- INCOMPRESSIBILITY OF BOUNDARY TORI 253
PROOF. The geodesic loop [, : [O, l] ---+ T defines a translation T.c (deck
transformation) of the universal covering space 7r : JR.^2 ---+ T given by the vector
.C (1) - .C (0) E JR.^2 , where the path (geodesic line segment) .C : [O, l] ---+ JR.^2 is t helift of£. The quotient JR.^2 / [T.c] is a cylinder of circumference Lo ~ Length(£) ~
1£(1)-L(O)I· That is, JR.^2 /[T.c] = (JR./L 0 Z) x R We have a natural covering
if : JR.^2 / [T.c] ---+ T so that
JR.2 ~ JR.2 I [T.c]7r\, /7r
T ,
where 1T is also a projection. The loop [, lifts to a "meridian circle" l in the
cylinder JR.^2 / [T.c], i.e., a geodesic lo op which is perpendicular to each geodesic line
in JR.^2 / [T.c].
Since 5i C T is a smooth embedded loop representing the same homotopy
class as £ , it lifts to a smooth embedded loop 5i c JR.^2 / [T.c] of the same length
-i iLi ~ Length(5 ) = Length(5 ).
Clearly 5i is contained in a set of the form
D ~ (JR./ LoZ) x [a, a+ Li] c (JR./ LoZ) x JR.,
where a E R The area of this set is equal to Lo Li. Clearly there exists a geodesicloop £', parallel to £, such that it has a lift [,' which is contained in n (in fact,
we can prescribe any meridian circle in JR.^2 / [T.c] as a lift). Thus there exists animmersed annulus in JR.^2 / [T.c] bounded by 5i U £' with area at most L 0 Li. This
immersed annulus projects to an immersed annulus in T bounded by 51 U [,' with
area at most Lo Li. DRecall that the pointed hyperbolic limit is denoted by (1i, h, x 00 ). Corre-
sponding to TA (to) is a topological end E in 1i and an exact hyperbolic cusp
([O,oo) x 1], dr^2 + e-^2 rgfat)· Corresponding to SA,p• ' let [, denote the em-
bedded geodesic loop in the flat torus TA = {re,A} x Te of area A ; let Lo ~
e-rE , A Length 9 E (£). Since we are in an almost hyperbolic piece and by (33.59),
flat
for any€ > 0 there exists t~ < oo such that the length of SA,p• satisfies
(33.82)Hence, by Lemma 33 .39,
(33.83) Area 9 (to)(AA,p.)::; (l+c)^2 e-P•LoL(p•) provided t o 2.':t!.
Using again that we are in an almost hyperbolic piece and now using (33.60), we
obtain the estimate
(33.84)
Area 9 (to) (BA,p.)::; (1 +c)
2
(1-e-P•) Lo::; (1 +c)
2
Lo provided to 2: t!.Hence, by (33.81),
A (P•)::; (1 + c)^2 (1 + e-P• L (p.)) Lo.
On the other hand, since P• 2: P#, by (33.78) we also have t hat at time to,
(33.85)