1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 185

In case (B2), the torus T lies in B. By the Jordan- Brouwer separation theorem,

we have that B - T is the disjoint union of two connected open 3-dimensional

submanifolds U1 and U2, where U 1 is compact and 8U 1 = T Thus case (III) holds.

This completes the proof of the lemma. D

The following, which pertains to case (III) of Lemma 31.27, is an example of

an embedded compressible torus in any hyperbolic 3-manifold not bounding a solid

torus.

EXAMPLE 31.28. Take a knotted torus T2 in 53 bounding a solid torus X in

53. Now take an open 3-ball Y inside X and let B^3 = 53 - Y. Inside B lies T

Now embedding this B into any hyperbolic 3-manifold H^3 results in a compressible

torus T in 'H. This T does not bound a solid torus.

Motivated by the noncompact hyperbolic limit case of nonsingular solut ions on
3-manifolds , we consider the following.

DEFINITION 31.29. We say that a compact submanifold 'H~ of a closed 3-
manifold is a topological hyperbolic piece^5 if 8'He is a disjoint union of em-
bedded tori and the interior of 'He admits a complete finite-volume hyperbolic
metric.

Let 'H~ be a topological hyperbolic piece in a closed 3-manifold M^3. By Lemma
31.25, 8'He is incompressible in 'He. From this and Lemma 31.21 we immediately
obtain the following, which is relevant to 3-dimensional nonsingular solut ions of
Ricci fl.ow (see Proposition 33.9 in Chapter 33).

COROLLARY 31.30. If 8'He is incompressible in M -'He, then 8'He is incom-
pressible in M.

3. The Margulis lemma and hyperbolic cusps

The main topics in this section are the Margulis lemma and its consequences
for the geomefry of ends of finite-volume hyperbolic 3-manifolds.

3.1. Topological ends.

Let Mn be a noncompact differentiable manifold. Given a compact set KC M,

define
E (K) = {E : E is a connec~e.d component of}.
· M - K and E 1s noncompact

If K 1 and K 2 are compact sets with K 1 C K2, then we have a natural map

(31.12)

defined by if> ( E 2 ) ~ E 1 , where E 1 is the connected component of M -K1 containing
E2 EE (K2).^6

DEFINITION 31.31 (Topolo gical end). We say that a compact set K C M is
end-complementary if for every compact set K' :::> K the map

if> : E (K') ---+ E (K)

(^5) Topologists call such a manifold a "simple 3-manifold with torus boundary".
60bserve that E 1 is noncompact since any set containing a noncompact closed set is non-
compact. Thus Ei E [ (K1), so that¢ is well defined.