178 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS( 2) Take the Riemann surface S = <C - { 0, 1}, the twice-punctured plane. ThenSis biholomorphic to IHI^2 /f(2), where f(2) is the principal congruence subgroup of
level 2; i.e.,
r(2) ={[A] E PSL(2, Z) I A= id mod 2}.
The biholomorphism is given by a modular function arising from classical elliptic
function theory.
(3) Since PSL(2, Z) is a discrete subgroup of PSL(2, JR), any torsion-free sub-
group r of PSL(2, Z) produces a complete hyperbolic surface IHI/f. This includes
the example above.
( 4) One can generalize the construction to dimension 3 by taking the Bianchi
group. Take a positive square-free integer d (i.e., d is not divisble by any perfect
square except for 1) and let 0 d be the ring of integers in the quadratic imaginary
number field Q( R,). Then PSL(2, Od) is a discrete subgroup of PSL(2, <C), so that
IHI^3 / PSL(2, Od) has finite volume. Therefore, any finite-index torsion-free subgroup
r of PSL(2, Od) gives rise to a finite-volume hyperbolic 3-manifold IHI^3 /r.
(5) In Thurston's famous 1978 - 1980 notes [401], he was able to see that the
figure-eight knot complement in S^3 has a complete finite-volume hyperbolic metric.
He achieved this by decomposing the knot complement as a union of two tetrahedra
without vertices (the so-called ideal tetrahedra), realizing each of the tetrahedron by
the regular ideal hyperbolic tetrahedron and gluing them by isometries. The associ-
ated discrete subgroup of PSL(2, <C) is a subgroup of the Bianchi group PSL(2, 03).
This example of Thurston initiated the geometrization conjecture in dimension 3.
(6) The first known closed hyperbolic 3 -manifold is called the Seifert- Weber
space, constructed in 1933 by H. Seifert and C. Weber. It is motivated by Poincare's
construction of the Poincare homology sphere, which is to glue each face of a regular
dodecahedron to its opposite face by a 7r /5 rotation. (A picture of the identification
can be found in Chapter 9 of Seifert and Threlfall's classical book on topology
[361].) As realized by Poincare, if one chooses the dihedral angle of a regular
spherical dodecahedron carefully, then the gluing maps can be realized by spherical
isometries, so that the quotient Poincare homology sphere has a Riemannian metric
of constant sectional curvature l. In the case of Seifert and Weber, they take the
regular dodecahedron and glue each face to its opposite face by a 37r /5 rotation and
realize that such gluing maps can be realized by hyperbolic isometries on a regular
dodecahedron. The quotient space is then a closed hyperbolic 3-manifold.
- Topology and geometry of hyperbolic 3-manifolds
In this section we discuss aspects of the topology and geometry of 3-manifolds,
especially hyperbolic 3-manifolds. For simplicity, we shall assume that all the 3-
manifolds that we consider in the rest of this chapter are orientable.
2.1. The loop theorem and its consequences.
Recall that a 3-manifold is irreducible if every embedded 2-sphere bounds
an embedded 3-ball. By the Schonflies theorem, JR^3 and IHI^3 are irreducible. As a
consequence, we have
LEMMA 31.13. Any complete hyperbolic 3 -manifold is irreducible.