180 31. HYPERBOLIC GEOMETRY AND^3 -MANIFOLDSrespectively. By the classification of surfaces with boundary, there exists a homeo-
morphism
¢ : E,,. ---+ Ea,
mapping the two boundary copies of rJ to the two boundary copies of a 1 , with both
copies mapped in t he same way. This homeomorphism induces a self-homeomor-
phism of E which maps rJ to a 1. Since [a 1 ] is primitive, we conclude that [<J] is also
primitive.
(2) Suppose that rJ is separating. Cut E along rJ to obtain two connected
compact surfaces Ei and E §, each with boundary consisting of t he single circle
rJ. Since [<J] i= l, there exists 1 ::; k ::; g - 1 such that E 1 h as ge nus k and
E 2 has genus g - k. Now consider E again as a 4g-sided polygon with sides
ai, b 1,a-1 1 , b-1 1 , ... ,a 9 , b 9 ,a 9 -1 , b-1 9.
Define the commutator [a , b] ~ aba-^1 b-^1. Cut E along a simple loop 0-which is
homotopic to the product [a 1 , b 1 ] [a2, b2] · · · [ak, bk] to obtain two connected compact
surfaces ti and t§, each with boundary consisting of 0-. We h ave that t 1 h as genus
k and t 2 has genus g-k. Hence there exist homeomorphisms </Ji : E i---+ ti, i = 1, 2,such that </Ji maps rJ to 0-for i = 1, 2 (and we may assume they map rJ to 0-the
same way). Hence we can glue the homeomorphisms ¢ 1 and ¢ 2 together to obtain
a homeomorphism ¢ 1 ......., ¢2 : E ---+ E which maps rJ to 0-. Since [a-] is primitive, we
again conclude that [<J] is primitive.^1 0
REMARK 31.17. When t he genus of t he surface is at lea.st two, another way
to prove that [<J] is primitive is to endow the surface with a hyperbolic metric.
Recall that if a is a homotopically nontrivial loop in a closed hyperbolic surface,
then a is freely homotopic to a unique closed geodesic a*. Furthermore, if a is a
simple loop, then so is a*; see for instan ce Chapter 3 in Casson and Bleiler's b ook
[54] for a proof. Now suppose otherwise that rJ is an essential simple loop that is
homotopic to j3k for some loop /3 and k > 1. Since rJ is essential, f3 is essential.
Then rJ* = (/3*)k by the uniqueness of t he geodesic representative. This contradicts
the fact that rJ* is simple.
In view of the fact that we a re interested in incompressible tori, we consider
t he (opposing) compressible case.
LEMMA 31.18 (Map on n1 induced by inclusion for compressible torus). Let
M^3 be a compact 3-manifold (with possibly nonempty boundary) such that n 1 (M)
contains no torsion element (e.g., M is a closed hyperbolic 3-manifold or th e trun-
cation of a finite-volume hyperbolic 3-manifold). If i : E^2 '---+ M is a compressible
torus, then either(1) ker (i*) = n 1 (E) or
(2) ker (i*) is an infinit e cyclic subgroup of n 1 (E) generated by a primitive
element.
REMARK 31.19. The lemma is not true for lens spaces.
PROOF. Since i : E '---+ M is compressible, it follows that ker (i*) f= {O}.
Suppose ker (i*) i= n1 (E) ~ Z x Z. We claim that ker (i*) must b e an infinitecyclic subgroup of 7r1 (E). Since 7r 1 (M) contains no torsion element, the quotient
group ni(E)/ker(i) ~ image(i) C ni(M) contains no torsion element. Thus
(^1) T hat [u] is primitive is a nontrivia l fact, unlike the case of [a 1 ] above.