# Hyperbolic 3-manifolds

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## 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 ${\Bbb H}^3$${{Stub}} == 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 {\Bbb H}^3,
- it is homeomorphic to \Gamma\backslash{\Bbb H}^3, for some discrete, torsion-free group of isometries of hyperbolic 3-space.

Here, hyperbolic 3-space {\Bbb H}^3 is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, {\Bbb H}^3 is unique up to isometry. There are different models for {\Bbb H}^3, like the upper half-space model, the Poincaré disc model or the hyperboloid model.
The ideal boundary \partial_\infty{\Bbb H}^3 can be identified with projective space P^1{\Bbb C}\cong S^2. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group Isom\left({\Bbb H}^3\right) with the group of conformal automorphisms Conf\left(S^2\right).

The group SO\left(3,1\right) acts on the hyperboloid model and one can use this action to identify SO\left(3,1\right) with the index two subgroup Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}\right) of orientation-preserving isometries. The action is transitive and has SO\left(3\right) as a point stabilizer, thus {\Bbb H}^3 is isometric to the homogeneous space SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right).
The group PSL\left(2,{\Bbb C}\right) acts by fractional-linear automorphisms on P^1{\Bbb C}. This action on \partial_\infty{\Bbb H}^3 uniquely extends to an action on {\Bbb H}^3 by orientation-preserving isometries. One can use this action to identify PSL\left(2,{\Bbb C}\right) with Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right). The action is transitive and has PSU\left(2\right) as a point stabilizer, thus {\Bbb H}^3 is isometric to the homogeneous space PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right).

Thus, if M is oriented, then there are two more equivalent conditions:
An oriented 3-manifold is hyperbolic if and only if
- it is homeomorphic to \Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right) for some discrete, torsion-free subgroup \Gamma\subset PSL\left(2,{\Bbb C}\right),
- it is homeomorphic to \Gamma\backslash SO\left(3,1\right)/SO\left(3\right) for some discrete, torsion-free subgroup \Gamma\subset SO\left(3,1\right).
== Construction and examples == ; ... == 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. ... == 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 M is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then \partial M is a (possibly empty) union of incompressible tori.

Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....
== Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]{\Bbb H}^3$
,
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$$\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.

Here, hyperbolic 3-space ${\Bbb H}^3$${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartans Theorem, ${\Bbb H}^3$${\Bbb H}^3$ is unique up to isometry. There are different models for ${\Bbb H}^3$${\Bbb H}^3$, like the upper half-space model, the Poincaré disc model or the hyperboloid model.
The ideal boundary $\partial_\infty{\Bbb H}^3$$\partial_\infty{\Bbb H}^3$ can be identified with projective space $P^1{\Bbb C}\cong S^2$$P^1{\Bbb C}\cong S^2$. Isometries of hyperbolic 3-space act as conformal automorphisms of the ideal boundary. Thus we can identify the isometry group $Isom\left({\Bbb H}^3\right)$$Isom\left({\Bbb H}^3\right)$ with the group of conformal automorphisms $Conf\left(S^2\right)$$Conf\left(S^2\right)$.

The group $SO\left(3,1\right)$$SO\left(3,1\right)$ acts on the hyperboloid model and one can use this action to identify $SO\left(3,1\right)$$SO\left(3,1\right)$ with the index two subgroup $Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}\right)$$Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$$SO\left(3\right)$ as a point stabilizer, thus ${\Bbb H}^3$${\Bbb H}^3$ is isometric to the homogeneous space $SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$$SO\left(3,1\right)/SO\left(3\right)\cong Spin\left(3,1\right)/Spin\left(3\right)$.
The group $PSL\left(2,{\Bbb C}\right)$$PSL\left(2,{\Bbb C}\right)$ acts by fractional-linear automorphisms on $P^1{\Bbb C}$$P^1{\Bbb C}$. This action on $\partial_\infty{\Bbb H}^3$$\partial_\infty{\Bbb H}^3$ uniquely extends to an action on ${\Bbb H}^3$${\Bbb H}^3$ by orientation-preserving isometries. One can use this action to identify $PSL\left(2,{\Bbb C}\right)$$PSL\left(2,{\Bbb C}\right)$ with $Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$$Conf^+\left(S^2\right)=Isom^+\left({\Bbb H}^3\right)$. The action is transitive and has $PSU\left(2\right)$$PSU\left(2\right)$ as a point stabilizer, thus ${\Bbb H}^3$${\Bbb H}^3$ is isometric to the homogeneous space $PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$$PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)\cong SL\left(2,{\Bbb C}\right)/SU\left(2\right)$.

Thus, if $M$$M$ is oriented, then there are two more equivalent conditions:
An oriented 3-manifold is hyperbolic if and only if
- it is homeomorphic to $\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$$\Gamma\backslash PSL\left(2,{\Bbb C}\right)/PSU\left(2\right)$ for some discrete, torsion-free subgroup $\Gamma\subset PSL\left(2,{\Bbb C}\right)$$\Gamma\subset PSL\left(2,{\Bbb C}\right)$,
- it is homeomorphic to $\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$$\Gamma\backslash SO\left(3,1\right)/SO\left(3\right)$ for some discrete, torsion-free subgroup $\Gamma\subset SO\left(3,1\right)$$\Gamma\subset SO\left(3,1\right)$.

...

## 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. ...

## 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 $M$$M$ is an orientable 3-manifold with boundary, whose interior admits a complete hyperbolic metric of finite volume, then $\partial M$$\partial M$ is a (possibly empty) union of incompressible tori.

Ends of infinite volume fall into two classes, geometrically finite ends and geometrically infinite ends....

...