1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 189

set 1-l[e,oo) is not always end-complementary. This happens when there is a compact

connected component of 1-l - 1-l[E,oo). On the other hand, we have the following.

LEMMA 31.42 (Topological ends are essentially c:-thin). Let (1-ln, h) be a finite-

volume hyperbolic manifold. Then we have the following.

(1) Let JC be an end-complementary compact set in 1-l. If E E E (JC) is a

topological end, then for any c: > 0 there exists E' C E with E - E'

compact such that E' is an c:-thin end.

(2) There exists c:o > 0 depending on h such that if c: ::; c: 0 , then any c:-thin

end is a topological end.

The above result is actua lly a consequence of Theorem 31.44 below, which we

shall prove from the following geometric consequence of the Margulis lemma.

THEOREM 31.43. Let (1-ln, h), n ~ 3, be a finite-volume hyperbolic manifold

and let c: E (0, En], where En is the Margulis constant. The c:-thin part 1-l(o ,e] is the

disjoint union of subsets, positive distances apart, of the following forms:

(1) (Hyperbolic cusp end) A subset S homeomorphic to [O, oo) x vn-^1 , where

V is a closed manifold admitting a fiat metric, with the interior being

diffeomorphic to (0, oo) x V. Moreover there exists a closed set S' c S

with S - S' compact and such that (S', hl 8 ,) is isometric to

([O, oo) x V , dr^2 + e-^2 r gflat) ,

where gflat is a fiat metric on V.

(2) (Tubular neighborhood of geodesic loop- Margulis tube) A subset S home-

omorphic to f3n-l (1) x 51 whose interior is diffeomorphic to Bn-l (1) x

51. Moreover there exists a unique smooth geodesic loop / c S such that

for o > 0 sufficiently small,

(31.14)

(a) the o-neighborhood N 0 (r) of/ is contained in S,

(b) the submanifold (N 0 (r), hlN 6 (1)) is isometric to Bn-l (r) x 51 (£)

with the metric

n-l

gtube ~ "'"""' ~ (x'ds. + dx' ·)^2 + ds^2

i=l

for some r , £ > 0, where Bn-l (r) denotes the Euclidean (n -1)-ball

of radius r and 51 (£) denotes the circle of length£.
(3) (Circle) A smooth geodesic loop of length c:.
Regarding part (2), see Example 31.45 below.
3.3. Geometry of finite-volume hyperbolic ends.
The next classical fact follows from the above geometric consequence of the
Margulis lemma.

THEOREM 31.44 (Finite-volume hyperbolic ends are standard cusps). Let (1-ln, h),
n ~ 2, be a finite-volume hyperbolic manifold and let JC be an end-complementary
compact set in 1-l. For every topological end E E E (JC) there exists a closed subset
U c E such that E - int (U) is compact and (U, hlu) is isometric to [O, oo) x vn-l
with a metric of the form

gcusp = d r^2 + e -2r gflat'

where (V, gflat) is a closed fiat manifold.