| 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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