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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetSparseGrowth
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A]
def boundedConjugates (S F : Set A) (n : ℕ) : Set A :=
{x | ∃ g∈S^n, ∃ f∈F, g*f*g⁻¹=x}
def conjugateStage (S F : Set A) (n : ℕ) : Subgroup A :=
Subgroup.closure (boundedConjugates S F n : Set A)
lemma mem_boundedConjugates (S F : Set A) (n : ℕ) {x : A} :
x∈boundedConjugates S F n ↔ ∃ g∈S^n, ∃ f∈F, g*f*g⁻¹=x := by
rfl
lemma boundedConjugates_mono (S F : Set A) (h1 : (1:A)∈S) :
Monotone (boundedConjugates S F) := by
intro m n h x hx
obtain ⟨g,hg,f,hf,rfl⟩ := (mem_boundedConjugates S F m).mp hx
exact (mem_boundedConjugates S F n).mpr
⟨g,Set.pow_subset_pow_right h1 h hg,f,hf,rfl⟩
lemma conjugateStage_mono (S F : Set A) (h1 : (1:A)∈S) :
Monotone (conjugateStage S F) := by
intro m n h
exact Subgroup.closure_mono (boundedConjugates_mono S F h1 h)
lemma boundedConjugates_subset_pow (S F : Set A) (hsym : S⁻¹=S)
(L : ℕ) (hF : F⊆S^L) (n : ℕ) :
boundedConjugates S F n ⊆ S^(2*n+L) := by
intro x hx
obtain ⟨g,hg,f,hf,rfl⟩ := (mem_boundedConjugates S F n).mp hx
have hgi : g⁻¹∈S^n := by
rw [←hsym,inv_pow]
exact Set.inv_mem_inv.mpr hg
have he : 2*n+L=n+L+n := by omega
rw [he,pow_add,pow_add]
exact Set.mul_mem_mul (Set.mul_mem_mul hg (hF hf)) hgi
lemma conjugateStage_le_normalClosure (S F : Set A) (n : ℕ) :
conjugateStage S F n ≤ Subgroup.normalClosure (F : Set A) := by
apply (Subgroup.closure_le _).mpr
intro x hx
obtain ⟨g,hg,f,hf,rfl⟩ := (mem_boundedConjugates S F n).mp hx
exact Subgroup.normalClosure_normal.conj_mem f (Subgroup.subset_normalClosure hf) g
lemma conjugateStage_conj_generator (S F : Set A) (n : ℕ)
(s : A) (hs : s∈S) {x : A} (hx : x∈conjugateStage S F n) :
s*x*s⁻¹∈conjugateStage S F (n+1) := by
change x∈Subgroup.closure _ at hx
have hmap : conjugateStage S F n ≤
(conjugateStage S F (n+1)).comap (MulAut.conj s) := by
apply (Subgroup.closure_le _).mpr
intro y hy
obtain ⟨g,hg,f,hf,rfl⟩ := (mem_boundedConjugates S F n).mp hy
apply Subgroup.subset_closure
apply (mem_boundedConjugates S F (n+1)).mpr
refine ⟨s*g,?_,f,hf,?_⟩
· rw [pow_succ']
exact Set.mul_mem_mul hs hg
· simp [MulAut.conj_apply,mul_assoc]
exact hmap hx
lemma conjugateStage_normal_of_stationary (S F : Set A) (hsym : S⁻¹=S)
(hgen : Subgroup.closure (S : Set A)=⊤) (n : ℕ)
(he : conjugateStage S F (n+1)=conjugateStage S F n) :
(conjugateStage S F n).Normal := by
rw [←Subgroup.normalizer_eq_top_iff]
apply top_unique
rw [←hgen]
apply (Subgroup.closure_le _).mpr
intro s hs
change ∀ x, x∈conjugateStage S F n ↔ s*x*s⁻¹∈conjugateStage S F n
intro x
constructor
· intro hx
rw [←he]
exact conjugateStage_conj_generator S F n s hs hx
· intro hx
have hsi : s⁻¹∈S := by rw [←hsym]; exact Set.inv_mem_inv.mpr hs
have hy := conjugateStage_conj_generator S F n s⁻¹ hsi hx
rw [he] at hy
simpa only [inv_inv, mul_assoc, inv_mul_cancel_left, mul_inv_cancel_right, inv_mul_cancel, mul_one] using hy
lemma normalClosure_eq_conjugateStage_of_stationary (S F : Set A)
(h1 : (1:A)∈S) (hsym : S⁻¹=S) (hgen : Subgroup.closure (S : Set A)=⊤)
(n : ℕ) (he : conjugateStage S F (n+1)=conjugateStage S F n) :
Subgroup.normalClosure (F : Set A)=conjugateStage S F n := by
let _ := conjugateStage_normal_of_stationary S F hsym hgen n he
apply le_antisymm
· apply Subgroup.normalClosure_le_normal
intro f hf
apply Subgroup.subset_closure
exact (mem_boundedConjugates S F n).mpr
⟨1,Set.one_mem_pow h1,f,hf,by simp⟩
· exact conjugateStage_le_normalClosure S F n
theorem exists_conjugateStage_stationary (S F : Set A)
(h1 : (1:A)∈S) (hsym : S⁻¹=S) (K : Subgroup A) (hK : K≤Subgroup.closure F) (d : ℕ) (hg : HasDegree K S d)
(L : ℕ) (hF : F⊆S^L) :
∃ n : ℕ, conjugateStage S F (n+1)=conjugateStage S F n := by
by_contra! hn
have hw : ∀ n : ℕ, ∃ x∈boundedConjugates S F (n+1), x∉conjugateStage S F n := by
intro n
by_contra! h
apply hn n
apply le_antisymm ((Subgroup.closure_le _).mpr h)
exact conjugateStage_mono S F h1 (Nat.le_succ n)
choose g hgin hgout using hw
have hbase : K≤conjugateStage S F 0 := hK.trans (Subgroup.closure_mono (by
intro f hf
exact ⟨1,by simp,f,hf,by simp⟩))
apply no_linear_strict_chain K S h1 d hg (conjugateStage S F)
(conjugateStage_mono S F h1) hbase g (fun n => Subgroup.subset_closure (hgin n))
hgout (L+2) (by omega)
intro n
apply Set.pow_subset_pow_right h1 (show 2*(n+1)+L≤(L+2)*(n+1) by nlinarith)
exact boundedConjugates_subset_pow S F hsym L hF (n+1) (hgin n)
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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