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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetBounded
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A]
def subgroupCosetEmbedding (K H : Subgroup A) : H ⧸ K.subgroupOf H ↪ A ⧸ K where
toFun := Quotient.map' Subtype.val (by
intro a b h
simpa only [QuotientGroup.leftRel_apply,Subgroup.mem_subgroupOf,Subgroup.coe_mul,Subgroup.coe_inv] using h)
inj' := Quotient.ind₂' (by
intro a b h
change (QuotientGroup.mk a.val : A ⧸ K)=QuotientGroup.mk b.val at h
apply QuotientGroup.eq.mpr
change a.val⁻¹*b.val∈K
exact QuotientGroup.eq.mp h)
@[simp] lemma subgroupCosetEmbedding_mk (K H : Subgroup A) (h : H) :
subgroupCosetEmbedding K H (QuotientGroup.mk h)=QuotientGroup.mk h.val := rfl
lemma subgroupCosetEmbedding_range (K H : Subgroup A) :
Set.range (subgroupCosetEmbedding K H)=(QuotientGroup.mk : A → A ⧸ K) '' (H:Set A) := by
ext q
constructor
· rintro ⟨x,rfl⟩
obtain ⟨h,rfl⟩ := QuotientGroup.mk_surjective x
exact ⟨h,h.property,rfl⟩
· rintro ⟨h,hh,rfl⟩
exact ⟨QuotientGroup.mk (⟨h,hh⟩ : H),rfl⟩
lemma finite_coset_image_subgroup_iff (K H : Subgroup A) :
((QuotientGroup.mk : A → A ⧸ K) '' (H:Set A)).Finite ↔ K.relIndex H≠0 := by
rw [←subgroupCosetEmbedding_range]
rw [Set.finite_range_iff (subgroupCosetEmbedding K H).injective]
change Finite (H ⧸ K.subgroupOf H) ↔ Nat.card (H ⧸ K.subgroupOf H)≠0
exact ⟨fun h => by let := h; exact Nat.card_pos.ne',
fun h => (Nat.finite_of_card_ne_zero h)⟩
lemma bounded_subgroup_iff (K H : Subgroup A) (S : Set A)
(h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤)
(hK : Bounded S (K:Set A)) (hf : ∀ n, (ball K S n).Finite) :
Bounded S (H:Set A) ↔ K.relIndex H≠0 := by
rw [←finite_coset_image_subgroup_iff]
exact ⟨finite_coset_image_of_bounded K S (H:Set A) hf,
bounded_of_finite_coset_image S K h1 hi hgen hK (H:Set A)⟩
lemma subgroup_ball_image (K H : Subgroup A) (T : Set H) (n : ℕ) :
(subgroupCosetEmbedding K H) '' ball (K.subgroupOf H) T n=
ball K (H.subtype '' T) n := by
rw [ball,Set.image_image,ball,←Set.image_pow]
rw [Set.image_image]
rfl
lemma HasDegree.subgroup_bounded {K : Subgroup A} {S : Set A} {d : ℕ}
(h1 : (1:A)∈S) (hg : HasDegree K S d) (H : Subgroup A) (T : Set H)
(hT : Bounded S (H.subtype '' T)) : HasDegree (K.subgroupOf H) T d := by
have ht := hg.bounded_generators h1 hT
have hf (n : ℕ) : (ball (K.subgroupOf H) T n).Finite := by
apply Set.Finite.of_finite_image (f:=subgroupCosetEmbedding K H)
rw [subgroup_ball_image]
exact ht.1 n
exact (subgroupCosetEmbedding K H).injective.injOn
refine ⟨hf,?_⟩
obtain ⟨C,hC⟩ := ht.2
refine ⟨C,?_⟩
intro R
obtain ⟨n,hn,hb⟩ := hC R
refine ⟨n,hn,?_⟩
rw [←subgroup_ball_image,Set.ncard_image_of_injective _ (subgroupCosetEmbedding K H).injective] at hb
exact hb
lemma finite_image_of_finite_coset_image {B : Type*} [Group B]
(K : Subgroup A) (f : A →* B) (hK : (f '' (K:Set A)).Finite)
(T : Set A) (hT : ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite) :
(f '' T).Finite := by
obtain ⟨F,hF,hcov⟩ := (finite_coset_image_iff_cover K T).mp hT
apply ((hF.image f).mul hK).subset
rintro _ ⟨t,ht,rfl⟩
obtain ⟨a,ha,k,hk,rfl⟩ := hcov ht
rw [map_mul]
exact Set.mul_mem_mul ⟨a,ha,rfl⟩ ⟨k,hk,rfl⟩
lemma finiteIndex_bounded_generators (S : Set A)
(h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤)
(H : Subgroup A) [H.FiniteIndex] :
∃ T : Set H, (1:H)∈T ∧ T⁻¹=T ∧ Subgroup.closure T=⊤ ∧ Bounded S (H.subtype '' T) := by
let := H.fintypeQuotientOfFiniteIndex
obtain ⟨R,hR,hR1⟩ := H.exists_isComplement_right 1
have hRf : R.Finite := by
let : Fintype R := Fintype.ofEquiv _ hR.rightQuotientEquiv
exact Set.toFinite R
let U : Set H := (R*S).image (fun g => ⟨g*(hR.toRightFun g:A)⁻¹,hR.mul_inv_toRightFun_mem g⟩)
have hUgen : Subgroup.closure U=⊤ := Subgroup.closure_mul_image_eq_top hR hR1 hgen
have hUb : Bounded S (H.subtype '' U) := by
have hRb := bounded_finite S h1 hi hgen hRf
have hSb : Bounded S S := ⟨1,by simp⟩
exact ((hRb.mul hSb).mul (hRb.inv hi)).mono (by
rintro _ ⟨_,⟨g,hg,rfl⟩,rfl⟩
exact Set.mul_mem_mul hg (Set.inv_mem_inv.mpr (hR.toRightFun g).property))
let T := {1}∪(U∪U⁻¹)
refine ⟨T,by simp [T],?_,?_,?_⟩
· simp [T,Set.union_inv,Set.union_comm]
· apply top_unique
rw [←hUgen]
exact Subgroup.closure_mono (by intro u hu; exact Or.inr (Or.inl hu))
· have he : H.subtype '' T={1}∪((H.subtype '' U)∪(H.subtype '' U)⁻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 (hUb.union h1 (hUb.inv hi))
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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