hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

, the ending lamination theorem, originally conjectured by , states that

hyperbolic 3-manifold In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...

s with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic

lamination Lamination is the technique/process of manufacturing a material in multiple layers, so that the composite material achieves improved strength, stability, sound insulation, appearance, or other properties from the use of the differing materia ...

s on some surfaces in the boundary of the manifold. The ending lamination theorem is a generalization of the Mostow rigidity theorem to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces. and proved the ending lamination conjecture for Kleinian surface groups. In view of the Tameness theorem this implies the ending lamination conjecture for all finitely generated Kleinian groups, from which the general case of ELT follows.

Ending laminations

Ending laminations were introduced by . Suppose that a hyperbolic 3-manifold has a geometrically tame end of the form ''S''×[0,1) for some compact surface ''S'' without boundary, so that ''S'' can be thought of as the "points at infinity" of the end. The ending lamination of this end is (roughly) a lamination on the surface ''S'', in other words a closed subset of ''S'' that is written as the disjoint union of geodesics of ''S''. It is characterized by the following property. Suppose that there is a sequence of closed geodesics on ''S'' whose lifts tends to infinity in the end. Then the limit of these simple geodesics is the ending lamination.

References

* * * * * * * *{{Citation , last1=Thurston , first1=William P. , author1-link=William Thurston , title=Three-dimensional manifolds, Kleinian groups and hyperbolic geometry , doi=10.1090/S0273-0979-1982-15003-0 , mr=648524 , year=1982 , journal=Bulletin of the American Mathematical Society , series=New Series , volume=6 , issue=3 , pages=357–381, doi-access=free Hyperbolic geometry 3-manifolds Kleinian groups