# Hyperbolic 3-manifolds

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}{\Bbb H}^3$,
- it is homeomorphic to $\Gamma\backslash{\Bbb H}^3$$\Gamma\backslash{\Bbb H}^3$, for some discrete, torsion-free group of isometries of hyperbolic 3-space.

Here, hyperbolic 3-space ${\Bbb H}^3$${\Bbb H}^3$ is the simply connected, complete, Riemannian manifold with sectional curvature constant -1. By Cartan's Theorem, ${\Bbb H}^3$${\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 the projective line $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}^3\right)$$Isom^+\left({\Bbb H}^3\right)\subset Isom\left({\Bbb H}^3\right)$ of orientation-preserving isometries. The action is transitive and has $SO\left(3\right)$$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....

Geometrically finite ends are classified by the corresponding points in Teichmüller space of $\partial M$$\partial M$. (Ahlfors-Bers) ...

Geometrically infinite ends are classified by the corresponding ending laminations. (Brock-Canary-Minsky) ....

...