l. INTRODUCTION TO HYPERBOLIC SPACE 177LEMMA 31.11 (Nilpotent subgroups of Isom (lH!n)). If G c Isom (lH!n) is a
nilpotent subgroup, then either
(1) G ={id},
( 2) there exists <p E G - {id} which is elliptic,
(3) every element of G - {id} is hyperbolic and has the same two fixed points
in olHin' or
( 4) every element of G - {id} is parabolic and has the same fixed point in
olHin.In the study of cusps of hyperbolic manifolds the following is useful.DEFINITION 31.12. A horoball in lH!n, using the disk model D n, is an inter-
section Dn n B n, where Bn is a closed ball in IRn tangent to 8Dn. The boundary
of a horoball is called a horosphere.A horoball is the closure of the limit of balls in hyperbolic space as the radiitend to infinity. Topologically, if n > 2, a horosphere is diffeomorphic to a Euclidean
space and hence is simply connected.
In the upper half-space model un, a horoball either is an intersection un n Bn,
where B n is a closed ball in IRn tangent to aun, or is the region above a horizontalhyperplane: {x: Xn 2: c}, where c > 0.
1.4. Hyperbolic manifolds and some examples.
If (11.n, h) is a complete hyperbolic manifold, then its universal covering space
(it", h) with the lifted metric is a complete, simply-connected Riemannian manifold
with constant sectional curvature -1. Hence (H, h) is isometric to lHin. Let p : lH!n -t- denote the projection map and let
(31.6)denote the group of covering transformations. For a ny x E 1-1., the fundamental
group 7r 1 (1-1., x) is naturally isomorphic to r. The group r acts freely and prop-erly discontinuously on lHin; that is, for any compact sets K 1 , K 2 c lHin, the
set {IE r: ')' (K 1 ) n K 2 f= 0} is finite and 1(x) =I x for all x E lH!n and for all
/EI' - {id}. (In fact, I' is a torsion-free discrete subgroup.) Hence (1-1., h) is iso-
metric to lHin /r. Thus the study of hyperbolic manifolds is the same as the study
of discrete torsion-free subgroups of Isom (lH!n).
Henceforth , a finite-volume hyperbolic manifold is assumed to b e con-
nected, complete, and noncompact. Finite-volume hyperbolic manifolds in dimen-
sion at least 3 are very difficult to construct. All known examples before the work
of Thurston are either expressed in the form lHin /r by describing the discrete group
r explicitly or by taking convex hyperbolic polyhedra and gluing their faces iso-
metrically.
We conclude with some examples.
(1) If S is a Riemann surface with nonabelian fundamental group, then the
uniformization theorem says there exists a unique complete hyperbolic metric g
on S conformal to the complex structure. This fundamental theorem provides the
key link between geometry, a n alysis, and algebraic geometry for surfaces.