- APPLICATION OF MOSTOW RIGIDITY TO THE EXISTENCE OF ISOMETRIES 277
PROOF OF STEP 2. We first show the following.
C la im. For each topological end E, Te,A = P ( {re,A} x Te) is contained in the
€3/2-thin part 1-l(o,c: 3 ; 2 J of (1-l, h), where E3 is the 3-dimensional Margulis constant.
Proof of the claim. Since A:::; A , we have {re,A} x Te C 1-l[o,c: 3 ; 2 )· So, given
y E Te,A, there exists a piecewise smooth geodesic loop "Y based at x =i= P-^1 (y)
with ["Y] E ?T1 (1-l, x) - {1} and L ("'!) :::; c3/2. From A:::; A/2, we have that re,A 2:
re,A. + ~ ln 2 and hence "Y C 1-l -1-l;,,. Therefore there exists a loop f3 c {re,A} x Te
which is homotopic to "Yin 1-l. Since {re,A} x Te is incompressible in 1-l, we obtain[/3] E ?T1 ( {re,A} x Te, x) - {1}.Moreover, since P is an isometry, P o "Y is a piecewise smooth geodesic loop
based at y with L(F o "Y) = L ("'!):::; c 3 /2. That [F o "Y] '/-1 E ?T 1 (1-l, y) follows from
[Fo/3] '/-1 E ?T 1 (Te,A, y) and from the fact (from Step 1) that Te,A is incompressible
in 1-l. Hence y E 1-l(o, 03 ; 2 ]. D
By the claim, P maps truncated cusp ends in 1-lA to truncated cusp ends in
1-l. Using the fact that the hyperbolic metric h in a cusp end is a standard warped
product, we may show that P may be extended to a global isometry I : 1-l --+ 1-l. In
particular, any cusp end is a warped product of the form ([O, oo) x T2, gcusp), where
gcusp = dr^2 + e-^2 rgflat and (T, gflat) is a fiat 2-torus. The slices {r} x T foliating
the cusp are intrinsically fl.at with constant second fundamental form. Since P is
an isometry on 1-lA, the hypersurfaces P ( {r} x T) share the same properties as
{ r} x T. From the uniqueness of such foliations, we conclude that the P ( { r} x T)
are also standard slices in one of the cusp ends. In particular, for each end E there
is an end E such that
F: ([O,re,A] X Te,gcusp)--+ ([a,b] X 'Jf,9cusp)
is a product map of the form P (r, y) = (£ (r), <p (y)), where C is linear and <p
(Te, gflat) --+ ('fe, 9flat) is a homothety. Now it is clear how to extend P to an
isometry I on each cusp. This completes the proof of Step 2.
STEP 3. Existence of an isometry I near F. Suppose that Theorem 34.22is false. Then there exist C E N, E > 0, a sequence ki E N with ki --+ oo, and
embeddings Fi : 1-lA --+ 1-l such that(34.68) llFth - hi 1 Cki(1iA) < ~i
and for every isometry I: (1-l, h)--+ (1-l, h) we have
(34.69) dce(1-lA,h) (Fi, I) 2: E.
By (34.64) and (34.59), for i E N sufficiently large, we have that the surface
Fi(81-lA) in 1-l consists of 2-tori which are approximately intrinsically fl.at and ap-
proximately totally umbillic. In fact, Fi maps the foliation of a collar of 81-lA in
1-lA by intrinsically fiat, totally umbillic 2-torus slices to a foliation of a collar of
Fi(81-lA) in Fi(1-lA) by approximately intrinsically fiat, approximately totally um-
billic 2-tori. By the Margulis lemma (more aptly, by Corollary 31.47), for A E (0, A]
and for i sufficiently large, such image 2-tori must lie in the hyperbolic cusp ends
of 1-l and must be close to intrinsically fl.at, totally umbillic 2-tori.