃ (H : Subgroup G), ∀ a ∈ A, a • ↑H = ↑A ∧ MulOpposite.op a • ↑H = ↑A

A non-empty set A with no doubling is the coset of a subgroup H.

Precisely, H can be taken to be the stabiliser of A and A then is both a left and right coset of H.

theorem GrowthInGroups.Lecture1.lemma_1_11 {G : Type u_1} [Group G] [DecidableEq G] {A : Finset G} (hA : ↑(A * A).card < 3 / 2 * ↑A.card) : ∃ (H : Subgroup G) (x : Fintype ↥H), ↑(Fintype.card ↥H) < 3 / 2 * ↑A.card ∧ ∀ a ∈ A, ↑A ⊆ a • ↑H ∧ a • ↑H = MulOpposite.op a • ↑H

A set A of tripling strictly less than three halves can be contained in a coset of a subgroup H of size strictly |H| < 3/2 |A|.

One can furthermore arrange for A to lie in the centraliser of H.