184 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDSWe claim that case (Al) cannot happen. Suppose ¢ 1 and </>2 are two hyperbolic
isometries that ge nerate i* (7r 1 (T)) ~ Z x Z and that fix the sam e two points on
81Hl^3 ~ C U { oo}. We may assum e, without loss of generality, that the two fixed
points on 81Hl^3 are 0 and oo. Then, as sp ecia l cases of (31.5),
(31.10) </>i (z, t) = (ciz, lcil t),
where 0 < lcil ;fa 1, for i = 1, 2, and where for a ll r, s E Z - {O} we h ave
(31.11)
Let L b e the geodes ic in the upper half-space (U^3 , gv) j oining the two fixed
points 0 and oo; i.e., let L b e the positive z-axis. Then L is invariant under </> 1 and
¢ 2 , so that </>i, i = 1, 2, acts by translation on L with respect to the metric ~ on
L induced by the hyperbolic metric 91!J· Let ai denote the translation distance on
L for </>i, i = 1, 2. By the incompressibility assumption, we see that the subgroup
of (JR., +), i.e., of the real numbers under addition, generated by a 1 and a 2 , is
isomorphic to Z x Z. Now it is well known that a subgroup of JR. isomorphic to
Z x Z is dense. Thus we see that the subgroup generated by ¢ 1 and ¢ 2 is not
discrete. This completes the proof of ruling out case (Al).
Hence we h ave that case (A2) holds and that i (7r 1 (T)) is generated by a
p a ir of pa r abolic isometries h aving the same fixed point in 81Hl^3. By Theorem
31.44, the quotient space 1Hl^3 / i ( ?r 1 (T)) is isometric to a hyperbolic cusp (JR. x V^2 ,
dr^2 + e-^2 r 9flat). From the discreteness of r we see that for ro large enough,
([ro, oo) x V , dr^2 + e-^2 r 9flat )
is isometric to a subset of (H, h ). In particular, i ( 7r 1 (T)) is equa l to t he j ( 7r 1 (V))
for some standard torus slice j : V-+ H emb edded in a hyp erbolic cusp of H. We
conclude, from Waldhausen's theorem that homotopic incompressible surfaces a re
isotopic in Haken m anifolds (see Corollary 5.5 in [426]), that i (T) CH is isotopic
to j (V) c H. That is , case (I) in the statem ent of the lemma holds. This finishes
the ana lysis of possibility (A).
If (B), then there exists a nontrivial element in
ker ( i* : 7r1 (T) -+ 7r 1 (H))which is represented by an embedded loop a in T By the Loop Theorem (Corollary
31.15), there exists an embedded disk D 1 C H whose boundary is a and such that
D1 n T = a (the last property follows from a cut and p ast e argument simila r to
that in the alternate proof of Lemm a 31.21 given in the notes and commenta ry
at the end of this chapter). Push the disk D 1 to one side to obtain a nearby
parallel embedded disk D 2 with 8D 2 C T and such that the region in H bounded
by D1, D2, and the annulus in T between 8D 1 and 8D 2 is a compact 3-manifold
homeo morphic to D^2 x I , where D is a 2-ba ll and I is a closed interval. Now in
T cut out the annulus between 8D 1 and 8D 2 and glue in D 1 and D 2 to obtain an
embedded 2-sphere S^2. Since H^3 is irreducible (see Lemma 31.13), S^2 bounds a n
embedded 3-ball B^3. Now there are two cases:
(Bl) (D x I) n B = D 1 U D 2,
(B2) D x I c B.
In case (Bl) (D x I) U Bis an embedded solid torus with boundary T ; that is ,
case (II) holds.