Hyperbolic 3manifolds
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Contents 
1 Introduction
A 3manifold 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 pullback metric) is isometric to hyperbolic 3space ,
 it is homeomorphic to , for some discrete, torsionfree group of isometries of hyperbolic 3space.
Here, hyperbolic 3space 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 halfspace model, the Poincaré disc model or the hyperboloid model.
The ideal boundary can be identified with the projective line . Isometries of hyperbolic 3space 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 orientationpreserving isometries. The action is transitive and has as a point stabilizer, thus is isometric to the homogeneous space .
The group acts by fractionallinear automorphisms on . This action on uniquely extends to an action on by orientationpreserving 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 3manifold is hyperbolic if and only if
 it is homeomorphic to for some discrete, torsionfree subgroup ,

it is homeomorphic to
for some discrete, torsionfree subgroup .
2 Construction and examples
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3 Invariants
By Mostow rigidity, complete hyperbolic metrics of finite volume on a 3manifold are unique up to isometry. This implies that geometric invariants of the hyperbolic metric, such as the volume and the ChernSimonsinvariant, are topological invariants. ...
4 Classification/Characterization
By the Marden tameness conjecture (proved by Agol and CalegariGabai) each hyperbolic 3manifold with finitely generated fundamental group is the interior of a compact 3manifold with boundary.
If is an orientable 3manifold 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 . (AhlforsBers) ...
Geometrically infinite ends are classified by the corresponding ending laminations. (BrockCanaryMinsky) ....
5 Further discussion
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