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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.StrongAbstractSpan
import OAI.Probability.CriticalPercolation.LinearGroups.InvariantSubspaceDensity
import OAI.Probability.CriticalPercolation.LinearGroups.InvariantBlockSolvability
import OAI.Probability.CriticalPercolation.Nilpotent.CharacteristicNilpotentCore
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
lemma finiteIndex_comap_surjective {A B : Type*} [Group A] [Group B]
(f : A →* B) (hf : Surjective f) (P : Subgroup B) [P.FiniteIndex] :
(P.comap f).FiniteIndex := by
constructor
rw [Subgroup.index_comap,f.range_eq_top_of_surjective hf,Subgroup.relIndex_top_right]
exact Subgroup.FiniteIndex.index_ne_zero
lemma solvable_range_of_mem {A B : Type*} [Group A] [Group B]
(f : A →* B) (P : Subgroup B) [Group.IsSolvable P] (hp : ∀ a, f a∈P) :
Group.IsSolvable f.range := by
have hh : f.range≤P := by rintro _ ⟨a,rfl⟩; exact hp a
exact Group.isSolvable_of_isSolvable_injective (f:=Subgroup.inclusion hh)
(Subgroup.inclusion_injective hh)
lemma solvable_restricted_range {A B : Type*} [Group A] [Group B]
(f : A →* B) (P : Subgroup f.range) [Group.IsSolvable P]
(J : Subgroup A) (hJ : J≤P.comap f.rangeRestrict) :
Group.IsSolvable (f.comp J.subtype).range := by
let Q := P.map f.range.subtype
let e := P.equivMapOfInjective f.range.subtype Subtype.val_injective
let : Group.IsSolvable Q := Group.isSolvable_of_surjective (f:=e.toMonoidHom) e.surjective
apply solvable_range_of_mem (f.comp J.subtype) Q
intro j
exact ⟨f.rangeRestrict j.val,hJ j.property,rfl⟩
lemma virtually_solvable_linear_of_rank (K : Type) [Field K] [CharZero K]
[Countable K] [IsAlgClosed K] (r : ℕ) :
∀ (V G : Type) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
[Group G] [DecidableEq G], finrank K V=r →
∀ (ρ : G →* (V ≃ₗ[K] V)), Injective ρ →
∀ d, SparseGrowth.HasPolynomialDegree G d →
∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by
induction r using Nat.strong_induction_on with
| h r ih =>
intro V G _ _ _ _ _ hr ρ hρ d hg
rcases subsingleton_or_nontrivial V with hv | hv
· let := hv
have hc : ∀ g h : G, g*h=h*g := by
intro g h
apply hρ
exact Subsingleton.elim _ _
let : Group.IsSolvable G := Group.isSolvable_of_comm hc
exact ⟨⊤,inferInstance,inferInstance⟩
let := hv
by_cases hd : StrongFullSpan ρ
· obtain ⟨H,hHi,hH⟩ := hd.virtually_comm_range ρ hg
let : Group.IsSolvable (ρ.comp H.subtype).range := Group.isSolvable_of_comm hH
have hinj : Injective (ρ.comp H.subtype).rangeRestrict := by
intro x y he
apply Subtype.ext
apply hρ
exact congrArg Subtype.val he
exact ⟨H,hHi,Group.isSolvable_of_isSolvable_injective hinj⟩
obtain ⟨H,hHi,hH⟩ : ∃ H : Subgroup G, H.FiniteIndex ∧ ¬FullSpan (ρ.comp H.subtype) := by
simp only [StrongFullSpan,not_forall] at hd
obtain ⟨H,hHi,hH⟩ := hd
exact ⟨H,hHi,hH⟩
let := hHi
let ρH := ρ.comp H.subtype
have hgH := hg.finiteIndex_subgroup H
obtain ⟨W,hW0,hW1,hW⟩ := invariant_subspace_of_not_fullSpan ρH hH
let σ := subspaceRepresentation ρH W hW
let τ := quotientRepresentation ρH W hW
have hWrank : finrank K W<r := hr ▸ Submodule.finrank_lt hW1
have hQrank : finrank K (V ⧸ W)<r := by
have he := W.finrank_quotient_add_finrank
have hp : 0 < finrank K W := by
let : Nontrivial W := Submodule.nontrivial_iff_ne_bot.mpr hW0
exact Module.finrank_pos (R:=K) (M:=W)
omega
have hgσ := hgH.of_surjective σ.rangeRestrict σ.rangeRestrict_surjective
have hgτ := hgH.of_surjective τ.rangeRestrict τ.rangeRestrict_surjective
obtain ⟨P,hPi,hPs⟩ := ih _ hWrank W σ.range rfl σ.range.subtype Subtype.val_injective d hgσ
obtain ⟨Q,hQi,hQs⟩ := ih _ hQrank (V ⧸ W) τ.range rfl τ.range.subtype Subtype.val_injective d hgτ
let := hPi
let := hQi
let := hPs
let := hQs
let J := P.comap σ.rangeRestrict ⊓ Q.comap τ.rangeRestrict
let : (P.comap σ.rangeRestrict).FiniteIndex :=
finiteIndex_comap_surjective σ.rangeRestrict σ.rangeRestrict_surjective P
let : (Q.comap τ.rangeRestrict).FiniteIndex :=
finiteIndex_comap_surjective τ.rangeRestrict τ.rangeRestrict_surjective Q
let : J.FiniteIndex := inferInstance
have hWJ (j : J) : W.map (ρH j.val).toLinearMap=W := hW j.val
have hsJ : subspaceRepresentation (ρH.comp J.subtype) W hWJ=σ.comp J.subtype := by
ext j w; rfl
have htJ : quotientRepresentation (ρH.comp J.subtype) W hWJ=τ.comp J.subtype := by
ext j v
induction v using Submodule.Quotient.induction_on
rfl
let : Group.IsSolvable (subspaceRepresentation (ρH.comp J.subtype) W hWJ).range :=
hsJ.symm ▸ solvable_restricted_range σ P J inf_le_left
let : Group.IsSolvable (quotientRepresentation (ρH.comp J.subtype) W hWJ).range :=
htJ.symm ▸ solvable_restricted_range τ Q J inf_le_right
let : Group.IsSolvable J := solvable_of_subspace_quotient (ρH.comp J.subtype)
((hρ.comp Subtype.val_injective).comp Subtype.val_injective) W hWJ
let L := J.map H.subtype
let e := J.equivMapOfInjective H.subtype Subtype.val_injective
have hLi : L.FiniteIndex := by
constructor
rw [Subgroup.index_map_subtype]
exact mul_ne_zero (inferInstance : J.FiniteIndex).index_ne_zero hHi.index_ne_zero
exact ⟨L,hLi,Group.isSolvable_of_surjective (f:=e.toMonoidHom) e.surjective⟩
lemma virtually_solvable_linear (K V G : Type) [Field K] [CharZero K]
[Countable K] [IsAlgClosed K] [AddCommGroup V] [Module K V] [FiniteDimensional K V]
[Group G] [DecidableEq G] (ρ : G →* (V ≃ₗ[K] V)) (hρ : Injective ρ)
{degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) :
∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H :=
virtually_solvable_linear_of_rank K (finrank K V) V G rfl ρ hρ degree hg
end CriticalPercolation.LinearGrowth
end
end
end OAI
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