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A 3-manifold is hyperbolic if it satisfies the following (equivalent) conditions:
- it admits a complete Riemannian metric of sectional curvature constant -1,
- it admits a Riemannian metric such that its universal covering (with the pull-back metric) is isometric to hyperbolic 3-space ,
- it is homeomorphic to , for some discrete, torsion-free group of isometries of hyperbolic 3-space.
Here, hyperbolic 3-space is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartan's Theorem, is unique up to isometry. There are different models for , like the upper half-space model, the Poincaré disc model or the hyperboloid model.
The ideal boundary can be identified with the projective line . Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group with the group of conformal automorphisms .
The group acts on the hyperboloid model and one can use this action to identify with the index two subgroup of orientation-preserving isometries. The action is transitive and has as a point stabilizer, thus is isometric to the homogeneous space .
The group acts by fractional-linear automorphisms on . This action on uniquely extends to an action on by orientation-preserving isometries. One can use this action to identify with . The action is transitive and has as a point stabilizer, thus is isometric to the homogeneous space .
Thus, if is oriented, then there are two more equivalent conditions:
An oriented 3-manifold is hyperbolic if and only if
- it is homeomorphic to for some discrete, torsion-free subgroup ,
- it is homeomorphic to for some discrete, torsion-free subgroup .
2 Construction and examples
By Mostow rigidity, complete hyperbolic metrics of finite volume on a 3-manifold are unique up to isometry. This implies that geometric invariants of the hyperbolic metric, such as the volume and the Chern-Simons-invariant, are topological invariants. ...
By the Marden tameness conjecture (proved by Agol and Calegari-Gabai) each hyperbolic 3-manifold with finitely generated fundamental group is the interior of a compact 3-manifold with boundary.
If is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then is a (possibly empty) union of incompressible tori.
Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....
Geometrically finite ends are classified by the corresponding points in Teichmüller space of . (Ahlfors-Bers) ...
Geometrically infinite ends are classified by the corresponding ending laminations. (Brock-Canary-Minsky) ....
5 Further discussion