| import Mathlib |
| import OAI.Algebra.GroupRing.SplitModel |
| import OAI.Algebra.GroupRing.ConcreteTorsion |
|
|
| namespace OAI |
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| noncomputable section |
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| namespace TorsionFreeZeroDivisors |
| lemma HasFiniteTwoDimensionalClassifyingSpace.of_equiv {G H : Type} [Group G] [Group H] |
| (e : G ≃* H) (h : HasFiniteTwoDimensionalClassifyingSpace H) : |
| HasFiniteTwoDimensionalClassifyingSpace G := by |
| rcases h with ⟨X,topX,hT,hP,cw,hf,hh,h2,x,⟨k⟩,E,topE,hE,p,hp,hps⟩ |
| exact ⟨X,topX,hT,hP,cw,hf,hh,h2,x,⟨e.trans k⟩,E,topE,hE,p,hp,hps⟩ |
| end TorsionFreeZeroDivisors |
| namespace TorsionFreeZeroDivisors.ConcreteModel |
| open scoped Classical |
| open SampleGraph |
| variable (rep : ℕ) (r : ActualPatternEvent.Match rep) |
| (x : VA rep) (y : VB rep) |
| (hL : 2≤GirthAsymptotics.L (TypedGraphs.size rep)) (hd : GraphSample.Diameters rep r) |
| (hsample : r∈ActualPatternEvent.samples rep) |
| (havoid : IsEmpty (BoundedSystem (PlanarParameters.C PathSystem.ε ClosedWords.D) |
| (PlanarParameters.K PathSystem.ε ClosedWords.D) (PlanarParameters.sides PathSystem.ε ClosedWords.D) rep r x y)) |
| include hL hd hsample havoid in |
| /-- The finite exact two-dimensional model of the actual sampled graph group. |
| The cover contraction is built by absorbing physical disk-track obstructions |
| through the proved integral cell-boundary splitting. -/ |
| theorem has_model : HasFiniteTwoDimensionalClassifyingSpace (ConcreteGroup.G rep r) := by |
| have hex {O : Type} [Fintype O] [Nonempty O] |
| (a : SphericalPicture.Data (O:=O) (graph rep r) (ConcreteTorsion.signedLabels rep) WordPairing.inverseLetter) : ¬a.Reduced := by |
| intro ha |
| exact SpherePicture.excluded (ConcreteTorsion.unsigned rep r a) ha hL hd hsample havoid |
| obtain ⟨t,ht⟩ := Finset.card_pos.mp (show 0<(Types.outgoingA x).card by |
| have h := (Types.degreeA_bounds x).1 |
| omega) |
| let : Nonempty (GraphConePresentation.Edge (graph rep r)) := |
| ⟨GraphConePresentation.edge (graph rep r) (Sum.inl ⟨(x,t),ht⟩)⟩ |
| have hm := RelationModel.has_model |
| (ConeArea.words (graph rep r) (ConcreteGroup.routes rep r) (ConcreteTorsion.signedLabels rep)) |
| (ConeTorsion.total_zero (graph rep r) (ConcreteGroup.routes rep r) (ConcreteTorsion.signedLabels rep) |
| (ConcreteTorsion.signedLabels_reverse rep r) hex) |
| unfold RelationArea.G at hm |
| rw [ConeTorsion.relations_eq _ _ _ (ConcreteTorsion.signedLabels_reverse rep r)] at hm |
| rw [ConcreteTorsion.letters_eq] at hm |
| exact HasFiniteTwoDimensionalClassifyingSpace.of_equiv |
| (GraphConePresentation.equivalence (graph rep r) (ConcreteGroup.labels rep) (ConcreteGroup.routes rep r) |
| (ConcreteGroup.labels_reverse rep r) (GraphConePresentation.canonical_base _)) hm |
| end TorsionFreeZeroDivisors.ConcreteModel |
|
|
| end |
|
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| end OAI |
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|
