186 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDSis a bijection. If JC is an end-complementary compact set, then we call each element
of£ (JC) a topological end.
The set of topological ends is well defined. If JC 1 and JC2 are end-complementary
compact sets in M, then we have a natural bijection
(h^0 (P-1^1 : £ (JC1)--+ £ (JC2),where ¢ 1 : £(JC') --+ £ (JC 1 ) and <h : £(JC') --+ £ (JC2) are defined for JC' :J JC1 U JC2
in the same way as ¢ in ( 31.12). The bijection ¢ 2 o ¢1^1 is independent of the choice
of JC' :J JC 1 U JC 2. In this sense the definition of the set of topological ends £(JC)
is independent of the choice of end-complementary compact set JC.So, when we speak of a topological end, we mean an element E of£ (JC) for
some end-complementary compact set JC. Roughly speaking, one can think of a
topological end as a connected component of a neighborhood of infinity.
EXAMPLE 31.32 (Cylinder). Let Mn= Nn-l x JR., where N is a closed mani-
fold. The compact set N x [O, 1] is end-complementary and M has two topological
ends.
EXERCISE 31.33. Show that the number of topological ends is not necessarily
nondecreasing under pointed Cheeger- Gromov limits.HINT. One can easily construct a sequence of pointed Riemannian manifolds{(Mi, gi, xi)} of the same dimension, where Mi has i topological ends and whose
pointed Cheeger- Gromov limit is Euclidean n -space. In this case the limit has onlyone topological end, whereas the lim inf of the number of ends of Mi is equal to
infinity.3.2. The Margulis lemma and hyperbolic cusps.
Consider the horoball iun ~ { x E JR.n : Xn 2: 1} contained in the upper half-
space model and let r = lnxn. The hyperbolic metric may be written as. dxf + · · · + dx;, 2 -2r ( 2 2 )
9hyp =:= 2 = dr + e dx 1 + · · · + dxn-l.
Xn
Let L be a lattice in JR.n-^1 which is isomorphic to zn-l and which acts by trans-
lation. Then dxi + ... + dx;_l induces a flat metric 9flat on vn-l ~ JR.n-l IL. The
hyperbolic metric 9hyp on iun induces the hyperbolic metric 9cusp ~ dr^2 + e-^2 r 9flat
on iun / L = [O, oo) x V, where L acts on the first n-1 components. The Riemannian
manifold ([O,oo) x V,gcusp) is called a hyperbolic cusp.
EXAMPLE 31.34. When n = 3, V^2 is a torus and there exist two parabolic
isometries of the hyperbolic disk JHI^3 which fix a point x E 8JHI^3 at infinity and a
horoball B tangent to 81HI^3 at x such that ([O,oo) x V,gcusp) is isometric to the
quotient of B by the group generated by the two parabolic isometries (see Lemma
31.25 above).
If a closed subset U with the induced metric of a complete hyperbolic manifold
(1-ln, h) is isometric to a hyperbolic cusp, we call (U, hlu) a hyperbolic cusp end
(or cusp end) of (7-l, h). We call (U, hlu) a maximal cusp end if it is a cusp end
which is not properly contained in another cusp end.
Note that the pre-image of a maximal cusp under the projection map p : lHin --+
1-ln consists of the union of horoballs whose interiors are disjoint and are such that