1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

188 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

Now we turn to study the €-thin and €-thick parts of a hyperbolic manifold.

DEFINITION 31.39. The t:-thin part Mco,c] is the set of points x E M such

that there exists a piecewise smooth loop a based at x such that

[a] E 7r1 (M, x) - {1} and Lg (a) < €.

The €-thick part M[c,oo) is the set of points x E M such that for every piecewise

smooth loop a based at x with [a] E 7r 1 (M, x) - {1} we have

Lg (a) 2: t:.

The €-thick-thin decomposition of a Riemannian manifold (M, g) is

EXAMPLE 31.40.

(1) (Simply- connected Riemannian manifolds are all thick.) Ifwe have 7r 1 (M, x)

= {1}, then for every€> 0 we have

Mco, 0 ] = 0 and M[c,oo) = M.

(2) (Closed Riemannian manifolds are all €-thick for € small enough.) If

(Mn, g) is closed and if 7!' 1 (M, x) =I= {1}, then for€> 0 sufficiently small,

we have Mco, 0 ] = 0 and M[c,oo) = M. We may simply take any€< t:o,

where

t:o ~ infLg Q (a)

and where the infimum is taken over all piecewise smooth loops a in M

with [a] E 7r1 (M) - {l}.

(3) (Hyperbolic cusp ends are eventually €-thin.) If (7-l^3 , h) is a finite-volume

hyperbolic 3-manifold and if [O, oo) x V^2 c 7-l is a cusp end with h =

dr^2 + e-^2 r 9flat, then for every € > 0 there exists r < oo such that

([r, oo) x V , h) C 7-lco,c]·

Note that, by Lemma 31.25, the tori { r} x V C 7-l are incompressible.

Note that if a is a geodesic loop, which is smooth except possibly at its base-
point, in a complete hyperbolic manifold (7-ln, h), then [a] E 7!' 1 (7-l, x) - {l}. To
see this, suppose that [a] = l. Then a lifts to a geodesic ii in IH!n with a self-
intersection, which is a contradiction. As a consequence, we obtain part (1) of the
following; see Proposition D.2.6 of [24] for the proof of part (2).

LEMMA 31.41. Let (7-ln, h) be a complete hyperbolic manifold.

(1) Then the injectivity radius at any point in the €-thick part 1-l[c,oo) is at

least t:/2.

(2) If (7-l, h) has finite volume, then for any € > 0, the €-thick part 7-l[c,oo) is

compact.

Given a complete hyperbolic manifold (7-ln, h) and € E (0 , t:nJ, where €n is the
Margulis constant, an €-thin end of 7-l is defined as the closure of a connected
component of 'H - 7-l[c,oo). An €-thin end is not always a topological end; i.e., the