1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

190 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

PROOF. Let (Hn, h) b e a complete hyperbolic manifold. Since its fundamental

group is countable, the length spectrum of (H, h), i.e., the set of lengths of smooth

geodesic loops, is countable.^7 Therefore we may choose a "generic" E E (0, En] in

Theorem 31.43 so that case (3) does not occur for t his c, i.e., so that there are no

smooth geodesic loops of length E in H. Thus each component of the c-thin part

H (o,,,] for such an E is either a Margulis tube or a cusp end.

Now assume that (H, h) has finite volume. The c-thick part H[,,,=) is compact

due to the volume bound and the injectivity radius lower bound on H[,,,=J (see

Lemma 31.41 for the latter fact).

Since we have ruled out case (3) of Theorem 31.43, the boundary of t he c-thick

part H [,,,=) is equal to the boundary of the c-thin part H(o,c] , which is a smooth

embedded surface. Therefore H[,,,=) is a smooth compact manifold with boundary.

This implies that H[,,,=) has only a finite number of boundary components,

which in turn implies that the E-thin part H(o,,,] has only a finite number of compo-

nents. Thus there are only a finite number of Margulis tubes in H(o,,,]. Now choose

E to be smaller than the shortest closed geodesic in this finite set of Margulis tubes.

Then each component of H(o,,,] is a cusp end. It follows that each topological end

is a cusp end; i.e., Theorem 31.44 is proved. 0

EXAMPLE 31.45 (Margulis tube metric as a quotient). Consider the diffeomor-

phism

defined by

n-'±' (-x,s ) ::;= · ( e s x,e - s) ::;=. ( y,. (^1). .,y n-1 ,y n).
The hyperbolic metric on t he upper half-space is
°"n (d i) 2 °"n- l (d ( s i)) 2 (d (^5 ))2 n-1
ui=l Y = ui= l e x + e = L (xids + dxi)2 + ds2
(yn)2 (es)2 i=l '
which is the same formula as (31.14). Define the ball
B(r) ~ {zE JRn-l: IZJ < r}
and the cone

C(r) ~ {(y,yn) E JRn- 1 X JR+: IYJ < ryn}'

where y ~ (y^1 , ... , yn-l). We have

<P: B (r) x JR--+ C (r).

Given any f E JR+, consider the action of Z on JRn-^1 x JR generated by the isometry

(x, s) t-t (x, s + £). We then have that (B (r) x JR) /Z is the Margulis tube (31.14).

We have the following quantitative version of the Margulis lemma in relation

to 3-dimensional hyperbolic cusp ends due toColin Adams [3].

(^7) An aside: If (1-ln, h) is geometrically finite (in particular, if (1-l, h) has finite volume),
then the length spectrum is discrete.