| import OAI.GroupTheory.Hyperbolic.Links |
|
|
| namespace OAI |
|
|
| namespace Release075.MarkedLineData |
| attribute [local instance] Classical.propDecidable Classical.decEq |
| variable {q r : ℕ} [Fact q.Prime] (d : MarkedLineData q r) (hr : 0 < r) |
|
|
| theorem rectangle_turn_large (i : d.Label) (u u' v v' : Fin r) |
| (hu : u ≠ u') (hv : v ≠ v') |
| (a b : (d.presentation hr).OrientedEdge) |
| (ha : a ∈ [(d.presentation hr).orientedX i u v, |
| (d.presentation hr).flipEdge ((d.presentation hr).orientedX i u' v), |
| (d.presentation hr).orientedX i u' v', |
| (d.presentation hr).flipEdge ((d.presentation hr).orientedX i u v')]) |
| (hb : b ∈ [(d.presentation hr).orientedX i u v, |
| (d.presentation hr).flipEdge ((d.presentation hr).orientedX i u' v), |
| (d.presentation hr).orientedX i u' v', |
| (d.presentation hr).flipEdge ((d.presentation hr).orientedX i u v')]) |
| (ht : (d.presentation hr).edgeTarget ((d.presentation hr).flipEdge b) = |
| (d.presentation hr).edgeTarget a) |
| (p : (d.allLink hr).Walk (d.linkDirection hr ((d.presentation hr).flipEdge b)) |
| (d.linkDirection hr a)) (hp : ReducedWalk p) : |
| 5 ≤ TriangleRibbon.weight (d.linkColor hr (d.linkDirection hr a)) * p.length := by |
| simp only [List.mem_cons,List.not_mem_nil,or_false] at ha hb |
| rcases ha with rfl|rfl|rfl|rfl |
| all_goals rcases hb with rfl|rfl|rfl|rfl |
| all_goals simp only [BlockPresentation.orientedX,BlockPresentation.flipEdge, |
| BlockPresentation.edgeTarget,Bool.not_true,Bool.not_false] at ht |
| all_goals cases ht |
| all_goals |
| first |
| | exact d.x_turn_large hr i u v u' v true false rfl |
| (by intro he; exact hu (congrArg Prod.fst he)) p hp |
| | exact d.x_turn_large hr i u v u v' true false rfl |
| (by intro he; exact hv (congrArg Prod.snd he)) p hp |
| | exact d.x_turn_large hr i u' v u v false true rfl |
| (by intro he; exact hu.symm (congrArg Prod.fst he)) p hp |
| | exact d.x_turn_large hr i u' v u' v' false true rfl |
| (by intro he; exact hv (congrArg Prod.snd he)) p hp |
| | exact d.x_turn_large hr i u' v' u' v true false rfl |
| (by intro he; exact hv.symm (congrArg Prod.snd he)) p hp |
| | exact d.x_turn_large hr i u' v' u v' true false rfl |
| (by intro he; exact hu.symm (congrArg Prod.fst he)) p hp |
| | exact d.x_turn_large hr i u v' u v false true rfl |
| (by intro he; exact hv.symm (congrArg Prod.snd he)) p hp |
| | exact d.x_turn_large hr i u v' u' v' false true rfl |
| (by intro he; exact hu (congrArg Prod.fst he)) p hp |
| |
| /-- Every actual rectangular word in the fixed presentation is nontrivial: |
| nullhomotopy gives a minimal labelled planar filling, whose reduced link sectors |
| contradict its Euler-characteristic-one angular identity. -/ |
| theorem rectangular_nontriviality : d.RectangularNontriviality hr := by |
| intro i u u' v v' hu hv he |
| have hf := (d.presentation hr).rectangle_fillable_of_eq_one i u u' v v' he |
| obtain ⟨D,hD⟩ := TriangleFilling.exists_noDipole hf |
| have hn : D.boundary ≠ [] := by |
| intro hz |
| have hh := D.boundary_word |
| rw [hz] at hh |
| cases hh |
| obtain ⟨b,hb⟩ := List.exists_mem_of_ne_nil D.boundary hn |
| apply (d.concreteRibbon hr D b hb).not_all_turns_geodesic |
| (d.allLink hr) (d.dartDirection hr D) |
| (d.ribbon_adj hr D b hb) (d.ribbon_reduced hr D b hb hD) |
| (d.ribbon_closed hr D b hb) |
| intro a ha p hp |
| have ha' : a.val ∈ D.boundary := (d.ribbon_outer_iff hr D b hb a).mp ha |
| have hb' : D.next a.val ∈ D.boundary := (D.boundary_cycle.mem_iff a.val).mpr ha' |
| have hal : D.label a.val ∈ _ := List.mem_map.mpr ⟨a.val,ha',rfl⟩ |
| have hbl : D.label (D.next a.val) ∈ _ := List.mem_map.mpr ⟨D.next a.val,hb',rfl⟩ |
| rw [D.boundary_word] at hal hbl |
| rw [d.ribbon_color hr D b hb a] |
| have ht : (d.presentation hr).edgeTarget ((d.presentation hr).flipEdge (D.label (D.next a.val))) = |
| (d.presentation hr).edgeTarget (D.label a.val) := by |
| rw [← D.reverse_label] |
| exact D.color_next a.val |
| have hs := d.ribbon_direction_step hr D b hb a |
| let p' := p.copy hs rfl |
| have hp' : ReducedWalk p' := by |
| intro j hj |
| simpa only [p',SimpleGraph.Walk.getVert_copy] using |
| hp j (by simpa only [p',SimpleGraph.Walk.length_copy] using hj) |
| have hh := d.rectangle_turn_large hr i u u' v v' hu hv _ _ hal hbl ht p' hp' |
| have hlen : p'.length = p.length := SimpleGraph.Walk.length_copy p hs rfl |
| change 5 ≤ TriangleRibbon.weight (d.linkColor hr (d.linkDirection hr (D.label a.val))) * p'.length at hh |
| rw [hlen] at hh |
| exact hh |
|
|
| end Release075.MarkedLineData |
|
|
| namespace Release075.MarkedLineData |
| open Equiv PermCycles |
| attribute [local instance] Classical.propDecidable Classical.decEq |
| variable {q r : ℕ} [Fact q.Prime] (d : MarkedLineData q r) (hr : 0 < r) |
| variable {w : List (d.presentation hr).OrientedEdge} |
| variable (D : TriangleFilling (d.presentation hr).flipEdge (d.presentation hr).edgeTarget |
| (d.presentation hr).TriangleFace w) (b : D.Dart) (hb : b ∈ D.boundary) |
|
|
| theorem ribbon_closed_strict (a : D.ActiveDart) |
| (p : (d.allLink hr).Walk (d.dartDirection hr D a) (d.dartDirection hr D a)) |
| (hn : 0 < p.length) (hp : ReducedWalk p) : |
| 12 ≤ TriangleRibbon.weight ((d.concreteRibbon hr D b hb).color |
| (cl (d.concreteRibbon hr D b hb).vertexPerm a)) * p.length := by |
| rw [d.ribbon_color hr D b hb a] |
| exact d.allLink_closed hr _ p hn hp |
|
|
| theorem ribbon_boundary_length : (d.concreteRibbon hr D b hb).counts.boundaryLength = w.length := by |
| let R := d.concreteRibbon hr D b hb |
| change (∑ v : R.Vertex, R.visits v) = _ |
| rw [R.visits_sum] |
| let e : {a : R.Dart // cl R.face a = R.outer} ≃ {a : D.Dart // a ∈ D.boundary} := { |
| toFun a := ⟨a.val.val,(d.ribbon_outer_iff hr D b hb a.val).mp a.property⟩ |
| invFun a := ⟨⟨a.val,D.active_of_mem a.property⟩, |
| (d.ribbon_outer_iff hr D b hb _).mpr a.property⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl } |
| rw [Fintype.card_congr e,Fintype.card_subtype] |
| have hset : Finset.univ.filter (fun a => a ∈ D.boundary) = D.boundary.toFinset := by |
| ext a; simp |
| rw [hset,List.toFinset_card_of_nodup D.boundary_cycle.1] |
| exact (List.length_map D.label).symm.trans (congrArg List.length D.boundary_word) |
| |
| include b hb in |
| theorem reduced_linear_active_bound (hD : IsEmpty D.Dipole) : |
| D.activeCount + 36 ≤ 19*w.length := by |
| let R := d.concreteRibbon hr D b hb |
| have hstrict := R.interior_angle_lower (d.allLink hr) (d.dartDirection hr D) |
| (d.ribbon_adj hr D b hb) (d.ribbon_reduced hr D b hb hD) 12 |
| (d.ribbon_closed_strict hr D b hb) |
| have hface := R.counts.linear_face_bound hstrict |
| have hlen := d.ribbon_boundary_length hr D b hb |
| change R.counts.boundaryLength = w.length at hlen |
| rw [hlen] at hface |
| have hdart : Fintype.card R.Dart = D.activeCount := by |
| change Fintype.card D.ActiveDart = _ |
| exact Fintype.card_subtype _ |
| have he := R.exterior_count |
| rw [← R.dart_card,hdart] at he |
| change R.counts.boundaryLength + 3*R.counts.faces = D.activeCount at he |
| rw [hlen] at he |
| omega |
|
|
| /-- An actual finite nullhomotopy has a genuine planar filling of linearly |
| bounded size, not an assumed Dehn-function estimate. This estimate is a necessary |
| bridge to the uniform Cayley thinness argument below. -/ |
| theorem exists_linear_filling (hf : Fillable (d.presentation hr).flipEdge |
| (d.presentation hr).edgeTarget (d.presentation hr).TriangleFace w) : |
| ∃ D : TriangleFilling (d.presentation hr).flipEdge (d.presentation hr).edgeTarget |
| (d.presentation hr).TriangleFace w, D.activeCount ≤ 19*w.length := by |
| by_cases hw : w = [] |
| · subst w |
| refine ⟨TriangleFilling.empty,?_⟩ |
| unfold TriangleFilling.activeCount |
| apply (Finset.card_le_univ _).trans |
| change 0 ≤ _ |
| exact Nat.zero_le _ |
| · obtain ⟨D,hD⟩ := TriangleFilling.exists_noDipole hf |
| have hn : D.boundary ≠ [] := by |
| intro he |
| apply hw |
| rw [← D.boundary_word,he,List.map_nil] |
| obtain ⟨b,hb⟩ := List.exists_mem_of_ne_nil D.boundary hn |
| exact ⟨D,by have hh := d.reduced_linear_active_bound hr D b hb hD; omega⟩ |
| |
| /-- The fixed presentation's non-residual-finiteness is now unconditional in |
| its finite incidence data; no rectangular nontriviality premise remains. -/ |
| theorem actual_presentation_not_residuallyFinite (hq : r < q) (hq200 : 200 ≤ q) : |
| ¬ Group.ResiduallyFinite (d.presentation hr).GroupType := |
| d.presentation_not_residuallyFinite hr hq hq200 (d.rectangular_nontriviality hr) |
|
|
| end Release075.MarkedLineData |
|
|
| end OAI |
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|
