Hyperbolic 3-manifolds
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Contents |
[edit] 1 Introduction
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
Tex syntax erroris 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
.
[edit] 2 Construction and examples
...
[edit] 3 Invariants
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. ...
[edit] 4 Classification/Characterization
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
Tex syntax erroris an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then
![\partial M](/images/math/a/c/9/ac94b5170de807d6b59ca98c1d6e90ae.png)
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) ....
[edit] 5 Further discussion
...