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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 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.AlgebraicVirtualSolvability
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical Cardinal
namespace CriticalPercolation.LinearGrowth
lemma countable_finiteType_int (D : Type) [CommRing D] [Algebra ℤ D]
[Algebra.FiniteType ℤ D] : Countable D := by
obtain ⟨n,f,hf⟩ := (Algebra.FiniteType.iff_quotient_mvPolynomial).mp
(inferInstance : Algebra.FiniteType ℤ D)
let : Countable (MvPolynomial n ℤ) := by
apply Function.Injective.countable (f := AddMonoidAlgebra.coeff)
intro x y h
cases x; cases y; cases h; rfl
exact Function.Surjective.countable hf
lemma countable_fraction (D : Type) [CommRing D] [Countable D] : Countable (FractionRing D) := by
apply Cardinal.mk_le_aleph0_iff.mp
rw [Cardinal.mk_fractionRing]
exact Cardinal.mk_le_aleph0
lemma countable_algebraicClosure (F : Type) [Field F] [Countable F] :
Countable (AlgebraicClosure F) := by
have h := Algebraic.countable F (AlgebraicClosure F)
have he : {x : AlgebraicClosure F | IsAlgebraic F x}=Set.univ :=
Set.eq_univ_of_forall (fun _ => Algebra.IsAlgebraic.isAlgebraic _)
rw [he] at h
exact Set.countable_univ_iff.mp h
lemma matrixRepresentation_injective {K G n : Type} [Field K] [Group G]
[Fintype n] [DecidableEq n] (ρ : G →* GL n K) (hρ : Injective ρ) :
Injective (matrixRepresentation ρ) :=
(LinearMap.GeneralLinearGroup.generalLinearEquiv K (n → K)).injective.comp
(Matrix.GeneralLinearGroup.toLin.injective.comp hρ)
lemma virtually_solvable_real_matrix {G n : Type} [Group G] [DecidableEq G]
[Fintype n] [DecidableEq n] (ρ : G →* GL n ℝ) (hρ : Injective ρ)
{degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) :
∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by
let := hg.fg
obtain ⟨D,hD,hm⟩ := exists_finiteType_matrix_ring ρ
let := hD
let : Countable D := countable_finiteType_int D
let F := FractionRing D
let K := AlgebraicClosure F
let : Countable F := countable_fraction D
let : Countable K := countable_algebraicClosure F
let f : D →+* K := (algebraMap F K).comp (algebraMap D F)
have hf : Injective f := (algebraMap F K).injective.comp (IsFractionRing.injective D F)
let σ := liftMatrixGroup ρ D hm
let τ := (Matrix.GeneralLinearGroup.map f).comp σ
have hσ : Injective σ := liftMatrixGroup_injective ρ D hm hρ
have hτ : Injective τ := by
intro x y h
apply hσ
apply Units.ext
apply Matrix.ext
intro i j
apply hf
exact congrArg (fun M : GL n K => M i j) h
exact virtually_solvable_linear K (n → K) G (matrixRepresentation τ)
(matrixRepresentation_injective τ hτ) hg
lemma virtually_solvable_real_linear {G V : Type} [Group G] [DecidableEq G]
[AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V]
(ρ : G →* (V ≃ₗ[ℝ] V)) (hρ : Injective ρ)
{degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) :
∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by
let τ := (linearMatrixEquiv (K:=ℝ) (V:=V)).toMonoidHom.comp ρ
exact virtually_solvable_real_matrix τ (linearMatrixEquiv.injective.comp hρ) hg
end CriticalPercolation.LinearGrowth
namespace CriticalPercolation.LinearGrowth
lemma virtually_solvable_real_linear_general {G V : Type*} [Group G] [DecidableEq G]
[AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V]
(ρ : G →* (V ≃ₗ[ℝ] V)) (hρ : Injective ρ)
{degree : ℕ} (hg : SparseGrowth.HasPolynomialDegree G degree) :
∃ H : Subgroup G, H.FiniteIndex ∧ Group.IsSolvable H := by
let τ := (linearMatrixEquiv (K:=ℝ) (V:=V)).toMonoidHom.comp ρ
have hτ : Injective τ := linearMatrixEquiv.injective.comp hρ
have hgτ := hg.of_surjective τ.rangeRestrict τ.rangeRestrict_surjective
obtain ⟨P,hPi,hPs⟩ := virtually_solvable_real_matrix τ.range.subtype Subtype.val_injective hgτ
let := hPi
let := hPs
let H := P.comap τ.rangeRestrict
have hHi : H.FiniteIndex := finiteIndex_comap_surjective τ.rangeRestrict τ.rangeRestrict_surjective P
let ψ : H →* P := (τ.rangeRestrict.comp H.subtype).codRestrict P (fun h => h.property)
have hψ : Injective ψ := by
intro x y he
apply Subtype.ext
apply hτ
exact congrArg (fun z : P => (z.val : GL (Fin (finrank ℝ V)) ℝ)) he
exact ⟨H,hHi,Group.isSolvable_of_isSolvable_injective hψ⟩
end CriticalPercolation.LinearGrowth
end
end
end OAI
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