1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

192 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS

4. Mostow rigidity

In this section we state a special case of t he Mostow rigidity theorem and
some important consequences.

THEOREM 31.49 (Mostow rigidity). If (1-l'l, h 1 ) and (H!], h2), n 2 3, are finite-
volume hyperbolic manifolds and if

is an isomorphism, then there exists a unique isometry

such that i. = ¢.

This implies the volume of a finite-volume hyp erbolic m anifold is a topological
invariant. R ecall that two topological sp aces X and Y are homotopy equivalent

if there exist continuous maps f : X -+ Y and g : Y -+ X such that g o f is

homotopic to idx and fog is homotopic to idy.

COROLLARY 3 1. 50 (Homotopy equivalent{=? diffeomorphic {=?isometric). Let

(1-l'l, h 1 ) and (H!], h2) be finite-volume hyperbolic manifolds, where n 2 3. Then

the fallowing conditions are equivalent:

(1) (1-l1, h1) and (1-l2, h2) are isometric,

(2) 1-l 1 and 1-l 2 are diffeomorphic,

(3) 1-l 1 and 1-l 2 are homotopy equivalent.

Next we give a descript ion of the isometry group of a finite-volume hyperbolic
manifold, which is a consequence of the Mostow rigidity theorem. Given a group
G , let .Aut (G) denote the group of automorphisms of G , i.e., the group (where
multiplication is defined by composition) of isomorphisms of G onto itself. Recall
that the inner automorphism group of G is the normal subgroup of Aut ( G)
consisting of conjugations

Inn(G) ~ { g H aga-^1 : a E G}

and t he outer automorphism group is t he quot ient

Out (G) ~ Aut (G) /Inn (G).

This group is a quantitative measure of how many automorphisms there are which
are not given by conjugation.

COROLLA RY 31.51 (Isometry group of a hyperbolic manifold). If (1-ln, h), n 2
3, is a finite-volume hyperbolic manifold, then the isometry group of (1-l, h) is finite
and

Isom (1-l) ~ Out ( 7f1 (1-l)).