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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetTransfer
import OAI.Probability.CriticalPercolation.Growth.CosetNormalClosure
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A]
lemma integer_kernel_normalClosure (S : Set A)
(hgen : Subgroup.closure S=⊤)
(f : A →* Multiplicative ℤ) (a : A) (ha : f a=Multiplicative.ofAdd 1) :
f.ker=Subgroup.normalClosure ((fun s => s*a^(-Multiplicative.toAdd (f s))) '' S) := by
let F := (fun s => s*a^(-Multiplicative.toAdd (f s))) '' S
let N := Subgroup.normalClosure F
have hN : N≤f.ker := by
apply Subgroup.normalClosure_le_normal
rintro y ⟨s,hs,rfl⟩
change f (s*a^(-Multiplicative.toAdd (f s)))=1
rw [map_mul,map_zpow,ha]
apply Multiplicative.toAdd.injective
simp
let q := QuotientGroup.mk' N
have hqa : ∀ x : A, q x=(q a)^Multiplicative.toAdd (f x) := by
intro x
have hx : x∈Subgroup.closure S := hgen.symm ▸ Subgroup.mem_top x
induction hx using Subgroup.closure_induction with
| mem x hx =>
have hm : x*a^(-Multiplicative.toAdd (f x))∈N := Subgroup.subset_normalClosure ⟨x,hx,rfl⟩
have he := (QuotientGroup.eq_one_iff _).mpr hm
change q (x*a^(-Multiplicative.toAdd (f x)))=1 at he
rw [map_mul,map_zpow,zpow_neg] at he
exact mul_inv_eq_one.mp he
| one => simp
| mul x y _ _ hx hy => rw [map_mul,hx,hy,←zpow_add,map_mul,toAdd_mul]
| inv x _ hx => rw [map_inv,hx,map_inv,toAdd_inv,zpow_neg]
apply le_antisymm _ hN
intro x hx
apply (QuotientGroup.eq_one_iff _).mp
change q x=1
rw [hqa,MonoidHom.mem_ker.mp hx]
simp
lemma HasDegree.integer_kernel_bounded_generators {K : Subgroup A} {S : Set A} {d : ℕ}
(h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) (hg : HasDegree K S d)
(hKS : (K:Set A)⊆S) (f : A →* Multiplicative ℤ) (hf : Surjective f) (hKf : K≤f.ker) :
∃ T : Set f.ker, (1:f.ker)∈T ∧ T⁻¹=T ∧ Subgroup.closure T=⊤ ∧
Bounded S (f.ker.subtype '' T) ∧ (K.subgroupOf f.ker : Set f.ker)⊆T := by
obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1)
let F := (fun s => s*a^(-Multiplicative.toAdd (f s))) '' S
have hFker : F⊆f.ker := by
rintro y ⟨s,hs,rfl⟩
change f (s*a^(-Multiplicative.toAdd (f s)))=1
rw [map_mul,map_zpow,ha]
apply Multiplicative.toAdd.injective
simp
have hKF : (K:Set A)⊆F := by
intro k hk
refine ⟨k,hKS hk,?_⟩
dsimp only
rw [MonoidHom.mem_ker.mp (hKf hk)]
simp
have hfi : (f '' (K:Set A)).Finite := by
apply (Set.finite_singleton (1:Multiplicative ℤ)).subset
rintro _ ⟨k,hk,rfl⟩
exact MonoidHom.mem_ker.mp (hKf hk)
have hfs : (f '' S).Finite := finite_image_of_finite_coset_image K f hfi S (by simpa only [ball,pow_one] using hg.1 1)
have hFb : Bounded S F := by
have hp := bounded_finite S h1 hi hgen (hfs.image (fun z => a^(-Multiplicative.toAdd z)))
exact (Bounded.mul (show Bounded S S from ⟨1,by simp⟩) hp).mono (by
rintro y ⟨s,hs,rfl⟩
exact Set.mul_mem_mul hs ⟨f s,⟨s,hs,rfl⟩,rfl⟩)
obtain ⟨L,hL⟩ := hFb
obtain ⟨n,hn⟩ := exists_conjugateStage_stationary S F h1 hi K
(fun k hk => Subgroup.subset_closure (hKF hk)) d hg L hL
have hncl : f.ker=conjugateStage S F n :=
(integer_kernel_normalClosure S hgen f a ha).trans (normalClosure_eq_conjugateStage_of_stationary S F h1 hi hgen n hn)
let U := boundedConjugates S F n
have hUk : U⊆f.ker := by
intro u hu
rw [hncl]
exact Subgroup.subset_closure hu
let T0 : Set f.ker := f.ker.subtype ⁻¹' U
have hT0image : f.ker.subtype '' T0=U := by
ext u
exact ⟨fun ⟨x,hx,he⟩ => he ▸ hx,fun hu => ⟨⟨u,hUk hu⟩,hu,rfl⟩⟩
have hT0gen : Subgroup.closure T0=⊤ := by
apply Subgroup.map_injective f.ker.subtype_injective
rw [MonoidHom.map_closure,hT0image,←MonoidHom.range_eq_map,Subgroup.range_subtype]
exact hncl.symm
have hT0b : Bounded S (f.ker.subtype '' T0) := by
rw [hT0image]
exact ⟨2*n+L,boundedConjugates_subset_pow S F hi L hL n⟩
let T := {1}∪(T0∪T0⁻¹)
refine ⟨T,by simp [T],?_,?_,?_,?_⟩
· simp [T,Set.union_inv,Set.union_comm]
· apply top_unique
rw [←hT0gen]
exact Subgroup.closure_mono (fun t ht => Or.inr (Or.inl ht))
· have he : f.ker.subtype '' T={1}∪((f.ker.subtype '' T0)∪(f.ker.subtype '' T0)⁻¹) := by
simp only [T,Set.image_union,Set.image_singleton,map_one,Set.image_inv]
rw [he]
exact (bounded_singleton S h1 hi hgen 1).union h1 (hT0b.union h1 (hT0b.inv hi))
· intro k hk
apply Or.inr ∘ Or.inl
exact ⟨1,Set.one_mem_pow h1,k.val,hKF hk,by simp⟩
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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