2. TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3-MANIFOLDS 183
PROOF. Let i : Vr -+ 7-l denote the inclusion map. If the lemma is false, then
there exists a homotopically nontrivial loop f : 51 -+ Vr such that i o f : 51 -+ 7-l is
null-homotopic. Thus the loop i o f can be lifted to the universal cover 7r : JHI^3 -+ 7-l;
we denote the lift byso that 7r o j = i o f. Let
P ~ 7r-^1 (Vr) c lHI^3.
Since 7r is a Riemannian covering m ap and since the hypersurface Vr is complete
(in fact, compact), flat, and totally umbillic with principa l curvatures identically
-1, we have that its inverse image P is a complete, flat , totally umbillic surfacewith principal curvatures identically -1 embedded in JHI^3. Hence P is a horosphere;^4
in particular, P is diffeom orphic to ffi.^2. The image j ( 51 ) lies in P. Since P is
simply connected , the lifted m ap J: 51 -+ P , when considered as a map into P , is
null-homotopic in P. Thus f = 7r o J is null-homotopic in Vr. This contradicts the
assumption of f b eing homotopically nontrivial. D
REMARK 31.26. It is a well-known fact that if 7r : Y-+ Y is a universal covering
space and if X c Y is a path-connected subset, then the inclusion map i : X-+ Y
induces an injective homomorphism in 7r 1 if and only if each component of 7r-l (X)
is simply connected. The proof is the sam e as the a bove.
Now we recall the topological properties of embedded tori in finite-volume hy-
p erbolic 3-manifolds.
LEMMA 31.27 (Embedded tori in hyperbolic 3-manifolds). Let T2 be an em-
bedded 2-torus in a finite-volume hyperbolic 3-manifold (H^3 , h). Then either
(I) T is incompressible and T is isotopic to a standard torus slice in a hyper-
bolic cusp,
(II) T is compressible and T bounds an embedded solid torus B^2 x 51 , or
(III) T is compressible and T bounds a compact 3-manifold lying inside an
embedded 3-ball in 7-l.
PROOF. Clearly we h ave one of the following two possibilities:
(A) Tis incompressible or(B) Tis compressible.
If (A), then i* : 7r 1 (T) -+ 7r 1 (7-l) is injective and we h ave
Z x Z ~ i* (7r 1 (T)) c r c Isom (JHI^3 ) ,
where we have used the natural isomorphism between the group of covering trans-
formations r and 7r1 (7-l).
We claim that the abelian subgroup i* (7r 1 (T)) is generated by two parabolic
isometries. First, there are no elliptic elements in r since oth erwis e, by definition,
we would have a contradiction to the fact that r acts freely on JHI^3. Hence, by
Lemm a 31.11, either
(Al) every elem ent of i* (7r 1 (T)) - {id} is hyperbolic and h as the same two
fixed points in 8JHI^3 or
(A2) every element of i* (7r 1 (T)) - {id} is parabolic and has the same fixed
point in 8JHI^3.
(^4) This is a well-known fact (see for example [55] or [156]).