1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 18 7

some of the horoballs are tangent to each other. In a maximal cusp end, for each

r > 0, the slice {r} x v n-l c 1ln is an embedded hypersurface, whereas the slice

{O} x V C 1l h as self-intersections.

However , different maximal cusp ends may intersect. If this happens, then any

torus slice T of one cusp end cannot b e completely contained in another cusp end

E. This can be seen by lifting T and E to the universal cover !Hln. Indeed, if T

is contained in E, then co nsider the intersection of the pre-image of T under the

covering map p with a horoball B projecting onto E. The surface p-^1 (T) nB must

b e a horosphere inside B. Since all horospheres in B are tangent to fJB at the

infinity of IHin, we h ave that p-^1 (T) n B must b e one such horosphere. This implies

that Tis a torus slice of the cusp end E , which contradicts the assumption.

EXERCISE 31. 35 (Slices in a cusp are totally umbillic). Show that the slices

v;i-^1 ~ {r} x V c [O, oo) x V are totally umbillic with respect to gcusp· In particular,

for each r, with respect to the unit normal vector v ~ ffr, the second fundamental
form II of Vr with respect to gcusp is equal to the negative of the induced metric:

(31.13)

In other words, for the compact truncated cusp [O, r] x V , the boundary component
Vr is concave, whereas for the noncompact truncated cusp [r, oo) x V , the boundary
Vr is convex.

Given a subset of a group Sc G , (S) CG denotes the subgroup generated by
S. We have the following fundamental res ult.

THEOREM 31.36 (Margulis lemma- algebraic version). For any n E N there

exists a constant En > 0 such that if r C Isom (IHin) acts properly discontinuously

and if x E !Hln, then the group

I'c:n (x) ~({IE I': d (! (x), x)::::; En})

is virtually nilpotent; i.e., I'c:n (x) has a subgroup of finite index which is nilpo-
t ent, where the index is bounded by a constant depending only on n. The constant
En is call ed the (n-dimensional) Margulis constant.

REMARK 31.37. When n = 3, the group I'c:n (x) is virtually abelian; i.e.,
r C:3 ( x ) has a subgroup of finite index which is abelian.

We have the following local geometric consequence of the algebraic Margulis
lemma.

THEOREM 31.38 (Margulis lemma- local consequence). L et (Hn, h) be a com-

plete hyperbolic manifold so that (H, h) = IHin /I', where I' c Isom (!Hln) is a discret e

subgroup acting freely and properly discontinuously on IHin. Given E E (0, En] and

x E H , let f'c: (x) denote the subgroup of n1 (H,x) ~ I' generat ed by piecewise

smooth loops based at x with length ::::; E. Then:

(1) B ( x, E /2) is isometric to a geodesic ball of radius E /2 in IHin /f' c: ( x).

(2) f'c: (x ) is isomorphic to either {l}, Z, or a discret e subgroup of Isom (En-l)

acting freely on En-l.