Hyperbolic 3-manifolds

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This page has not been refereed. The information given here might be incomplete or provisional.

Contents

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,
- 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).




2 Construction and examples

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

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