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8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.PlaneTopology.HomeoPlane
import OAI.Geometry.Cannon.Boundary.BoundaryUniformComplementGraph
import OAI.Geometry.Cannon.Boundary.BoundaryUniformConnectedComplement
import OAI.Geometry.Cannon.PlaneTopology.ChartTameFaceCount
import OAI.Geometry.Cannon.PlaneTopology.BoundaryBallCover
import OAI.Geometry.Cannon.PlaneTopology.InteriorRadiusControl
import OAI.Geometry.Cannon.Boundary.BoundaryUniformRingPacking
import OAI.Geometry.Cannon.Uniformization.BoundaryUniformizationInputs
import OAI.Geometry.Cannon.Conformal.QuasiMobius
import OAI.Geometry.Cannon.Conformal.PlaneWeakQSBounded
import OAI.Geometry.Cannon.Actions.BoundaryAction
namespace OAI
/-! A group with a hyperbolic Cayley graph and spherical boundary admits a proper, cocompact action on hyperbolic three-space with finite kernel. -/
section
noncomputable section
namespace CannonRelease
universe u
theorem cannon (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G]
(D : CayleyData G) (hthin : D.ThinTriangles)
(hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) :
∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧
(ρ.ker : Set G).Finite := by
obtain ⟨δ,hδ⟩ := hthin
obtain ⟨φ⟩ := hsphere
let _ := visualMetricSpace D hδ
obtain ⟨ψ,hψ⟩ := boundary_quasiMobius_uniformizer D hδ φ
exact boundary_uniformized_action D hδ ψ hψ
end CannonRelease
end
end
end OAI
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