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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.CosetIndicability
import OAI.Probability.CriticalPercolation.Growth.CosetIntegerKernel
import OAI.Probability.CriticalPercolation.Growth.CosetKernelDescent
import OAI.Probability.CriticalPercolation.Growth.CosetNormalPromotion
import OAI.Probability.CriticalPercolation.Growth.CosetSubgroupGenerators
import OAI.Probability.CriticalPercolation.Growth.CosetFiniteCore
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise
namespace CriticalPercolation.RelativeGrowth
variable {A : Type*} [Group A]
lemma bounded_top_of_finiteIndex (K : Subgroup A) (S : Set A)
(h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤)
(hK : Bounded S (K:Set A)) [K.FiniteIndex] : Bounded S (⊤:Subgroup A) := by
let := K.fintypeQuotientOfFiniteIndex
apply bounded_of_finite_coset_image S K h1 hi hgen hK
exact Set.toFinite _
lemma relIndex_map_subgroupOf_of_le (K L : Subgroup A) (hKL : K≤L)
(N : Subgroup L) (hN : N.relIndex (K.subgroupOf L)≠0) :
(N.map L.subtype).relIndex K≠0 := by
have he : (K.subgroupOf L).map L.subtype=K := by
rw [Subgroup.subgroupOf,Subgroup.map_comap_eq,Subgroup.range_subtype,inf_eq_right.mpr hKL]
rw [←he,Subgroup.relIndex_map_map_of_injective N (K.subgroupOf L) L.subtype_injective]
exact hN
universe u
theorem bounded_normal_commensurable (d : ℕ) :
∀ (A : Type u) [Group A] (K : Subgroup A) (S : Set A),
(1:A)∈S → S⁻¹=S → Subgroup.closure S=⊤ → Bounded S (K:Set A) → HasDegree K S d →
∃ N : Subgroup A, N.Normal ∧ Bounded S (N:Set A) ∧ N.relIndex K≠0 := by
induction d with
| zero =>
intro A _ K S h1 hi hgen hK hg
let := hg.zero_finiteIndex h1 hi hgen
exact ⟨⊤,inferInstance,bounded_top_of_finiteIndex K S h1 hi hgen hK,by simp⟩
| succ d ih =>
intro A _ K S h1 hi hgen hK hg
rcases finite_or_infinite (A ⧸ K) with hfin | hinf
· let : K.FiniteIndex := ⟨by let := hfin; exact Nat.card_pos.ne'⟩
exact ⟨⊤,inferInstance,bounded_top_of_finiteIndex K S h1 hi hgen hK,by simp⟩
let := hinf
obtain ⟨H,hHi,f,hf,hKf⟩ := virtually_indicable h1 hi hgen hK hg
let := hHi
let J := K.subgroupOf H
obtain ⟨T,hT1,hTi,hTgen,hTb,hKT⟩ :=
finiteIndex_generators_containing_stabilizer K S h1 hi hgen hK H
have hgT : HasDegree J T (d+1) := hg.subgroup_bounded h1 H T hTb
obtain ⟨U,hU1,hUi,hUgen,hUb,hKU⟩ :=
hgT.integer_kernel_bounded_generators hT1 hTi hTgen hKT f hf hKf
have hgU : HasDegree (J.subgroupOf f.ker) U d :=
hgT.integer_kernel_descent hT1 hTi hTgen f hf hKf U hUb
obtain ⟨N,hNn,hNb,hNi⟩ := ih f.ker (J.subgroupOf f.ker) U hU1 hUi hUgen
⟨1,by simpa only [pow_one] using hKU⟩ hgU
let := hNn
let P := N.map f.ker.subtype
have hPL : P≤f.ker := by rintro _ ⟨n,hn,rfl⟩; exact n.property
have hPN : f.ker≤Subgroup.normalizer (P:Set H) := normalizer_of_normal_subgroup f.ker N
have hPb : Bounded T (P:Set H) := Bounded.image_trans f.ker.subtype hUb hNb
have hPi : P.relIndex J≠0 := relIndex_map_subgroupOf_of_le J f.ker hKf N hNi
obtain ⟨M,hMn,hMb,hMi⟩ := hgT.promote_integer_normal hT1 hTi hTgen f hf hKf P hPL hPN hPb hPi
let := hMn
let Q := M.map H.subtype
have hQn : H≤Subgroup.normalizer (Q:Set A) := normalizer_of_normal_subgroup H M
have hQb : Bounded S (Q:Set A) := Bounded.image_trans H.subtype hTb hMb
have hQi : Q.relIndex K≠0 := relIndex_map_subgroupOf_nonzero K H M hMi
exact hg.finiteIndex_normal_core h1 hi hgen hK H Q hQn hQb hQi
end CriticalPercolation.RelativeGrowth
end
end
end OAI
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