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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetFiniteIndex
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_eq_of_integer_value (f : A →* Multiplicative ℤ)
(a : A) (ha : f a=Multiplicative.ofAdd 1) (N : Subgroup A)
(hN : f.ker≤(Subgroup.normalizer (N:Set A))) (g : A) :
N.map (MulAut.conj g).toMonoidHom=N.map (MulAut.conj (a^Multiplicative.toAdd (f g))).toMonoidHom := by
let z := Multiplicative.toAdd (f g)
have hl : (a^z)⁻¹*g∈f.ker := by
change f ((a^z)⁻¹*g)=1
rw [map_mul,map_inv,map_zpow,ha]
apply Multiplicative.toAdd.injective
simp [z]
have he : N.map (MulAut.conj ((a^z)⁻¹*g)).toMonoidHom=N :=
Subgroup.mem_normalizer_iff_map_conj_eq.mp (hN hl)
have hg : g=a^z*((a^z)⁻¹*g) := by simp
conv_lhs => rw [hg]
rw [←conjugate_map_map,he]
lemma finite_bounded_conjugate_family (K : Subgroup A) (S : Set A)
(hf : ∀ n, (ball K S n).Finite) (f : A →* Multiplicative ℤ) (hK : K≤f.ker)
(a : A) (ha : f a=Multiplicative.ofAdd 1) (N : Subgroup A)
(hN : f.ker≤(Subgroup.normalizer (N:Set A))) (n : ℕ) :
((fun g => N.map (MulAut.conj g).toMonoidHom) '' S^n).Finite := by
have hKi : (f '' (K:Set A)).Finite := by
apply (Set.finite_singleton (1:Multiplicative ℤ)).subset
rintro _ ⟨k,hk,rfl⟩
exact MonoidHom.mem_ker.mp (hK hk)
have hfi := finite_image_of_finite_coset_image K f hKi (S^n) (hf n)
apply (hfi.image (fun z => N.map (MulAut.conj (a^Multiplicative.toAdd z)).toMonoidHom)).subset
rintro _ ⟨g,hg,rfl⟩
exact ⟨f g,⟨g,hg,rfl⟩,(conjugate_eq_of_integer_value f a ha N hN g).symm⟩
lemma HasDegree.promote_integer_normal {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)
(f : A →* Multiplicative ℤ) (hf : Surjective f) (hK : K≤f.ker)
(N : Subgroup A) (hNL : N≤f.ker) (hLN : f.ker≤(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
let J := K⊓N
have hJi : J.relIndex K≠0 := by simpa only [J,Subgroup.inf_relIndex_left] using hNi
let : Finite (K ⧸ J.subgroupOf K) := Nat.finite_of_card_ne_zero hJi
have hgJ := hg.smaller_stabilizer (show J≤K from inf_le_left)
obtain ⟨L,hL⟩ := hNb
obtain ⟨n,hn⟩ := exists_conjugateStage_stationary S (N:Set A) h1 hi J
(by simpa only [Subgroup.closure_eq] using (show J≤N from inf_le_right)) d hgJ L hL
let M := Subgroup.normalClosure (N:Set A)
have hM : M=conjugateStage S (N:Set A) n := normalClosure_eq_conjugateStage_of_stationary S (N:Set A) h1 hi hgen n hn
obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1)
refine ⟨M,inferInstance,?_,?_⟩
· rw [hM,conjugateStage_eq_join]
apply finite_normalized_join_bounded S h1 f.ker _
(finite_bounded_conjugate_family K S hg.1 f hK a ha N hLN n)
· rintro Q ⟨g,hg,rfl⟩
have hh := Subgroup.map_mono hNL (f:=(MulAut.conj g).toMonoidHom)
have he : f.ker.map (MulAut.conj g).toMonoidHom=f.ker := Subgroup.Normal.map_conj_eq f.ker g
rwa [he] at hh
· rintro Q ⟨g,hg,rfl⟩
exact normalized_conjugate f.ker N hLN g
· rintro Q ⟨g,hg,rfl⟩
exact (show Bounded S (N:Set A) from ⟨L,hL⟩).conj h1 hi hgen g
· exact fun he => hNi (Subgroup.relIndex_eq_zero_of_le_left (fun x hx => Subgroup.subset_normalClosure hx) he)
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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