| import Mathlib | |
| import OAI.Geometry.HyperbolicGroups.DevelopmentGeometricAction | |
| import OAI.Geometry.HyperbolicGroups.LocalGeodesic | |
| import OAI.Geometry.HyperbolicGroups.GeometricSpaceTransport | |
| namespace OAI | |
| section | |
| /-! Compact locally CAT(-1) geodesic metrics yield genuine geometric CAT(0) | |
| actions of the actual fundamental group. -/ | |
| noncomputable section | |
| namespace HyperbolicObstruction | |
| open Set Metric | |
| open scoped Topology ContinuousMap | |
| universe u | |
| lemma not_noGeometricCATZeroAction_of_compact_localCATNegOne | |
| {X : Type} [MetricSpace X] [CompactSpace X] (o : X) (g : IsGeodesic X) | |
| (hl : ∀ x : X, ∃ r : ℝ, 0<r ∧ CATNegOne (ball x r)) : | |
| ¬ NoGeometricCATZeroAction.{u} (FundamentalGroup X o) := by | |
| intro h | |
| obtain ⟨r,hr,hc⟩:=uniformLocalCATZero_of_compact hl | |
| let e := LocalGeodesic.deckPiOneEquiv (o:=o) hr hc g | |
| let A := LocalGeodesic.deckGeometricAction (o:=o) hr hc g | |
| let := LocalGeodesic.intrinsicMetric (o:=o) hr hc g | |
| let := LocalGeodesic.developed_proper (o:=o) hr hc g | |
| let : Nonempty (LocalGeodesic o) := ⟨LocalGeodesic.constant o⟩ | |
| let Y := ULift.{u} (LocalGeodesic o) | |
| let : ProperSpace Y := ulift_proper | |
| have hcat : CATZero Y := CATZero.of_isometryEquiv (uliftIsometry (LocalGeodesic o)) | |
| (LocalGeodesic.developed_catZero hr hc g) | |
| have : IsEmpty (GeometricAction (FundamentalGroup X o) Y) := | |
| h Y inferInstance inferInstance inferInstance inferInstance hcat | |
| exact isEmptyElim ((A.pullback e).spaceTransport (uliftIsometry (LocalGeodesic o))) | |
| lemma incompatibleLocalCATNegOne_of_noAction (L : FiniteComplex) (y : L.Carrier) | |
| (h : NoGeometricCATZeroAction.{u} (FundamentalGroup L.Carrier y)) : | |
| ¬ CompatibleLocallyCATNegOne L := by | |
| rintro ⟨m,hm,hg,hl⟩ | |
| have hcompact : CompactSpace L.Carrier := | |
| inferInstanceAs (CompactSpace (FinitePolyhedron.Carrier L.complex)) | |
| let P : TopologicalSpace L.Carrier → Prop := fun t => | |
| letI := t | |
| NoGeometricCATZeroAction.{u} (FundamentalGroup L.Carrier y) | |
| let := m | |
| let : TopologicalSpace L.Carrier := m.toUniformSpace.toTopologicalSpace | |
| let : CompactSpace L.Carrier := by | |
| change @CompactSpace L.Carrier m.toUniformSpace.toTopologicalSpace | |
| rw [hm] | |
| exact hcompact | |
| apply not_noGeometricCATZeroAction_of_compact_localCATNegOne y hg hl | |
| exact (congrArg P hm).mpr h | |
| lemma homotopy_incompatibleLocalCATNegOne {K L : FiniteComplex} (x : K.Carrier) | |
| (h : NoGeometricCATZeroAction.{u} (FundamentalGroup K.Carrier x)) | |
| (e : K.Carrier ≃ₕ L.Carrier) : ¬ CompatibleLocallyCATNegOne L := | |
| incompatibleLocalCATNegOne_of_noAction L (e x) | |
| (h.of_mulEquiv (fundamentalGroupHomotopyEquiv e x)) | |
| end HyperbolicObstruction | |
| end | |
| end | |
| section | |
| end | |
| end OAI | |