| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.NormalClosureGrowth |
|
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| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.SparseGrowth |
| variable {A : Type*} [Group A] [DecidableEq A] |
|
|
| lemma quotient_word_lower (S : Finset A) (h1 : (1:A)∈S) |
| (hgen : Subgroup.closure (S : Set A)=⊤) |
| (H : Subgroup A) [H.Normal] [Infinite (A ⧸ H)] (n : ℕ) : |
| n+1 ≤ ((S^n).image (QuotientGroup.mk' H)).card := by |
| let q := QuotientGroup.mk' H |
| have hq1 : (1:A ⧸ H)∈S.image q := Finset.mem_image.mpr ⟨1,h1,map_one q⟩ |
| have hqgen : Subgroup.closure (S.image q : Set (A ⧸ H))=⊤ := by |
| rw [Finset.coe_image,←MonoidHom.map_closure,hgen] |
| exact Subgroup.map_top_of_surjective q (QuotientGroup.mk'_surjective H) |
| have hinf : (Subgroup.closure (S.image q : Set (A ⧸ H)) : Set (A ⧸ H)).Infinite := by |
| rw [hqgen,Subgroup.coe_top] |
| exact Set.infinite_univ |
| rw [Finset.image_pow] |
| exact Finset.add_one_le_card_pow hq1 hinf n |
|
|
| lemma subgroup_linear_packing (S : Finset A) (h1 : (1:A)∈S) |
| (hgen : Subgroup.closure (S : Set A)=⊤) |
| (H : Subgroup A) [H.Normal] [Infinite (A ⧸ H)] |
| (T : Finset H) (L : ℕ) (hT : T.image H.subtype ⊆ S^L) (n : ℕ) : |
| (T^n).card*(n+1) ≤ (S^((L+1)*n)).card := by |
| have hTin : (T^n).image H.subtype ⊆ (S^(L*n)).filter (fun a => a∈H) := by |
| intro a ha |
| obtain ⟨b,hb,rfl⟩ := Finset.mem_image.mp ha |
| refine Finset.mem_filter.mpr ⟨?_,b.property⟩ |
| have he : (b : A)∈(T.image H.subtype)^n := by |
| rw [←Finset.image_pow] |
| exact Finset.mem_image.mpr ⟨b,hb,rfl⟩ |
| rw [pow_mul] |
| exact Finset.pow_subset_pow_left hT he |
| calc |
| _ = (n+1)*((T^n).image H.subtype).card := by |
| rw [Finset.card_image_of_injective _ H.subtype_injective,Nat.mul_comm] |
| _ ≤ ((S^n).image (QuotientGroup.mk' H)).card * |
| ((S^(L*n)).filter (fun a => a∈H)).card := |
| Nat.mul_le_mul (quotient_word_lower S h1 hgen H n) (Finset.card_le_card hTin) |
| _ ≤ (S^(n+L*n)).card := Finset.card_pow_quotient_mul_pow_inter_subgroup_le |
| _ = (S^((L+1)*n)).card := by congr 2; ring |
| |
| theorem subgroup_degree_descent (S : Finset A) (h1 : (1:A)∈S) |
| (hgen : Subgroup.closure (S : Set A)=⊤) |
| (H : Subgroup A) [H.Normal] [Infinite (A ⧸ H)] |
| (T : Finset H) (L : ℕ) (hT : T.image H.subtype ⊆ S^L) |
| (d : ℕ) (hg : HasDegree S (d+1)) : HasDegree T d := by |
| obtain ⟨C,hC⟩ := hg |
| refine ⟨C*(2*(L+1))^(d+1),?_⟩ |
| intro R |
| obtain ⟨r,hr,hbound⟩ := hC ((L+1)*max R 1) |
| let n := r/(L+1) |
| have hL : 0<L+1 := by omega |
| have hnr : (L+1)*n≤r := by |
| simpa only [n,Nat.mul_comm] using Nat.div_mul_le_self r (L+1) |
| have hn : max R 1≤n := (Nat.le_div_iff_mul_le hL).mpr (by simpa [Nat.mul_comm] using hr) |
| have hn1 : 1≤n := (le_max_right R 1).trans hn |
| have hrn : r≤(2*(L+1))*n := by |
| have hlt := Nat.lt_mul_div_succ r hL |
| change r<(L+1)*(n+1) at hlt |
| have hh : (L+1)*(n+1)≤(2*(L+1))*n := by nlinarith |
| exact hlt.le.trans hh |
| refine ⟨n,(le_max_left R 1).trans hn,?_⟩ |
| have hb : (T^n).card*n ≤ (C*(2*(L+1))^(d+1)*n^d)*n := by |
| calc |
| _ ≤ (T^n).card*(n+1) := Nat.mul_le_mul_left _ (Nat.le_succ n) |
| _ ≤ (S^((L+1)*n)).card := subgroup_linear_packing S h1 hgen H T L hT n |
| _ ≤ (S^r).card := Finset.card_le_card (Finset.pow_subset_pow_right h1 hnr) |
| _ ≤ C*r^(d+1) := hbound |
| _ ≤ C*((2*(L+1))*n)^(d+1) := Nat.mul_le_mul_left C (Nat.pow_le_pow_left hrn (d+1)) |
| _ = (C*(2*(L+1))^(d+1)*n^d)*n := by rw [mul_pow,pow_succ n]; ac_rfl |
| exact Nat.le_of_mul_le_mul_right hb (by omega : 0<n) |
|
|
| theorem HasPolynomialDegree.normalClosure_descent {d : ℕ} |
| (h : HasPolynomialDegree A (d+1)) (F : Finset A) |
| [Infinite (A ⧸ Subgroup.normalClosure (F : Set A))] : |
| HasPolynomialDegree (Subgroup.normalClosure (F : Set A)) d := by |
| obtain ⟨S,h1,hs,hgen,hg⟩ := (hasPolynomialDegree_iff (d+1)).mp h |
| let H := Subgroup.normalClosure (F : Set A) |
| let _ : Group.FG H := (Group.fg_iff_subgroup_fg H).mpr |
| (normalClosure_fg_of_sparse_growth S F h1 hs hgen (d+1) hg) |
| obtain ⟨T,hT1,hTs,hTgen⟩ := exists_symmetric_generators (A := H) |
| obtain ⟨L,hL⟩ := finite_subset_word_ball S h1 hs hgen (T.image H.subtype) |
| exact (hasPolynomialDegree_iff d).mpr ⟨T,hT1,hTs,hTgen, |
| subgroup_degree_descent S h1 hgen H T L hL d hg⟩ |
|
|
| theorem HasPolynomialDegree.normal_subgroup_descent {d : ℕ} |
| (h : HasPolynomialDegree A (d+1)) (H : Subgroup A) [H.Normal] |
| [Infinite (A ⧸ H)] [Group.FG H] : HasPolynomialDegree H d := by |
| obtain ⟨S,h1,hs,hgen,hg⟩ := (hasPolynomialDegree_iff (d+1)).mp h |
| obtain ⟨T,hT1,hTs,hTgen⟩ := exists_symmetric_generators (A := H) |
| obtain ⟨L,hL⟩ := finite_subset_word_ball S h1 hs hgen (T.image H.subtype) |
| exact (hasPolynomialDegree_iff d).mpr ⟨T,hT1,hTs,hTgen, |
| subgroup_degree_descent S h1 hgen H T L hL d hg⟩ |
|
|
| end CriticalPercolation.SparseGrowth |
| end |
| end |
|
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| end OAI |
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