1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

176 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

is a parabolic isometry. When n = 2, 3, every parabolic isometry is con-

jugate to an isometry of this form.

(3) In the upper h alf-space model, if A E SO (n - 1, IR) and A> 0 with A =I-1,

then

X f----+ (.AA (x1,... , Xn-1), AXn)

is a hyp erbolic isometry. In all dimensions any hyperbolic isornetry is

conjugate to an isometry of this form.

(4) The composition of two inversions about disjoint (n - 1)-spheres is a hy-

perbolic isometry.

When n = 3, we can give criteria for when [M] E PSL (2, <C) is elliptic, para-

bolic, or hyp erbolic.

LEMMA 31.9. Given M = ( ~ ~ ) E SL (2, <C) - {±id}, corresponding to

[M] E PSL (2, <C) ~Isom+ (1HI^3 ) , we have that

(1) [M] is elliptic if and only if tr (M) E (-2, 2) CIR,

(2) [M] is parabolic if and only if tr (M) = ±2,

(3) [M] is hyperbolic if and only if tr (M) E (-oo, -2) U (2, oo) C IR or

tr (M) E C - IR.

As a special case, if [M] fixes oo E S^2 ~ IR^2 U {oo}, then c = 0, a =I-0, and

d = a-^1. We have, as an isometry of the upper h alf-space (U^3 ,gu),

(31.5) </JM (z, t) = (a^2 z +ab, lal^2 t)

for z EC and t E (0, oo). Since

tr (M) =a+ a-^1 = a (1 - ~) + 2Re (~),

lal lal

we h ave

(1) Mis elliptic if and only if lal = 1 and a =I-±1,

(2) M is parabolic if and only if a= ±1,

(3) M is hyperbolic if and only if lal =I-1.

Since any parabolic isometry fixing the point at infinity is of the form

(z , t) f----+ (z + b, t) , where b EC,

the subgroup of Isom+ (1HI^3 ) of parabolic isometries fixing a given point on the

sphere at infinity is isomorphic to IR^2.

R ecall t he following.

DEFINITION 31.10 (Nilpotent group). Let G be a group. Given subgroups

H and J of G, let [H, J] denote the subgroup of G generated by the subset

{hjh-^1 F^1 : h EH, j E J}. Define inductively Gk~ [G,ck-l] fork EN, where

G^0 ~ G. We say that G is nilpotent if Ge = {e} for some f EN.

The following result applies to abelian subgroups , e.g., subgroups isomorphic

to Z x Z. In particular, it is useful in the study of embedded tori in complete

hyperbolic 3-manifolds (see the proof of Lemma 31.27).