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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetTransfer
import OAI.Probability.CriticalPercolation.Criticality.BoundedNormalJoins
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A]
lemma conjugate_map_one (N : Subgroup A) : N.map (MulAut.conj (1:A)).toMonoidHom=N :=
Subgroup.mem_normalizer_iff_map_conj_eq.mp (Subgroup.normalizer (N:Set A)).one_mem
lemma conjugate_eq_of_normalizer (N : Subgroup A) (g r : A) (h : r⁻¹*g∈(Subgroup.normalizer (N:Set A))) :
N.map (MulAut.conj g).toMonoidHom=N.map (MulAut.conj r).toMonoidHom := by
have he : N.map (MulAut.conj (r⁻¹*g)).toMonoidHom=N := Subgroup.mem_normalizer_iff_map_conj_eq.mp h
have hh := congrArg (fun P : Subgroup A => P.map (MulAut.conj r).toMonoidHom) he
rw [conjugate_map_map,mul_inv_cancel_left] at hh
exact hh
lemma conjugate_stabilizer_relIndex (K : 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) (g : A) :
(K.map (MulAut.conj g).toMonoidHom).relIndex K≠0 := by
have hk := (bounded_subgroup_iff K (K.map (MulAut.conj g⁻¹).toMonoidHom) S h1 hi hgen hK hf).mp (hK.conj h1 hi hgen g⁻¹)
have he := Subgroup.relIndex_map_map_of_injective K (K.map (MulAut.conj g⁻¹).toMonoidHom) (f:=(MulAut.conj g).toMonoidHom) (MulAut.conj g).injective
rw [conjugate_map_map,mul_inv_cancel,conjugate_map_one] at he
rwa [he]
lemma conjugate_commensurable_index (K N : 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)
(hN : N.relIndex K≠0) (g : A) :
(N.map (MulAut.conj g).toMonoidHom).relIndex K≠0 := by
apply Subgroup.relIndex_ne_zero_trans (K:=K.map (MulAut.conj g).toMonoidHom)
· rwa [Subgroup.relIndex_map_map_of_injective N K (MulAut.conj g).injective]
· exact conjugate_stabilizer_relIndex K S h1 hi hgen hK hf g
lemma finite_conjugate_family_of_finiteIndex_normalizer (H N : Subgroup A)
[H.FiniteIndex] (hHN : H≤(Subgroup.normalizer (N:Set A))) :
(Set.range (fun g : A => N.map (MulAut.conj g).toMonoidHom)).Finite := by
let := H.fintypeQuotientOfFiniteIndex
obtain ⟨R,hR,hR1⟩ := H.exists_isComplement_left 1
have hRf : R.Finite := by
let : Fintype R := Fintype.ofEquiv _ hR.leftQuotientEquiv
exact Set.toFinite R
apply (hRf.image (fun g => N.map (MulAut.conj g).toMonoidHom)).subset
rintro _ ⟨g,rfl⟩
refine ⟨hR.toLeftFun g,(hR.toLeftFun g).property,?_⟩
exact (conjugate_eq_of_normalizer N g (hR.toLeftFun g) (hHN (hR.inv_toLeftFun_mul_mem g))).symm
lemma HasDegree.finiteIndex_normal_core {K : Subgroup A} {S : Set A} {d : ℕ}
(h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤)
(hK : Bounded S (K:Set A)) (hg : HasDegree K S d)
(H N : Subgroup A) [H.FiniteIndex] (hHN : H≤(Subgroup.normalizer (N:Set A)))
(hNb : Bounded S (N:Set A)) (hNi : N.relIndex K≠0) :
∃ M : Subgroup A, M.Normal ∧ Bounded S (M:Set A) ∧ M.relIndex K≠0 := by
refine ⟨N.normalCore,inferInstance,hNb.mono N.normalCore_le,?_⟩
let F := Set.range (fun g : A => N.map (MulAut.conj g).toMonoidHom)
let : Finite F := (finite_conjugate_family_of_finiteIndex_normalizer H N hHN).to_subtype
have he : N.normalCore=⨅ P : F, P.val := by
rw [Subgroup.normalCore_eq_iInf_map_conj]
apply le_antisymm
· apply le_iInf
rintro ⟨P,⟨g,rfl⟩⟩
exact iInf_le _ g
· apply le_iInf
intro g
exact iInf_le (fun P : F => P.val) (⟨N.map (MulAut.conj g).toMonoidHom,⟨g,rfl⟩⟩ : F)
rw [he]
apply Subgroup.relIndex_iInf_ne_zero
rintro ⟨P,⟨g,rfl⟩⟩
exact conjugate_commensurable_index K N S h1 hi hgen hK hg.1 hNi g
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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