hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, the ending lamination theorem, originally

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d by as the eleventh problem out of his twenty-four questions, 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 group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

s are determined by their

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

together with certain "end invariants", which are geodesic

lamination Simulated flight (using image stack created by μCT scanning) through the length of a knitting needle that consists of laminated wooden layers: the layers can be differentiated by the change of direction of the wood's vessels Shattered windshi ...

s on some surfaces in the boundary of the

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

. 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 group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s. 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 Conjectures that have been proved Theorems in differential geometry

Ending laminations

See also

References