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544609e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | 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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