Strong Rigidity of Locally Symmetric Spaces. (AM-78), Volume 78 9781400881833

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Annals of Mathematics Studies Number 78

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES BY

G. D. MOSTOW

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON NEW JERSEY 1973

Copyright © 1973 by Princeton University Press ALL RIGHTS RESERVED

L C C : 73-13003 ISB N : 0-691-08136-0

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

Contents

§1.

Introduction ..............................................................................................

3

§2.

A lg e b ra ic P r e lim in a r i e s .........................................................................

10

§3.

T h e Geom etry of

P re lim in a rie s ...................................................

20

§4.

A Metric D efin itio n of the M axim al B o u n d a r y ..................................

31

§5.

P o la r P a rts

..............................................................................................

35

§6.

A B a s ic Inequality .................................................................................

44

§7.

Geometry o f N eigh b o rin g F la t s ............................................................

52

§8.

D en sity P ro p e rtie s of D isc re te Subgroups

......................................

62

§9.

P s e u d o -Iso m e trie s

.................................................................................

66

P s e u d o -Iso m e trie s of Simply C on nected S p ace s with N e g a t iv e C u r v a t u r e ...................................................................................

71

§10.

§11. §12.

X:

P o la r R e g u la r E lem ents in C o-C o m p act

F ......................................

76

P s e u d o -Iso m e tric In varian ce of Sem i-Sim ple and U nipotent E lem en ts .................................................................................

80

§13.

T h e B a s ic A pproxim ation

§14.

T h e Map

§15.

T h e B oundary Map

§16.

T it s Geom etry ............................................................................................. 120

§17.

R -R an k G reater than O n e ..........................................................................125

§18.

R ed u ctio n to Sim ple G roups

§19.

S p a c e s of R -R a n k 1 .................................................................................. 134

§20.

T h e Boundary S e m i-M e tric ........................................................................142

§21.

Q u a si-C o n fo rm al M appin gs O ver K and A b s o lu te Continuity on A lm ost A ll R -C ir c le s .................................... 156

§22.

T h e E ffe c t o f E rgo d icity ..........................................................................169

§23.

R -R an k 1 R igid ity P ro o f C on clu ded .................................................... 180

§24.

C on clu d in g R e m a r k s .................................................................................. 187

0

.................... ................................................

96

.............................................................................................. 103 ........................................................................... 107

................................................................... 128

B ib lio g r a p h y .......................................................................................................... 193

v

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

S T R O N G R IG ID IT Y O F L O C A L L Y S Y M M E T R IC S P A C E S G. D. M o sto w 1

§1. Introduction

T h e phenomenon of strong rigid ity that w e e s ta b lis h is the fo llo w in g : Let point

p

Y

be a lo c a lly sym m etric Riem annian s p a c e ; that is , given any

in

Y,

som e b a ll in through

p

Y

the symmetry map o f center

p,

w ith tangent vector

ture at each point of

Y

com pact and connected.

THEO RE M A.

: exp ty -» exp-ty

w here y

t -> exp ty

at

p.

is n o n -p o sitive.

is an isom etry on

d enotes the g e o d e s ic

A ssu m e that the s e c t io n a l c u rv a­ A ssu m e moreover that

Y

is

We prove (c f. Theorem 2 4 .1 0

The fundamental group

n ^ (Y )

determ ines

Y

uniquely up

to an isom etry and a c h o ic e of norm alizing constants, provid ed that Y has no c lo s e d one or two dim ensiona l g e o d e s ic s u b sp a c e s which are direct factors locally.

T h e norm alizin g co n stan ts referred to in our theorem in v o lv e s ch an gin g the metric o f

Y

changed for a ll let

X

in such a w ay that the symmetry maps p e Y.

Y.

sym m etric R iem annian s p a c e , that is the symmetry map

group

G

X

onto

X.

remain un­

T h e con stan ts a r is e in a c a n o n ic a l fash io n .

denote the sim p ly connected c o v e rin g s p a c e o f

isom etry of

o^

T h e s e t o f sym m etries

of iso m etries w hich act tran sitiv ely on

a

[a ^ ; p e X } X,

T h en

X

is a

is an g lo b a l gen erate a

and the connected

Supported in part by the National Science Foundation Grant G P 33893X.

3

For

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

4

component of w here

1 of the group

G Q is its center and

G

is a d irect product

{ G 1, G 2 , . . . , G n S is the s et of a ll its non-

a b e lia n normal, sim p le, an aly tic su bgro u p s. product decom position of

G °,

X = X q x X j x ... x X n , where space

G ° = G QxG^^ x ... x G n

C orresp o n d in g to th is direct

there is a direct product d ecom position of X^

is an-.orbit of

G -,

i = o, . .. ,n .

X Q is a E u c lid e a n s p a c e and thus, by h y p oth esis on

to a point.

N o w each

stant factor

c 1 for

X-

h as a unique

i= l,2 ,...,n .

norm alizing co nstants.

red u ces

G --in varian t metric up to a co n ­

Thfese facto rs

In the c a s e that

Y,

The

G°

c 1, . . . ^ 11 are the

is sim p le, they reduce to a

s in g le m u ltiplicative constant. Our theorem can be reform ulated a s a s s e rtin g the strong rigid ity o f d is c re te subgro up s of sem i-sim p le groups. group

r

K

^ (Y ) = T,

the

of co v erin g transform ations c o n s is t s of iso m etries and can be

regarded as a subgroup of w here

F o r if w e set

G °;

w e have

Y = T\X.

is a maxim al com pact subgroup o f

subgroup of

G”

if and only if

G /T

G.

M oreover,

We c a ll

is com pact.

T

X = G /K

a “ co-com pact

T h u s, our Theorem is

eq u iv alen t to

T H E O R E M A'.

Let

G

be a s e m i-sim p le analytic group having no center

and no com pact normal subgroup other than (1 ). G.

c o -co m p a ct subgroup of by

r,

provid ed

Then the pair

P S L (2 , R )

Let

(G ,F )

V

be a d iscre te

is uniquely determ ined

is not a direct factor of

G

which is c lo s e d

modulo r . That is, g iv e n two such pair 0 : r -* r '

and (G ', T ' )

there e x is ts an analytic isomorphism

the restriction of 6 P S L (2 , R )

(G ,T )

to

T,

such that V G^

and an isomorphism

0 : G -> G ' such that 6 is

provid ed there is no factor is a c lo s e d subgroup of

G-

isom orphic to

G.

T h e re ason for the p ro v iso is w e ll-k n o w n from uniform ization theory. Set

G = P S L (2 ,R ).

X = \z e C ; Im z > Oi

T h en

G

o perates on the upper h a lf-p la n e

v ia z

^J_b cz + d

§1. INTRODUCTION

5

and it p re s e rv e s the metric given by Riem ann s u rfa c e

ds2 = — z- . G iv en a com pact (Im z ) 2 o f gen us greater than one, then its sim ply connected

Y

c o v erin g s p a c e is a n a ly tic a lly e q u iv a le n t to id en tified w ith a subgroup

T

of

G;

X.

c le a r ly

T

be a n a ly tic a lly eq u iv a le n t; that is under an autom orphism o f

G.

and

V

may be

is d is c re te and co-com pact.

It is w e ll-k n o w n that two com pact R iem ann s u rfa c e s sam e gen us have isom orphic fundam ental groups

77^ ( Y )

T h ere fo re

T

and

Y

and

Y ' o f the

Y ' but need not

Y ' are not in gen e ra l co n ju ga te

T h u s stron g rigid ity fa ils for

P S L (2 ,R ).

H o w ev er, it is the only factor c a u s in g the fa ilu re of stron g rigid ity for any s em i-sim p le a n a ly tic group — or e q u iv a le n tly , for a lo c a lly sym m etric space. T h e ch ron ology of rigid ity b e g in s with the theorem of A . S e lb e rg ([1 6 ]) that a d is c re te co-com p act subgro up

T

of

S L (n , R )

cannot b e continu­

o u sly deform ed e x c e p t triv ia lly , that is , by inner autom orphism s of S L (n , R ),

if

elem en ts in

n > 2; T

S e lb e r g ’s proof rested on sh o w in g that the trace of

are p reserved under d eform ations o f T .

ap p lie d to the other c l a s s i c a l groups o f rank greater than

S e lb e r g ’s method 1.

A t about the

sam e time, E . C a la b i and E . V e s e n t in i proved the rigid ity of com plex structure under in fin itesim al deform ations o f com pact quotients o f bounded sym m etric dom ains ([3 b ]), and later C a la b i proved the metric a n a lo gu e for com pact h y p erb o lic n -sp a c e forms for

n > 2

([3 a ]).

Th ereup on A . W eil

([2 1 ]) g e n e ra liz e d S e lb e r g ’s and C a l a b i ’s re su lts to sem i-sim p le groups h av in g no com pact or 3 dim en sion al sim p le fac to rs. the rigid ity o f

T

cohom ology group

in

G

W e il’s proof d ed u ces

under d eform ations from the v a n ish in g of the

H 1^ , G )

w here

G

is the L i e a lg e b r a o f

as a T -m o d u le under the ad joint represen tation . su b gro u p s, w hich are la ttic e s (that is ,

T

G

regarded

In the c a s e o f arithmetic

is d is c re te in

G

and

G/r

h as fin ite H aar m easu re) but not ge n e ra lly co-com pact, the rigid ity under deform ations w a s proved in dependen tly by A . B o re l (u n p u b lis h e d ) in the c a s e of Q -s im p le groups o f Q -rank at le a s t tw o, and H. G arland (in the s p lit c a s e ), and by M. S. R aghunathan in the rem aining Q -rank one c a s e s , a g a in by s h o w in g that

H 1( F , G ) = 0.

6

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

T h e phenomenon of strong rigid ity for arbitrary la ttic e s firs t turned up in 1965 in my search for a geom etric exp lan atio n of d eform ation-rigidity. T h e point of v ie w adopted here can be e x p la in e d in terms of the counterexam ple cited abo ve.

When w e regard the fundam ental groups o f

two com plete Riem ann s u rfa c e s fin ite volum e a s su bgro u p s h a lf plan e

X,

and

and

l\

Y ' of the sam e genus

p

and of

of the isometry group of the upper

they are isom orphic transform ation groups; that is , there is

a diffeom orphism of phism

T

Y

Y

onto

Y ' and its lift p ro vid es a T -s p a c e isom or­

0 : X -> X ' such that for a ll

y eV

and

x e X,

0 ( y x ) = 0 (y ) 0 ( x ) w here

is giv en by the isom orphism of 77^ ( Y )

0 \T -> V '

transform ation grou ps, what d is tin g u is h e s On

X

they are in d istin g u is h a b le .

boundary of morphism

0

X,

T

But on

is co n ju ga te to

is smooth at

V ' in

X Q or not.

T

from

T '?

to

T h e an sw er is:

X U X Q, w here G

As

X Q is the

acc ord in g a s the T -s p a c e

M oreover this re su lt is true w ith­

out e x c ep tio n s for any s em i-sim p le an aly tic group h avin g no com pact normal sub gro u p s,

(c f. [1 2 f]; a ls o [1 2 i] for a more d e ta ile d a c c o u n t.)

In v ie w of the known deform ation-rigidity theorem s, it w a s natural to co n jecture that the h y p oth eses of e x is te n c e of boundary va lu e s and sm ooth ness at the boundary are su p erflu o u s for a ll symmetric s p a c e s of n eg ativ e curvature h avin g no tw o -dim en sio n al fac to rs.

T h is con jecture

w a s confirm ed in [1 2 g ] for the c a s e of two d iffeom orphic com pact s p a c e s o f constant n e g a tiv e curvature of dim ension e x c e e d in g two.

T h e proof

re lied h e a v ily on an a ly tic to o ls — not s u rp risin g ly in v ie w o f the an a ly tic nature o f the problem .

F o rtu n a tely , the theory of q u asi-co n fo rm al m appings

in 3 -s p a c e w a s at hand; upon g e n e ra liz in g the theory to n -sp a c e , one could be certain that the map

0

took on continuous boundary v a lu e s

w hich w ere alm ost every w h ere d iffe re n tia b le . of r

U t iliz in g the ergo dic action

at infinity, I could prove that the boundary map

M obius transform ation if

n > 2.

w a s actu a lly a

§1. INTRODUCTION

7

T h e re le v a n c e o f the u s u a l theory of q uasi-co n fo rm al m appings is unfortunately lim ited to s p a c e s o f constant curvature only.

F o r the c a s e

o f arbitrary sym m etric s p a c e s , it w a s n e c e s sa ry to find an en tirely d iffe r­ ent method to e s t a b lis h that

4> ta k es on boundary v a lu e s .

T h e method adopted here r e lie s on the key notion o f a p s e u d o -is o m e try, w hich is d efin ed as fo llo w s : Let

k > 1 and

c a lle d a

(k, b )

b > 0.

A map

X ' betw een metric s p a c e s is

p seu d o-iso m etry if

(1 )

d (0 ( x ), 0 ( y ) ) < k d (x , y ) ,

(2 )

d (0 ( x ), 0 ( y ) ) > k- 1 d (x , y ) , X A map

such that

cf> : X

d (x ,y )> b ,

w here

x, y 6 X for a ll

d

in

den o tes d is ta n c e .

X ' is c a lle d a p seud o-iso m etry if it is a

isom etry for som e

x ,y

(k, b )

p se u d o-

(k, b).

Our proof of stron g rigid ity c o n s is t s of four main ste p s. (i)

Let

Y

and

Y ' b e as in Theorem A , let

sim ply connected c o v e rin g s p a c e s , and let there is a T -s p a c e p seu d o-iso m etry (ii) (iii)

T h e F - s p a c e p seu d o-iso m etry T h e map

X

and

X ' denote their

771( Y ) = T = 7r1( Y ') .

cf> : X -> X ' (cf> need not be in je c tiv e ). cf> h as continuous boundary v a lu e s 0 Q.

in d u ces an in cid en ce p re se rv in g isom orphism o f the

“ T it s geom etry’ ’ of

G

onto the T it s geometry o f

g e o d e s ic s u b s p a c e s o f d im ension greater than

1,

G'.

If

X

h as fla t

then the G e n e ra liz e d

Fundam ental Theorem o f P r o je c t iv e Geom etry sh o w s that by an an a ly tic isom orphism o f (iv )

If

X

X = H n^ ,

T hen

G

to

is induced

G'.

has no flat g e o d e s ic s u b s p a c e o f dim ension e x c e e d in g h y p erbo lic k n -sp a c e over the d iv is io n a lg e b ra

(q u atern io n s) or

0

(C a y le y num bers), w here

1, then

K = R, C, H

k = d im ^K .

In this c a s e the u s u a l theory o f q uasi-co n fo rm al m appings d o e s not w ork for

C,

H,

and

0

a s it d o e s for

R.

H o w ever, w e introduce the

notion o f a K -q u a si-co n fo rm s I mapping over a d iv is io n algebra

K,

w hich

STRONG RIGIDITY OF LO C ALLY SYMMETRIC SPACES

8

amounts to the u su a l theory if

K = R.

We then g e n e ra liz e som e a s p e c ts

of the u s u a l q uasi-co n fo rm al m apping theory to obtain the a b s o lu te co n ­ tinuity of the boundary map 21).

0 Q alo n g alm ost a ll “ R - c ir c le s ’ 1 (c f. Section

A g a in , as in the c a s e o f

the ergo dicity of T

us to prove that the boundary map to

g

at infinity a llo w s

0 Q is induced by an isom orphism o f

G

:

T h e proof presented here is an im plem entation o f the program that w a s first form ulated in 1965 and announced in [1 2 f ]. o f our strategy for the c a s e o f no R -rank

T h e s u c c e s s fu l executio n

1 fa c to rs w a s announced at the

International C o n g re s s o f 1970 (c f. [1 2 j]). D u rin g that intervenin g period, stron g rigid ity had been proved for c e r­ tain arithm etic (non co -com p act) su bgro u p s on the one hand in 1967 by B a s s -M iln o r-S e rre (c f. [1 ]) and on the other by M. S. R aghunathan [1 4 ], u s in g a lg e b ra ic and arithm etic methods. A ls o , G. A. M argu lis h as announced re su lts statin g that for R-rank > 1, every non co-com p act irred u c ible lattice is nearly arithmetic (c f. [1 0 ]) and in 1971 M argu lis announced stron g rigid ity for such non co-com pact irre­ d u c ib le la ttic e s . It is appropriate to mention prior p artial re su lts on the extent to w hich r

determ ines

G.

In 1967 H. F u rs te n b e rg stu died this q uestio n from a

p ro b a b ilis tic point of v ie w and w a s led to in terestin g q u estio n s about the P o is s o n boundary (c f. [6 b ]); his method y ie ld e d that a lattice in (n > 3 )

co uld not be a lattice in

that if

G/r

S O (l,n ).

S L (n , R )

In 1962 J. W olf pointed out

is com pact, the rank o f the symmetric s p a c e a s s o c ia t e d to

w a s determ ined by

T

G

(c f. [2 2 ]).

A lth o u gh w e have stated Theorem A ' for co-com pact la ttic e s only, our method a p p lie s for most part to arbitrary la ttic e s . co -c o m p actn ess of T isom etry

0 :X

X'.

Indeed w e require the

only to a s s u re the e x is te n c e o f a T -s p a c e p se u d o In th ose c a s e s such a s la ttic e s in R -rank one groups

or arithm etic la ttic e s w here information is a v a ila b le about the cu sp s of a fundam ental domain for

V

a T -s p a c e p seud o-isom etry

in 0.

X,

it is p o s s ib le to prove the e x iste n c e o f

Indeed the e x iste n c e of such a

0

has

9

§1. INTRODUCTION

been recen tly proved by G o p a l P r a s a d for the c a s e of R -rank one s p a c e s ; this e s t a b lis h e s strong rigid ity for non-com pact la ttic e s in the R-rank one c a s e s other than

P S L (2 ,R )

T h u s com bin ing our re su lts w ith M a rg u lis ’ and P r a s a d ’s , one can w eak en the h y p o th esis of co -c o m p actn ess o f r

b e a la ttic e in

G .”

T

in Theorem A

to:

“ Let

T h at is to s a y , strong rigid ity h o ld s for arbitrary

la ttice s in c e n te rle s s a n a ly tic s em i-sim p le groups without com pact facto rs apart from the aforem entioned exce p tio n . In Theorem A this amounts to a ssu m in g m erely that m easure rather than

Y

Y

has fin ite

is com pact.

In c o n c lu s io n , som e s e c tio n s of our proof h ave independent in terest. T h e theory of p se u d o -is o m e trie s w hich p la y s such a cen tral role in our method may b e u s e fu l in other con texts (c f. S e ction s 9, 10, 12). theory of q u asi-co n fo rm al m appings over the d iv is io n a lg e b ra C, H,

or

0 )

The

K (K = R ,

w h ich in terven es im p licitly in S ection s 20, 21 d e s e rv e s

further attention.

§2.

2.1.

We denote by

A lg e b ra ic P re lim in a rie s

M(n, R )

the s e t of a ll

by

G L (n ,R )

the group of in vertible

of

G L (n , R )

is c a lle d alg ebra ic if

G L (n , R )

of

P j,

n x n re al m atrices.

n x n G

where each

c o e ffic ie n ts w ith re a l c o e ffic ie n ts .

re a l m atrices.

We denote

A subgroup

G

is the set o f common ze ro e s in P-

is a polynom ial in the matrix

A n elem ent of

G L (n ,R )

is c a lle d

s e m i-sim p le if its minimal polynom ial h as no repeated facto rs — or e q u iv a ­ len tly , if it can be d ia g o n a liz e d over the com p lex numbers. is c a lle d unipotent if

(u —l ) n = 0 ;

Jordan normal form for elem ents of there are elem ents

s

and

s

is se m i-sim p le ,

u

is unipotent.

s

and

u

un iquely.

u

in

One c a lls

A n elem ent u

i.e ., a ll its e ig e n v a lu e s are G L (n , R )

G L (n ,R )

s

G iv e n

s a tis fy in g :

The

g e G L (n , R ),

g = su,

su = u s ,

M oreover, th ese cond itions determ ine and

Jordan com ponents re sp e c tiv e ly .

im p lies:

1.

u the s em i-sim p le and unipotent

A ny elem ent commuting with

g

com­

mutes with its Jordan com ponents. Let

s

be a sem i-sim p le elem ent in

in g the e ig e n s p a c e s of sim p le elem ents e ig e n v a lu e s o f

k

s

and

p

in

G L (n , R ) 1,

s.

F o r any

commutes with

g e G L (n , R )

sem i-sim p le component of If

G

exp t lo g u g.

If

g

w e denote by

g = po l g,

t e R,

s = kp, kp = pk, the

and

p

p.

pol g

p

are p o s itiv e .

un iquely, and any elem ent

We c a ll

p

the polar part of

the p olar part of the Jordan

g.

(c f. [4 ]) and

for a ll

k

k and

is an a lg e b ra ic group, and

com ponents of

s a tis fy in g :

the e ig e n v a lu e s of

M oreover, th ese co n d itio n s determ ine s

T hen by su ita b ly p air­

over the com plex num bers, one can find sem i­

k h ave modulus

commuting w ith

G L (n , R ).

g e G,

pol g

where

u

then

G

contains the Jordan

(c f. [1 2 a ]), a s w e ll as

u* =

is the unipotent Jordan component of

then for any re al number

10

t,

g^ = exp t lo g g

is a w e ll

§2. ALGEBRAIC PRELIMINARIES

11

d efin ed s em i-sim p le elem ent w ith p o s itiv e e ig e n v a lu e s in b e lo n g s to ev ery a lg e b r a ic group co n tain in g

for a ll

is isom orphic to If

V

fy in g

g eA Rs

A

of

G L (n , R )

is c a lle d a polar subgroup.

for som e

and

A ut V

such that

A n y polar subgroup

s.

is a fin ite d im en sio n al vecto r s p a c e over

H o m ^ (V , V )

and

g.

A n a r c w is e connected a b e lia n subgro up g = pol g

G L (n , R )

with

M(n, R )

and

R,

then upon id en ti­

G L (n , R )

v ia a ch o ice

o f b a s e , one can d e fin e a lg e b r a ic su bgro u p s and p o la r su bgro u p s of in an unam biguous w ay.

C le a rly any p o lar subgroup of

in d ia g o n a l form v ia a s u ita b le c h o ic e o f b a s e in 2.2.

Let

G

a lg e b ra o f

b e a L i e subgroup o f

G,

and w e id en tify

G

G L ( n , R ). with

Aut V

We denote by G

the L ie

{Y ; Y e M (n ,R ), exp R Y C G l;

G

adjoin t rep resentation is given by

A d g ( Y ) = g Y g - 1 , w here

A

can b e put

V.

that is , w ith the tangent s p a c e to

Let

A ut V

at the identity elem ent

b e a polar subgroup of the L i e subgroup

G.

1.

The

g e G , Y e G.

T h en

Ad A

can

b e d ia g o n a liz e d and therefore

G = ^

w here eac h

a

Ga

is an a n a ly tic homomorphism of

group o f p o s itiv e re a l numbers and s e t of a space of

A

Ga in

su ch that

Gq ^ 0

is s t a b le under G.

(d ire c t)

If a

eac h elem ent in

and

Gq

/3

A

G a = { Y e G; A d g ( Y ) = a ( g ) Y|.

are c a lle d R -ro o ts o f Ad Z (A )

into the m u ltip lic ative

where

are roots, then

Z (A )

G

on

A.

The

Each su b ­

d enotes the c e n tra liz e r

[G a ,G^g] C Ga ^g.

M oreover,

is nilpotent if a £ 1.

T h e m axim al p o lar su bgro u p s of

G

are c o n ju ga te v ia inner automor­

phism s [1 2 d ]. 2.3.

Let

G

be an an a ly tic subgro up o f

nected L i e subgroup. G

G L (n , R );

A s su m e m oreover that

G

that is ,

is a co n ­

is s e m i-s im p le ; that is ,

h as no norm al a n a ly tic a b e lia n subgroup other than

its own commutator su bgro u p , each elem ent o f

G

G

(1 ).

S ince

G

is

has determinant one.

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

12

Let

A

be a m axim al polar subgroup of

o f a n a ly tic homomorphisms of p o s itiv e re a l numbers. m ultiplication in

T h en

A

into

y

G,

T h e R -ro o ts of

G

on

A

x

denote the group

R + , the m u ltip lic ativ e group of

is an a b e lia n vecto r group.

a d d itiv e ly , w e have

y

and let

W riting the

(a + / 3 )(a ) = a (a )(3 (a )

for

a e A.

s a tis fy alm ost a ll o f the w ell-k n o w n proper­

tie s o f root sy stem s on C artan s u b a lg e b r a s of com plex sem i-sim p le L i e a lg e b ra s .

N am ely

(i)

If a

is an R -root, then

(ii)

If a, i3, a+/3

(iii)

T h ere is a s u b s e t of lin e a rly independent R -roots

ctj, . . . , a f

is an R-root.

are R -ro o ts, then

that any root h as the form subset

—a

± X- n-

[G a >G^g] ^ 0.

with

n-

such

n o n -n egative in tegers.

A

with this property is c a lle d a fundamental s y ste m of

R -ro o ts. (i v )

Let

N (A )

denote the norm alizer o f

the c e n tra liz e r o f

A

in

G.

T h en

A

in

N (A )/ Z (A )

G

and let

Z (A )

denote

o perates (v ia inner auto­

m orphism s) sim p ly -tra n s itiv e ly on the s e t of fundam ental sy stem s of R -ro o ts. (v )

T h e R -ro o ts se p a ra te the p oints o f A sem i-sim p le a n a ly tic group

A.

Thus

A « R r.

is the to p o lo g ic a lly connected com* ponent o f the identity o f the s m a lle s t a lg e b ra ic group G in G L (n , R ) containing

G.

T h e re fo re for any

G

g e G,

ta in s the one param eter groups

( pol g)*

unipotent Jordan component o f

g.

polar subgroup of DEFINITION.

G

and

the sem i-sim p le group and

u*

In p articu lar,

G

(t e R ) w here u

con­ is the

pol g lie s in a m axim al

u e G.

A n elem ent

g

in

G

is c a lle d polar regular if

dim Z ( pol g ) < dim Z ( pol h) for a ll Let g

g

w here

Z (h )

be an elem ent of

denotes the cen tra lize r of G

with

is polar re gu lar, and R -s in g u la r if Let

G.

h e G,

A

g = pol g. g

h

in

We c a ll

g

G. R -re g u la r if

is not p o lar regular.

b e a m axim al p o lar subgroup of the se m i-sim p le a n a ly tic group

Inasm uch a s a ll the m axim al polar subgroup s are co n ju ga te in

G,

A

§2. ALGEBRAIC PRELIMINARIES

con tain s R -re g u la r elem en ts.

Indeed an elem ent

and only if a ( a ) = 1 for som e R -root of

y.

A n elem ent

g e A

w here

a

13

a 0

a £ ^B.

if a ( a ) > 1 for a ll

a e ^B;

A.

F o r any R -root

a (^ B ) = 0

a

we

if a ( a ) = 1 for

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

14

The subspace

Ga ,

ia ; a ( " * B ) > Oj,

is a L i e s u b a lg e b r a o f n ilp o -

tent elem en ts s ta b le under inner automorphisms from U (^ B )

denote the co rresp o n d in g an a ly tic group.

Z (B )

T h en

(c f. 2.3).

U (^ B )

is a group

o f unipotent elem ents and is in fact unipotent, it is co n ju ga te in to a group o f trian gular m atrices.

Let

G L (n , R )

Set

P (^ B ) = Z (^ B ) U (^ B ) T hen

P (^ B )

is a group,

N (U ( ^ B ) ) = N ( P ( ^ B ) ) , From

U (^ B )

w here

P ( ^ B ) = N (P (^ B ))

T h e unipotent group for any chamber

is normal in

N(

)

it fo llo w s that U (^ A )

P (* B )

^Bj

P ( ^ B 2)

and

and

P (^ B ) = (

is c lo s e d in

)

in

G.

G.

is a maximal unipotent subgro up o f

G

and m oreover a ll m axim al unipotent su bgro u p s o f

are co n ju ga te v ia an inner automorphism o f If

P (* * B )

denotes the norm alizer o f

"*B 2

G [1 2 d].

are cham bers or cham ber w a lls , then

if and only if

**B2

is a fa c e of

^ B j.

c a lle d the parabolic subgroup a s s o c ia t e d to p a ra b o lic subgroup if and only if

^B

co n ju ga cy of a ll the cham bers in

G,

G

P ( ->P(n, is a bianalytic homeomorphism. For any g f G, the positive definite symmetric element g^g lies in 1_

G and hence by 2.3, (g*g)2 e G H P(n,R). It follows at once that that (i) G = (Gfl P(n, R)) • (G D 0(n, R)) this decomposition being a direct product topologically. Since G is closed in GL(n, R), (ii) G fl 0(n, R) is a compact subgroup of G. Moreover (cf. [12b]) (iii) Any compact subgroup of G is conjugate via an inner automorphism to a subgroup of G fl 0(n, R). In fact properties (i), (ii), and (iii) are valid for any self-adjoint group which is of finite index in an algebraic group. Thus if SC GG (1 0(n, 0(n, R) R) or S C G H P(n, R), then lZ(S) = Z(S) and Z(S) = (Z(S) n P(n> R)) • (Z(S) fl 0(n, R)) We consider two special cases of this observation. Set K = G H 0(n, R). Let A be a maximal abelian subgroup of P(n, R). By definition, Z(A) fl P(n, R) = A and therefore by (i) (iv) Z(A) = (Z(A) n K) • A . This implies that A is a maximal polar subgroup of G. As a second case, let s be a semi-simple element of G and let k • p = s be its polar decomposition with p = pol s. We wish to show that Z(s) is conjugate to a self-adjoint group. By 2.2, p is conjugate to an element of A. Without loss of generality we can assume that p f G D P(n, R). Hence lZ(p) = Z(p) andZ(p) Z(p)== (Z(p) fl P(n, R) • • (Z(p) fl 0(n, R)). Since k lies in a compactsubgroup subgroupofof Z(p),we we cancan assume after conjugation by an element of Z(p) that k e Z(p) fl 0(n, R),

STRONG RIGIDITY OF LO C ALLY SYMMETRIC SPACES

16

by (i i i ) .

Thus

Z ( s ) = Z (k ) H Z ( p ) = ^ ( s ) .

T h u s w e s e e that

Z (s )

oper­

ates sem i-sim p le ( “ re d u c tiv e ly ” ) on the un derlying vector s p a c e for any sem i-sim p le elem ent

s

of the s em i-sim p le group

G.

Such a group is

c a lle d red u ctive and its L ie a lg e b ra is e a s ily s e e n to be the direct sum of its center and a sem i-sim p le id eal. From the argum ents u sed above one d ed u ces e a s ily (v )

A ny elem ents o f

by an elem ent of 2.7.

Let

A,

that are co n ju ga te in

be a s abo ve.

F o r any R-root

a e A, Xa e Ga .

get

G = K + P (^ B )

is ,

G = K P (^ B )

Thus

and

G/P( ^B )

co ntaining G L (n , C ) Let

G/P

is com pact.

in

A.

T h at

Since a ll cham bers in

G

is com pact for any p a ra b o lic subgroup P .

G '/ P '

k:

An a lg e b r a ic subgroup

is a com plete variety.

P ' of

G ' is

In this context, the

are taken w ith co o rd in ates in an a lg e b r a ic a lly c lo s e d fie ld k.

In our situatio n , let

G ' b e the s m a lle s t a lg e b r a ic group in

containing the sem i-sim p le a n a ly tic subgroup

P ' denote the s m a lle s t a lg e b r a ic subgroup in

P ( ^ B ).

Xfl — *XQ e K, w e

T h e notion of p a ra b o lic subgroup can be d efin ed for a lg e b ra ic

p a ra b o lic if and only if G

\

Inasm uch as

e

G ' over an arbitrary fie ld

points o f

on the m axim al polar

for any cham ber or cham ber w a ll

are co n ju ga te, w e s e e that

groups

a

=a(a)Xa

a " 1 tXa a = a(a )

R E M A R K 1.

are co n ju gate

w e have

a Xfl a -1

for a ll

G

G fl 0 (n , R ).

G, A, K

subgroup

G fl P (n , R )

T h en

P ' is a p a ra b o lic subgroup o f

G '.

G C G L (n , R ).

G L (n , C ) T h e group

co ntaining P'

is n e c e s ­

s a rily connected and

P ( ^ B ) = P ^ * Thus P ' is connected in the Z a r is k i K K topology but not n e c e s s a rily in the E u c lid e a n to p o lo gy . O ne can c h a ra c ­ te riz e a p a ra b o lic subgroup of such that

G ' / P ' is com pact.

G

a s a Z a r is k i-c lo s e d subgroup

P

of

G

17

§2. ALGEBRAIC PRELIMINARIES

2.8.

Let

G

subgroup o f

be a s e m i-sim p le a n a ly tic group, let G,

let

^A

be a cham ber in

Bruhat d eco m p o sition of

G

G = U ( ^ B ) N (A ) P ( ^ B ) 2.9.

Let

§

zero and let

A,

A

be a m axim al polar

and let

U = U (^ A ).

a s s e rt s G = U N ( A ) U . A s

for any cham ber w a ll

^B

The

a c o n seq u en ce

in

A.

b e a sem i-sim p le L i e a lg e b ra o ver a fie ld o f c h a ra c te ristic n b e an elem ent of

J a c o b so n -M o ro zo v lemma s a y s : that

§ §

such that

ad n

is nilpotent.

co n tain s elem ents

h and

r -i [h, n _ ] = —2 n _ , [n, n_J =

[h, n] = 2n,

The

n_

su ch

h.

A s a d ire ct c o n s e q u e n c e , any (n o n -a b e lia n ) redu ctive a n a ly tic group in G L (n , R )

w h ich co n tain s a unipotent elem ent

u(u ^ 1)

d im en sion al a n a ly tic group with L i e a lg e b r a gen erators ab o v e and w ith

F o r any sym m etric matrix

have

T r X Y = —T r Y X = 0.

1,

H en ce

and a fo rtio ri the L i e a lg e b ra of G H P (n , R )

at

1.

sem i-sim p le a n a ly tic group. the orthogonal complement to Set

E = G fl S(n, R).

and skew -sym m etric matrix S(n, R ),

2

G H 0 (n , R )

Y

we P (n , R )

with re sp ect to

Tr XY,

is orthogonal to the tangent

T h is is true for any representation o f the T h u s one can c h a racterize K

w ith re sp ect to

G

G fl S(n, R )

as

T r ad X ad Y .

x -> *x- 1

of

G L (n , R )

and in du ces on it an automorphism

sta­ o

such that g y *g

of

and for any

G L (n , R )

and therefore in du ces a c a n o n ic a l action of space

.

denote the p ro je ctive s p a c e of lin e s through the origin of the

lin ear s p a c e

[S ].

Since n ° fi Let

on

X

f°r

is a morphism o f the G -s p a c e

X

denote the to p o lo g ic a l c lo su re of

the

is lin ea r in

y

on the p ro je c tiv e g, x 6 G ,

we se e

into the

G L (n , R )-

° f i (X )

in

[S ].

is a com pact G -s p a c e , the Satake p -com p a ctifica tion

of

X

X

[1 5 ].

Satake h as shown that

dim ension le s s than

dim X.

X

tt

is a fin ite union of G -o rb its, each of

Am ong th ese G -o rb its, there is a unique

com pact G -orb it; w e denote this orbit by

X Q. T h e orbit

e q u a lly w e ll ch a racterized a s the G -orbit in isotropy subgroup o f a point in

X

X Q may b e

of lo w e st dim ension. T h e

X Q is a p a ra b o lic subgroup of

Satake p -co m p actific atio n d epends on the represen tation (fo r ex am p le, if the “ high est R -w e ig h t ,J o f

then the isotropy subgroup o f a point in group.

w e s e e that

y e S(n, R ),

S(n, R )

G L (n , R )

jx (g x K ) = p (g ) / i(x K ) V ( g )

T h en

p

X =

We have

n (x K ) = p (x ) V ( x ) Let

V

Set

T h en one can s e le c t a b a s e in

P (G ) = (p (G ) fl P (n , R ) ) • p (K ) Let

G,

G.

p

p.

G.

The

F o r s u ita b le

lie s in sid e a cham ber)

X Q is a minimal p a ra b o lic s u b ­

F o r su ch a co m p actification w e c a ll

X Q the Fu rsten berg maximal

§2. ALGEBRAIC PRELIMINARIES

19

boundary (c f. A P o is s o n form ula fo r sem i-sim p le groups [ 6 ]. ) m axim al boundary,

XQ = G/P( ^ A )

for some cham ber

T h u s for the

**A.

We s h a ll g iv e in Section 4 b e lo w an a ltern ativ e d efin itio n of terms of the m etric p ro perties of

X.

X Q in

§3.

Let

G

sentation .

The Geom etry o f

X:

b e a sem i-sim p le an a ly tic group h avin g a fa ith fu l matrix repre­ Let

K

b e a m aximal com pact subgroup of

One can d e fin e a metric on

X

invariant under

C h o o s e a faith fu l matrix represen tation and

P re lim in a rie s

p (K ) = G f"l 0 ( n , R ) .

T hen imbed

p

X

G

G.

Set

in the fo llo w in g w ay.

(c f. 2 .6 ) such that in

X = G /K .

P (n , R )

*p (G ) = p (G )

v ia the map

p.:

/ i(x K ) = p (x ) lp (x ) for

x e G. On the s p a c e

P (n , R ),

the metric

( t ) 2 ■ Ti w here

p (t)

is a d iffe re n tia b le path in

the c a n o n ic a l

G L (n , R )

the induced metric on R E M A R K 1. X

If

G

action

X

P (n ,R )

y -> g y *g.

■ is c le a r ly invariant under

Since

f i ( g x K ) = p(g)/x (x K )*p (g ),

is G -in varian t.

is a sim p le an a ly tic group, any G -in varian t metric on

is unique up to a constant factor.

an a ly tic subgro up s of G|/K fl G -.

p>2

G,

then

If

G ^, . . . , G g

X = X 1x ...x X g

are the sim p le normal

(d ire c t)

T h e most gen eral G -in v arian t metric on

X

where

X- =

is the d irect

product metric and is therefore unique up to m ultiplication by a constant c-

in eac h factor Let

Z =

subgroup o f

fl G

X^,

i= l,...,s .

g g K g - 1 , ig e G i. and

G/Z

T h en

Z

is the maximum normal com pact

operates fa ith fu lly on

X.

T h e group

G /Z

has

no com pact normal subgroup and therefore, a s one may d edu ce e a s ily , h as no center.

C o n v e rs e ly , a s e m i-sim p le an aly tic group h avin g no com pact

direct facto rs and no center has no com pact normal subgro up other than (1 ) and thus operates fa ith fu lly on

X.

20

§3. THE GEOMETRY OF X: PRELIMINARIES

R E M A R K 2.

Let

r

denote the c a n o n ic a l map o f

group of iso m etries of r (G )

X

F o r eac h point 1 for

Y

and h as e ig e n -

are the e ig e n v a lu e s of

dg

on

P (n , R )

together

A e R.

T h e fo re go in g lemma h as an in terestin g interpretation. the metric

Y,

If w e d efin e

by the form ula

( | f )2 - T , (log p)2 then

dg < d,

Indeed

w here

d

denotes the

d g (p , q ) = |log p — lo g q|,

d efin ed on the lin e a r s p a c e Tr UV.

for any

S(n, R )

invariant metric on

p, q e P (n , R )

exp t Y ,

w here

P (n ,R ).

|Y|

is

by the E u c lid e a n inner product

Inasm uch a s one d im en sio n al s u b s p a c e s o f

in the E u c lid e a n m etric, and s in c e group

G L (n , R )

dg = d

S(n, R )

are g e o d e s ic s

alo n g the one param eter s u b ­

w e co n clu d e that the one param eter subgroup

p*

is the

unique g e o d e s ic jo in in g the identity elem ent to

p.

any two points there p a s s e s a unique g e o d e s ic .

M oreover, the a n g le s at

the identity matrix with re sp ect to both

d

L E M M A 3.3.

If

a, b, c

dg

and

C

c o in c id e .

a re the s id e s o f a g e o d e s ic triangle, then

c 2 > a 2 + b 2 — 2ab co s with equality if

T h e re fo re betw een

,

is at the identity, if and only if the triangle

the orbit o f an abelian subgroup of

P (n , R ).

lies in

STRONG RIGIDITY OF LO C ALLY SYMMETRIC SPACES

24

P ro o f.

P u t the vertex

B y Lem m a 3.2, the the

d

d istan ce.

3.4.

emma

a, b, c

dg

at the identity v ia an isom etry from

G L (n , R ).

d is ta n c e for the o p p o site s id e d oes not e x c e e d

T h e in eq uality now fo llo w s from the law of c o s in e s for

E u c lid e a n s p a c e .

L

C

T h e eq u a lity cond ition re su lts from Lem m a 3.2.

(i)

A, B, C

Let

be the v e r tic e s of a g e o d e s ic triangle and

the lengths of the corresponding s id es.

Then

180°


90°,

(i)

a, b, c.

lies

a > c s in A

.

b < c cos A

.

then

C o n sid e r a E u c lid e a n trian gle

A 1 Bj

Cj

w ith s id e s of length

B y Lem m a 3.3

2

cos

T h erefo re

£C^

^A

+ ^ C ^ ^ A j

+ ^B

> £C.

2

b

1

2

I ab

S im ilarly

~ c

< cos * C .

~

^A 1 > ^A

and

+ ^ B t + £ C X < 180°.

£ B 1> ^ B .

H ence

T h e e q u a lity condition

fo llo w s from Lem m a 3.3. (ii)

T h e firs t a s s e rtio n comes, from p la c in g the vertex

and com paring the R iem annian length of the s id e s dg-len gth .

BC

A

at the identity

with the

T h e seco n d a sse rtio n o f ( i i ) fo llo w s from

c 2 c o s 2 A = c 2 — c 2 s in 2 A > a 2 + b 2 — c 2 s in 2 A > b 2 . A s u b s e t of

P (n , R )

is c a lle d a g e o d e s ic su b sp a c e if it contains for

every p air o f d istin ct p oints through

px

and

any pair of points

p2

px

and

p2

(w ith re sp ect to the

P 1?P 2

w e denote by

the unique g e o d e s ic line p a s s in g G L (n , R ) [p 1, p 2l

invariant m etric). F o r

the unique c lo s e d

§3. THE GEOMETRY OF X: PRELIMINARIES

g e o d e s ic lin e segm ent from

p 1 to

p2 .

A su bset

c a lle d co n ve x if for every pair of points [p l t p 2 ^ ^ e s

P j,P 2

T h e to p o lo g ic a l c lo s u re in

C

of

*n C ,

25

P (n , R )

is

the segm ent

P (n , R )

of a convex set

is c le a r ly convex. We c o n sid er now the im bedding o f where

G

P (n , R )

o f the s p a c e

is a s e m i-sim p le a n a ly tic lin ea r group and

K

X = G /K

a m axim al com­

pact subgroup: fi(x K ) = p (x ) V ( x ) w here

p ( G ) = ^ p (G ).

p^ e jLi(X),

The image

w e h ave

p^ = g^ ^g*

d e s ic segm ent from w here

Y R ) ^ p (G ). for a l l

g x ex p s Y tg 1 -

for a l l

w ith

p2

is a g e o d e s ic s u b s p a c e .

n( X)

s e R.

H ence

( g x exp s/2 Y ) * (g x exp s/2 Y ) 6 p .(X )

contains the g e o d e s ic lin e p a s s in g through

p^

and

p2 . More g e n e ra lly , (3 .4 .1 )

if

G

is an analytic group such that G = ( G f l P ( n ,R )) • (G fl 0 ( n , R ) )

then

G fl P (n , R )

F o r any s u b s e t number

v,

S

is a g e o d e s ic s u b sp a c e of P (n ,R ).

o f a metric s p a c e and for any n o n -n ega tive real

w e denote b y

tance le s s than

LE M M A 3.5.

v

Let

of

C

T y (S )

the s u b s e t of points ly in g w ithin a d is ­

S.

be a co n ve x s u b s e t o f P (n , R ).

Then

T y( C )

is

co n ve x .

P ro o f. that

C

Sin ce

T y ( C ) = T y (C ),

is c lo s e d .

Let

w e can assu m e without l o s s of gen erality

p^ e T y ( C ) (i = 1,2).

p act, there e x is t s a point

q^eC

w ith

S in ce

C H T y (p p

is com­

d(p^, C ) = d(p^, q^) (i = 1, 2).

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

26

C h o o s e a point

p0 e [p 1? p2]

d (p 0 , U i > q 2] ) =

with or

Pq

su ch that

supp

d ( p ^ q ! , q 2] ) ^ p * t p i , p 2] l

>

taken a s an endpoint if the supremum is attained at either

p 2 . C h o o s e a point

q 0 f [ q 1, q 2]

su ch that

px

d (p Q, q Q) = d (p Q, [ q x , q2]) .

We c o n sid er three c a s e s C a s e 1.

pQ is an endpoint.

H ere

v > d (p 0 , [ q x , q 2] ) > d ( p , [ q 1 ; q 2] )

for a ll

p t [ p j , p 2]

C a s e 2.

s o that

pQ is not an endpoint and q Q is an endpoint of

d e fin ite n e s s , su p p o se p0

in the trian gle

P2Po^O - ^ If

[ p j , p 2] C T v ( [ q 1, q 2]).

qQ = q ^

p1 pQ q x

then

w e can s u p p o se

q Q,

O th erw ise by s e le c tin g a point

b V Lem m a 3.3 -

Set

^ p 0q q 0

contrad icting

We now c o n sid e r the q u ad rila teral 90°.

by Lem m a 3.3 and thus

T h is is im p o ssib le , by c h o ic e of

w e w o u ld get an obtuse an g le for

at le a s t

H en ce

pQ. T h u s

q x ^ q 2>

£ p 0q Qq2 > 90°.

d (P 0 >q ) < d (P (y

q0

the a n g le at

cannot b e obtu se by Lem m a 3.3.

d (p 2 , q x) > d (p Q, q j )

d(p 2 ^ cli >q 2^) = d ( P o ^ q r q 2])-

near

d (p Q, q x) > d (p x , q 1),

For

'

q 1 = q2 ,

We claim

S ince

[ q 1, q 2].

P2PoqQq2

q e [ q j , q 2]

and therefore

d (p 0 , q 0) = d (p Q, [ q x ,q 2l). with a n g le s at

c = d (q Q, p 2), 0 = £ p 2qoq 2 , 0 ' = * P 2qoPo*

Lem m a 3.4 (i i ), and noting that w e get

p2

as

A p p ly in g

0 + 0 ' > ^ p 0q 0q2 ,

d (p 0 , q0) < c c o s O ' = c sin (90 — 0 ' )
9 0 °

for

i = 1 ,2 ,

by the argument abo ve b a s e d on Lem m a 3.3. A ls o

^ p 1pQq 0 + ^ P 2Poclo = 1^0°,

at le a s t 90°. rila te ra l ab o v e

so that one of th e se tw o a n g le s is

A s su m e for d e fin ite n e s s that

p2pQq 0q 2

has a n g le s at

d (p 0 , q Q) < d (p 2 , q2)

^ p 2p0q Q > 90°.

pQ and

T hen the q u ad ­

qQ at le a s t 90°.

T h e re fo re , as

and w e get the sam e contradiction.

Thus

C a s e 3 d oe s not occur. Since only C a s e 1 o ccu rs, Lem m a 3.5 is proved. REMARK. point

Let

F

p e P (n , R )

b e a g e o d e s ic s u b s p a c e o f the s p a c e w e denote by

7r(p)

a point in

F

P (n , R ).

F o r any

(w h ic h is c lo s e d ) such

that d (p ,7 7 (p )) = d (p , F ) . T h en the g e o d e s ic segm en t g e o d e s ic s in

F

through

[ p , 77 (p )]

n (p),

forms a right an g le at

by Lem m a 3.3.

n {p )

with a ll

It fo llo w s from the fact

that the sum o f the a n g le s in a g e o d e s ic trian gle is at most 180° that the point

77(p)

is unique.

p ro jectio n o f

P (n , R )

the q u ad rila teral therefore

f

zatio n of and

b

in

F o r any points

p^

the orthogonal

and

^ a s f ight a n g le s at

L

p2 ^ (p j)

in

P (n ,R ) and

7r(p2 );

be a g e o d e s ic lin e in the R iem annian s p a c e P (n , R )

be a re a l va lu e d function on L R

by arc length. and

t

in

L.

T h e function

Let f

s -* p (s )

b e a param etri-

is c a lle d c o n v e x if for a ll

a

[0 , 1 ],

f ( p ( ( l ~ t ) a + t b ) ) < ( 1 - t ) f ( p (a )) + t f ( p (b ) )

L E M M A 3.6.

P ro o f.

F.

^ ( p j ) ^ (9 2 ^ 2

Let

(3 .6 .1 )

Then

onto

7 7 : P (n , R ) -> F

d (p x , p 2) > d (7r(p 1 ) , 7r(p2)).

DEFINITION. and let

We c a ll the map

Let

p -» d ( p , C )

L

he a g e o d e s ic line and C

is a c o n ve x function on

a c o n v e x s e t in

L.

We lo s e no gen erality in a ssu m in g that

C

is c lo se d .

P (n , R ).

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

28

G iv en any point

p

that

d (p , q ) = d (p , C ) .

and

q2 e C

w ith

in

P (n ,R ),

and

3.5, the c lo s e d b a ll of center fore co n tain s the segm ent for a ll

the segm ent

[p ,q ]

d (p, q x) = d (p , q 2) = d (p , C ). p

and ra d iu s

[ q 1 , q 2].

q e [ q 1, q 2].

s e le c t

qi e C

with eq u a lity at

(3 .6 .1 ) is v e rifie d for for

d (p , C ) .

q

to

p.

p

in

P (n , R ),

is co n vex and there­

in the interior o f

[ q 1, q 2l

[q ^ q ^ ,

[q , q x]

d (p , q x) > d (p , q )

C = [ q 1, q 2l. p = p2 ,

For

or

by Lem m a

It

d (p ,C )
q (p )

is a d iffe re n tia b le fu n c­

(n am ely, co n sid er it s e p a ra te ly on the

three connected com ponents o f the complement of w here

d (p , q ) =

C = [q 1, q 2l.

It is e a s y to v erify that

tion on the com plem ent o f

q1 € C

d (p ^ C ) = d (p i? q ^ ( i = 1, 2).

p = p1 and

d ( p , [ q 1, q 2l )

such

is unique.

s o that

T h u s w e assum e that

F o r eac h point [q 1, q 2]

q

T hen

s u ffic e s to prove Lem m a 3.6 for the c a s e d (p , [ q 1 , q 2l )

d (p , q x)

S e le c tin g

for d e fin ite n e s s .

p- £ L ,

C

B y Lemma

It fo llo w s therefore that

T h is contradiction im p lies that G iv en

in

forms an a n g le o f at le a s t 90° with either

“ say 3.3.

q

F o r su p p o se o th erw ise, then there w o u ld be

q 1 £ q2

d ( p , [ q 1, q 2l )

there is a unique point

U 7r“ 1(q 2),

is the orthogonal projection onto the g e o d e s ic lin e co ntaining

n

[ q ! , q 2]). Let

s

p (b ) = p 2 , if

p (s ) and

be a param etrization o f the lin e s et

f ( s Q) £ 0,w e h ave

q(p(so)))|s0 and q (p ), if

(3 .6 .2 )

w ith

p (a ) = p x

and

f ( s ) = d (p (s ), q ( p ( s ) )

w here

^ (s q ) =

and

^ d (p (s ),

= Jg d (p (s o ), q (p (s )))| s Q* From the d efin itio n o f

T h u s if

for a l l

s,

and hence

f ( s ) ^ 0,

f '(s ) = 0 (s )

T h en

f ' ( s Q) = ^ ( S q ) + f 2( s Q)

d (p ( s 0), q (p (s Q) ) ) < d (p (s Q), q ( p ( s ) ) )

f ( s 0) / 0.

w here

f ( s ) = d (p (s ), [ q t , q 2]).

L

is the an g le

f x( s ) = sin ( 0 ( s ) -

90 ) = - c o s 0 (s )

q ( p ( s ) ) p (s ) p2 .

f ' ( s ) = sin (9(s) • £

T h erefo re

f 2( s Q) = 0

§3. THE GEOMETRY OF X: PRELIMINARIES

C o n sid e r now the q u a d rila te ra l a n g le s at

q ( p ( s Q) )

and

q (p (s Q) ) p (s Q) p (s Q + A s ) q ( p ( s Q + A s ) ) .

q (p (s Q+ A s ))

are at le a s t

Lem m a 3.3 one co u ld find nearer points to

[^1 >^2^ ^ an q (s o ) and q ( s 0 + A s ). the q u ad rila teral d o e s not e x c e e d A s ) > 0

for a ll s and thus

f

is

a co n vex function. C a s e 2. f

L D [ q x , q 2]

is not empty.

v a n is h e s at a s in g le v a lu e

[sq ,

oo]

w ith

c re a s in g on on

R.

If

f ( s Q) = 0. [ —'oo, S q],

Since

and

f

f(s ) > 0

a s in g le point,

is co n vex on [ —, s Q]

for a ll

s,

f

m onotonically in c re a sin g on

L fl [ q 1, q 2]

and the co n v ex ity of

s Q,

T h en if it c o n s is t s of

is m onotonically d e ­ [ s q ,< »]

and co n vex

con tain s more than one point, then

f

and

[ q 1, q 2l C L

is clear.

T h e proof o f Lem m a 3.6 is now com plete.

L E M M A 3.7.

L

L C T y(F )

such that p e L.

Let

be a g e o d e s ic line and F /or som e fin ite

v.

M o re o ve r, for any d is tin ct points

a n g les of quadrilateral

p j ^ ( p j ) 7r(p2) p2

the orthogonal p rojection onto

P roof.

Let

f ( s ) = d (p (s ), F )

bounded co n v ex function on

Then

a g e o d e s ic s u b sp a c e d (p , F ) = d (L , F )

p ^ and p2

It fo llo w s at once that

fo llo w s by Lem m a 3.3 from the fa c t that

is

f

T h en

f

is a

is constant.

T h e seco n d a s s e rtio n

d (p .,7 r(p .)) = d (p -, F ) = d (L , 7r(p-)),

i = 1 ,2 ;

or alte rn a tiv e ly , it fo llo w s from (3 .6 .2 ).

Re

.

Inasm uch a s Lem m as 3.3 through 3.7 are v a lid for any g e o d e s ic

subspace of X

tt

F.

a s in the proof of Lem m a 3.6. R.

the four

are right a n g le s , where

From this the firs t a s s e rtio n o f the Lem m a fo llo w s .

m a r k

L,

in

for all

P (n , R ),

they are v a lid for the symmetric R iem annian s p a c e

a s s o c ia t e d to a s e m i-s im p le an alytic group.

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

30

DEFINITION.

A g e o d e s ic s u b s p a c e

F

of

P (n ,R )

is c a lle d flat if and

only if the sum of the three a n g le s of every g e o d e s ic trian gle in Let

ABC

sum 180°.

A

Then the triangle lie s in a flat su bsp a ce.

then giv en by

expsY

Y

and

and

Z

P (n , R )

containing

L e t now

ABCD

exp t Z

commute.

{e x p s Y + t Z , s , t f R i of

A

with

T h e s id e s Y

and

Z

AB in

and

AC

S(n, R ).

are

By

It fo llo w s at once that the s u b s e t

is isom etric to A ,B ,C ;

T o s e e th is, w e

is the identity m atrix, for w e can

into the identity by an isometry.

Lem m a 3.3,

is 180°.

be a non-degenerate g e o d e s ic trian gle w h o se a n g le s have

lo s e no gen erality in assu m in g that move

F

R2.

H en ce it is a fla t s u b s p a c e

w e c a ll such a s u b s p a c e a flat 2-plane.

be a q u a d rila teral the sum o f w h o se a n g le s is 360°.

Upon draw ing the d ia g o n a l

AC,

w e get two g e o d e s ic tria n g le s e a c h of

w h o se an g le sums is not le s s than 180°.

It fo llo w s that eac h an g le sum

is 180° by Lem m a 3.4 (i ), and that moreover £ B AC T h erefo re

ABC

and

ACD

+ £ C AD

= £B AD

lie in fla t two d im en sion al g e o d e s ic s u b s p a c e s .

Furtherm ore upon d ra w in g the d ia g o n a l in a fla t tw o -d im en sio n al plan e.

From

BD,

w e can in fer that

BAD

^CBAC + £ C A D = £ B A D ,

co n clud e that the three 2 -p la n e s co in cid e. 2-plane.

.

Thus

ABCD

lie s

we

lie s in a fla t

§4.

Let

G

A M etric D e fin itio n o f the Maximal Boundary

be a sem i-sim p le an aly tic group having a faith fu l matrix repre­

sentation , let

K

b e a m axim al com pact subgroup, and s e t

Xj^ denote the point of

X

fix e d by

find a faith fu l represen tatio n and

p (K ) = p (G ) fl 0 (n , R ). G

metric on

is G -in varian t.

w ith

of

G

T h e map

im bedding of X

p

K.

X = G /K .

Let

A s pointed out in 2.11, w e can into

G L (n , R )

s o that

f i : xK -» p (x ) *p (x )

onto a g e o d e s ic s u b s p a c e of

P (n , R )

p (G ) = *p (G )

then y ie ld s an and the induced

It is conven ien t som etim es to id en tify

X

/x(X).

A g e o d e s ic s u b s p a c e

F

of

X

is c a lle d flat if and only if the sum of

the three a n g le s o f every g e o d e s ic trian gle in REMARK.

A lthough

X

F

is

180°.

admits more than one G -in varian t metric (c f.

Rem ark 1 o f Section 3), the notion of flat is the sam e for a ll G -in varian t m etrics. T h e fla t s u b s p a c e s through

x^

are p re c is e ly the o rbits of

ab e lia n a n a ly tic su bgro u p s contained in p _ 1 ( P ( n , R ) fl p (G )); s u b s p a c e s through

gx^

subgro up s

w here

gA g-1

p - 1 (P (n , R ) fl p (G )).

are therefore the orbits of A

gx^

Inasm uch a s the maxim al p o lar su bgro u p s of

X.

G

p (G )

permutes

From 2.6 ( v ) w e co n ­

perm utes tra n s itiv e ly a ll the m axim al fla t s u b s p a c e s o f

p a s s in g through the point D EFIN ITIO N.

the flat

under the p olar

are co n ju ga te under an inner autom orphism , w e co n clud e that

K

under

is a p olar subgroup contained in

tran sitiv ely a ll the m axim al fla t s u b s p a c e s of c lu d e that

x^

T h e rank of

x^. X

is the dimension of a maximal fl at s u b ­

s pace.

31

X

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

32

Since

A x ^ = AK /K = A /A D K = A

for any p olar subgroup o f p ( G ),

w e s e e that rank X = T R -ran k G . M oreover any p olar subgroup operates fre e ly in with

p (p p € P (n , R ) fl p (G )

If rank

X = r,

im p lies

X.

p p 2p = P P 2P

Indeed

p xpK = p2pK

and hen ce

p x = P2 -

w e c a ll a maxim al fla t g e o d e s ic s u b s p a c e an r-flat. A

g e o d e s ic lin e is c a lle d regular if it lie s in only one r-flat; it is c a lle d singular if it is not regu lar.

Since the orbit of

subgroup o f p (G ) fl P ( n , R )

is an r-fla t of

lin e through

1

is s in g u la r in

/z(X)

1 under a maxim al polar

^t(X),

w e s e e that a g e o d e s ic

if and only if it is the orbit of an

T R -s in g u la r p olar subgro up o f p (G ) H P ( n , R ) . A ssu m e now that p = identity; identify Let

F

group o f

b e an r-fla t through

G fl P (n , R )

g e o d e s ic s in

F

such that

through

R -s in g u la r elem ents in ponent of

F — Sx^.

DEFINITION.

Let

Xj^

A.

T h en F

x^.

Let

F — XS

A

A x^ = F.

is the s et

Let

^F

with be the m axim al polar s u b ­

Then the union o f s in g u la r

Sx^

w here

S

is the set of

be a to p o lo g ic a lly connected com ­

^F =

be an r-fla t in

w here

^A

Let

x f X.

X.

union o f s in g u la r g e o d e s ic ra y s through component o f

X

x.

is a cham ber in Let

XS

A.

denote the

A to p o lo g ic a lly connected

is c a lle d a chamber o f origin

x.

T h e to p o lo g ic a l

c lo su re o f a cham ber is c a lle d a c lo s e d cham ber. From the fo re go in g, it is c le a r that every cham ber o f origin the form

g^Ax^

w here

is a cham ber in

G H P (n , R ).

and the la s t sen ten ce of (2 .2 ), a ll the cham bers in ju gate under

K.

cham bers in

X.

DEFINITION. w here NOTATIO N.

T h erefo re

G

is a w a ll o f the cham ber

the Hausdorff d istance betw een

B y (2 .6 )

has (v )

G U P (n , R ) are co n ­

o perates tran sitiv ely on the s e t o f a ll

A chamber w a ll of a cham ber

F o r any s u b s e t s

gx^

A

^A

in

G.

B

in

X

and A

h d (A , B ) = inf iv
d ( q ^ F )

qt

^F

q.

Sim ilarly

d (p , ^ F

q)

> d (q , " * F 0)

for a ll

qf

"*F. H en ce

h d ( * F 0 , * F ) = sup !d (p 0, ^ F ) , d ( ^ F 0 ,p)S < d (p 0 , p )

by ap ply in g the ab o v e co n vexity argument to the function stricted to

^F.

q -> d (p Qq )

re ­

§5.

We continue Section 4.

P o la r P a rts

G, K,

the notation

Let

and

X

F be a flat s u b s p a c e o f

cham ber w a ll in

F,

le t

E

X.

and the assu m ptions made Let

^F

b e a g e o d e s ic s u b s p a c e .

Ge

G -« p

L E M M A 5.1. Gp

gE = E

is a sem i-group and

for a ll

Gp

r = rank X.

Let

g e Gp

F

Let

and

map

F -> po l F

.

G *«p C G p .

be a flat su b sp a c e of

Gp

tains the polar part of e v e ry elem en t o f F.

Set

Thus

is a subgroup.

has a unique maximal polar subgroup, d enoted

tra n sitively on

be a cham ber or

= igeG; gECES

G < «p = I g e G ; g ^ F C ^ F i

One s e e s e a s ily that

in

M o re o ve r, if F

is an

pol F ,

X.

Then

which c o n -

and which operates sim ply

r-flat, then

G p = N (po/ F ). The

of the s e t of all r-fla ts to the s e t of a ll maximal polar

subgroups is b ije c tiv e .

Pro o f. G

Let

Xj^ b e the point of

is tran sitiv e on

F = Bxj£

w here

X,

B

infer

w e have

k B k ""1 = B

B

is tra n sitiv e on B

Gf .

Gp,

C le a rly

is

kBK C BK .

by (2 .6 ) (i).

it fo llo w s that N ( B ) fl K C G p .

Gp B

F,

xj^ e F .

B

w e get G p

Then

Gp.

by

= (G p fl K )B

kBk-1 C BK .

is normal in

Since

G fl P (n , R )

sim p ly tran sitiv e on

T h e re fo re ,

Thus

the m axim al polar su bgro u p s o f phism of

K = G fl 0 (n , R ).

no gen erality is lost in assu m in g

B fl K = (1), w e s e e that

k € G p fl K,

s ta b iliz e d by

is a p o lar subgroup con tain in g in

Lem m a 3.4 (i). Since s in c e

X

F.

For

any

From this w e Inasm uch a s

are co n ju ga te via an inner automor­

is the unique m axim al p o lar subgroup of

Thus

G p = ( N ( B ) fl K )B .

35

STRONG RIGIDITY OF LO C ALLY SYMMETRIC SPACES

36

T h e fla t s u b s p a c e subgroup of

G.

N (p o l F )

F

if

F

In that c a s e ,

N ( B ) = ( N ( B ) fl K )B

F

and

F^

H ence

Gp

elem ents in

is a m axim al polar

by (2 .6 ).

are r-fla ts s t a b iliz e d by

operates tra n sitiv e ly on the r -fla ts , w e have

G.

B

Thus

Gp =

p ol F .

Since

is an r-flat.

Su ppose now that G

is an r-flat if and only if

= G^p = g G p g - 1 .

Gp

pol F C Grp .

,

w e s e e that

Since

pol F

1 pol F 1 and therefore G p = N (p o l F ) ,

Since

F^ = g F

pol F x

for som e

B y h y p o th esis

is a m axim al p olar su bgro u p , w e get

pol F = g pol F g

w e get

g e Gp

p

q

and

;

in

is the s e t of a l l p olar

pol F 1 = g pol F g - 1 .

-l

g

that is ,

F x = g F = F.

pol F =

g f N (p o l F ) . Thus

Since

F -> pol F

is

a b ije c t iv e map. REMARK.

Let

and

be d istin ct points of

be the co rrespo n d in g sym m etries o f through

p

and

q.

w hich tra n s la te s N am ely, if

X.

Let

L

It is e a s y to v erify that

p

to a point

q = gp,

then

q

and

and let

is the midpoint o f

d (K , p K ) . x €X z eZ It now fo llo w s that

d (K , p K ) = in f„ d (x , g x ), i xe XS

with the infimum

2 attained if and only if

g

is s e m i-s im p le .

C le a rly

d (K , p K )2 = T r ( l o g p2) .

Lemm a 5.3 is now proved. Let

E

metry of

be a g e o d e s ic s u b s p a c e of

X

w ith re sp e c t to

G ' = Op G U G

(c f. 2.10).

p

X.

s t a b iliz e s

Inasm uch a s

pol E

lin e s

L

Gp

denote the subgroup gen erated by

in

E.

It is not hard to s e e that

gen erated by a ll its p o lar su bgro u p s, In p articu lar, F in a lly if

E,

G^

(c f. Rem ark 2 o f Section 3 ), w e s e e that Let

T h en for any

G

and

pol E C G p . Gp

s e lf-a d jo in t group and

p 6 E,

that is

the sym ­

e Gp,

is gen erated by

w here

ic ^ ; p 6 E|

o perates tra n s itiv e ly on pol L

pol E

E.

for a l l g e o d e s ic

is the subgroup of

Gp

(c f. Rem ark fo llo w in g Lem m a 5 .1 .)

A ls o , if

E C F,

then

pol E C p ol F .

are s e lf-a d jo in t groups, then the group

pol E

is a

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

42

pol E = ( pol E :fl P (n , R ) ) • (p o l E fl 0 (n , R ) ) (5 .4 .1 ) Ge

DEFINITION.

Let

n P ( n, R ))

= (p o l E

E

and F

a parallel translate of

F

• (G e

n 0 (n , R ) )

.

be g e o d e s ic s u b s p a c e s in

if and only if

E = gF

with

X.

We c a ll

E

g e Z (p o l F ).

T h is d efin itio n is an exten sio n of the defin ition for flat s u b s p a c e s giv en above.

L E M M A 5.4. S

Then

F

Let

S

and

is a parallel translate of a su b sp a c e of

S are

r-flats, then

P ro o f.

Let

tt :

X -> F

and

S.

q

g e o d e s ic s u b s p a c e tt( p )

in

into

S,

p.

T h en

w e find that

g e Z (p o l 77(S)).

F

If, m oreover,

denote the orthogonal p ro jection onto T h e function

lin e by Lem m a 3.6 and is bounded on p

F.

S C T V(F ).

with

and

F = S.

be a g e o d e s ic line in

any points

X

be g e o d e s ic su b sp a c es of

on

L,

E.

Let

g

sends

tt(S )

p -» d ( p , F ) L.

into

q.

p7r(p) 7r(q)q

Since

is a g e o d e s ic s u b s p a c e ,

L

is co n vex on every

denote the elem ent in n (q )

Let

T h erefo re it is constant.

the q u ad rila teral g

F.

L

For

lie s in a flat

pol E and

q

g77(S) = S,

se n d in g are arbitrary and

T h is p ro ves the first a s s e rtio n s .

T h e seco n d a ssertio n fo llo w s at once from the o bservatio n that Z (p o l F )

k e ep s

F

in variant if

F

is an r-flat sin c e

N (p o l F ) = G p

by

Lem m a 5.1.

L E M M A 5.5. pol F

Let

E

and

F

be g e o d e s ic s u b sp a c es with

commuting ele m e n t-w is e .

union of a ll g e o d e s ic lin es in

E

Let

x e E fl F ,

and let

perpendicular to

F

at

pol E

and

D

denote the

x.

Then

D

is

a g e o d e s ic su b sp a ce.

Proof.

Let

X

and

the L i e a lg e b ra o f

Y

be in the L i e a lg e b ra o f

p ol F .

T h en

pol E

and let

Z

be in

0 = T r [X Y ] Z + T r X [Z Y ] = T r [X Y ] Z ;

§5. POLAR PARTS

that is ,

Z

is orthogonal to the commutator s u b a lg e b ra of

lo s s o f gen erality w e may assu m e that group fix ed by

43

G fl 0 (n , R ).

T h is im p lies that

adjoint groups and that the s u b s e t of is a s u b a lg e b r a

H.

Let

It is e a s y to s e e that fo llo w s by (3 .4 ) that

H D

H

G

pol E

pol E

pol E.

Without

is s e lf-a d jo in t and and

pol F

x

are s e l f -

which is orthogonal to

pol F

denote the a n a ly tic group with L i e a lg e b ra

is a s e lf-a d jo in t an a ly tic group and is a g e o d e s ic s u b s p a c e .

is

H x Q = D.

K. It

§6.

L E M M A 6.1. a (t)

A

Let

A B a s ic In eq u ality

be a differentiable path in

en tia ble path in

P (n ,R )

be an abelian analytic subgroup in

P (n ,R ).

A

with

a (0 ) = 1.

Let

y (t )

and let

be a d iffer­

Set 2 Y (t ) = lo g y (t) H = 2 a (0 ) p (t ) = a (t ) y (t) a (t ) .

t = 0,

Then at

T r (p - 1 p )2 = T r ((c o s h ad Y ) ( H ) + (s in h ad Y )/ a d Y ) ( 2 Y ) ) 2 .

P ro o f.

D iffe re n tia tin g

p (t),

w e get

p = ay a + ay a + ay a . At

t= 0,

w e have

P

_ i_ 2

p (0 ) = y (0 )

- 2 PP

and thus at

t = 0,

_ i_ L _L _L 2 -2 2 - 2 = y ay + y yy

L 2 - 2 + y a y

= ( e ad Y + e “ ad Y ) ( a ) + r2 y ( w Y ) w here

r y ( Y ) = (e x p — Y / 2 ) (e x p Y ) ( e x p — Y / 2 ).

((s in h ad Y )/ a d Y ) ( 2 Y ) . Since

B y Lem m a 3.1,

( - -1. - l \2 1 p )2= T r \jp 2 p p 2 ) ,

T r (p

the lemma is now evident.

44

r2y ( 2 Y ) =

§6. A BASIC INEQUALITY

L E M M A 6.2.

G

Let

n : X -» F

and let

p e X,

K

let

F

K.

Let

X

denote the orthogonal projection of G

denote the s ta b iliz e r in

Y

denote

b e a flat s u b sp a c e of

of the point

77(p),

denote the orthogonal com plem ent in the L i e algebra of algebra of

X

be a s e m i-s im p le analytic linear group, let

the a s s o c ia te d sym m etric Riem annian sp a ce, let X,

45

denote the unique elem en t in

?

G

onto

F.

Let ?

and let

to the L i e s u b ­

such that

exp Y ( tt( p ) ) = p . j : ? -> X / \ denote the ca n on ica l map of

Let

fP onto the tangent s p a c e

L to

X

at 7r(p).

S et

f(t ) = (t co sh t/sinh t )2 .

IC 1 >

P roo f.

We can assu m e that

(c f. 2.6).

S in ce

G

|jf(a d Y ) j

G

Then for any tangent v e c to r

7rp (C)|

.

is a s e lf-a d jo in t su bgro u p o f

o perates tra n sitiv e ly on

X

G L (n , R )

and both s id e s of the in­

eq u a lity a b o v e are invariant under iso m e trie s, no g en erality is lo s t in assu m in g

K = G fl 0 (n , R ).

a su bset of

P (n , R )

T h en

v ia the map

fied w ith the identity matrix, subgroup

A = pol F ,

w ith the map through

1

a y a -> a

in

with A^

/x : gK -* g *g.

2

w here

orthogonal to C

A. to

p (0 ) = p, p (0 ) = C .

a e A

T h ereb y

n : X -> F

and

y e A

B y d efin itio n o f X

We id entify 7r(p)

X

with

is id en ti­

is id en tified w ith the a b e lia n a n a ly tic

and the p ro jection

G iv e n a tangent v ecto r p (t)

F

9 = S (n , R ) H G.

at

p,

X

b eco m es id en tified ,

Y,

the union of g e o d e s ic s w e h ave e x p Y * l * e x p Y =

w e s e le c t a d iffe re n t ia b le path

T h e re is a unique d iffe re n tia b le path

y (t)

such that p (t ) = a (t ) y (t) a (t ) .

D e fin e set

Y (t )

by the re latio n

Y = Y (0 ), H = 7T (C ) .

exp 2 Y (t ) = y (t )

T hen

w ith

H = a 2(0 ) = 2 a (0 ),

Y ( t ) e S (n ,R ), and

and

|C|2 = T r (p _ 1 p )2 .

A p p ly in g Lem m a 6.1, w e get

C|2 = T r ((c o s h ad Y ) ( H ) + ((s in h ad Y )/ a d Y ) ( 2 Y ) ) 2

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

46

(a d Y ) 2

T h e lin ear map Tr XY

(c f. Section 3). (a d Y ) 2

v ectors for v a lu e s .

Set

s t a b iliz e s Let

and is s e lf-a d jo in t w ith re sp e c t to

9

r/j, ...,ry

be an orthonormal s e t of e ig e n ­

v?, ..., v 2

denote the co rrespo n d in g e ig e n -

and let

c^ = cosh v i? s- = sinh v^/v-,i = 1,

m,

and

2Y = S ™ B i V Then |C |2 = T r ( £ / Ci ,m =

2

( c i A i + Si B i}

'

R earran ge the in d ic e s s o that

Set

Ai Bi > 0

i = 1, . . . , h

Ai Bi < 0

i = h+1 , . . m .

ei = ci A i + sj

s j -1 ( e — c- A - )

ici2

w i = c^/si (i = 1

,

m) .

T h en

=

and thus

> < 1= ? Af + ^

’ ^S 1 V i *

1

2 h”+ l

m

—

s in c e

and

^

A^ B^ = 2 T r H Y = 0.

W riting

1 sri A-Cej-CiA.)

\m

- 2 T

Ai B i = - 2 V h +l

h+l

w e get >h

„

^

|c|2 > S . c f A ^ s f B ?

+

t

S

^ *

S ^ 1

V

« i

Af * X " ^ h +1

b t i ( 2 . r , < = i * ? - 2 > r , " i A 1* e ? )

+1 T 'c i A ?

4

*

1

- 2c r S

A c e r 's , . ? )

s--c,(A ? -2 c-' ^ A ^ c ^ e ? ) '

§6. A BASIC INEQUALITY

1

_

inasm uch a s

s^c^

ic i2 ^

C le a rly

£

i

1jf(a d Y ) j

_

1

_

> c^

> c-

wi Ai

2

ark

1.

1 tt (C )l =

s^ > 1 and

2

c^ > 1.

9

C on se qu en tly

”1 w i A ? • 1

T h erefo re

* 1 -..

"*?

|jf(ad Y ) j - 1 77-p (C)|

In c a s e the s t a b iliz e r of a point

w e can com pose the maps s in c e

sin c e

” w i 2 n+1

> Re m

r\

47

-i-*

9

Xq

q

in

X

is

G H 0 (n ,R ),

and w e s e e that

° j ( Y ) = 2Y

exp Y ( q ) = exp Y • /x(q) • exp Y .

1_ R E M A R K 2. t,

f(t )

Inasm uch a s

f(t ) = (t c o sh t/sinh t) 2

is a pow er s e r ie s in

jf(a d Y ) j - 1

t2 .

S ince

(ad Y ) 2

is thus a w e ll d efin ed endomorphism of

e x p r e s s ib le in terms o f the curvature tensor s in c e [ [ Y 1# Y 2L Y 3] com pare (ad Y ) 2 Let

]C|

for any elem ents and

177p (C )| ,

Y 1( Y 2 , Y 3

9.

in

s ta b le , the map

9

^ (p )*

Indeed it is

R ( Y X, Y 2 , Y ^ ) = H o w ev er, in order to

it is convenient to co n sid er

ad Y

rather than

and to re state Lem m a 6.2 in a s lig h tly different w ay. M (n ,R )

and

S(n, R )

denote re sp e c tiv e ly the set o f a ll real

m atrices and a ll sym m etric re a l the inner product ad Y

is an even function of

keeps

nxn

m atrices.

< U , V > = T r U *V.

is s e lf-a d jo in t on

M(n, R )

On

M (n ,R )

We re c a ll that for any

(c f. Section 3).

nxn

w e introduce Y 2

n wi Ai i= l

[Y , 77^] = Vj 7?j, Wj = |v^ c o s h

(s in h V j)

2 j - 1(^p(C))=

Ai ^i -

Assum e

G

r/1 , ...,77 9 b e an n c o n sis t in g of e ig e n v e c t o rs for ad Y . Then

and

orthonormal b a s e in M(n, R )

and

Let

1 1 ( i = l , . . . , n 2),

and

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

48

T h is fo llo w s im m ediately from Lemma 6.2.

Proof.

F o r any tangent vector have denoted by

|C|

by the em bedding

C

to the symmetric Riem annian s p a c e

the length of

pi o f

X

into

C

X,

we

under the invariant metric induced

P (n , R ).

F o r any re a l

w e s et

nxn

matrix

Y,

1_ IIY|| = (T r Y l Y ) 2 .

F o r any

Y e 9,

w e h ave

/ ^ (jC Y )) = 2Y

and thus

||2Y|| = jj( Y ) j =

d (e x p Y K , K ) = d (p , n ( p )) .

L E M M A 6.3.

tt, p, Y ,

Let

G

s e m i-s im p le group Set

H = 77p (C ).

C

and

is s elf-a d jo in t in G L (n , R )

T h e n , identifying

1 i£ i

> ( 2 n)

B y Lem ma 6.2 b is ,

H

r

4

h

~

|j Y II2

|C | 2 > ^

w i A ?,

w here

[Y , r/-] = v- 17^ (i = 1, — , n2).

( s i n h v ^ - 1 ! and

77^ ..., 77 2 is orthonormal, w e h ave A s is w ell-k n o w n ,

Tr Z = 0

the e ig e n v a lu e s o f the elem ent

K = G fl 0 (n , R ).

/ijCH),

with

y

and

A s s u m e that the

LllYll ' I!h I!_

IHi

Proof.

be as in Lemma 6.2.

||H||2 =

for a ll Y.

Since

A i 7^

H =

and

A?.

Z e G.

T hen ^

w i = |vi co sh v i

Let

...,A .n denote

A- = T r Y = 0

and

T r (ad Y ) 2 = X ( A j - X -)2 = 2 n V X? = 2 n T r Y 2 "i,j= l ^ i= l

n2 T h erefo re ,

||Y||2 = (2n ) _ 1 T r (ad Y ) 2 = (2n ) _ 1

1 |VjJ/((2n ) 2 |1Y||) (i = 1, ..., n2).

|C | 2 > X

s in c e

v?.

Set

Uj =

i= l

wi A i

>

u 2 < u^ < 1.

S

Inasm uch a s

w ^ > | v -| ,

w e have

|vi | A i = (2 n ) 2 | | Y | | ^ U i A? > (2 n ) 2 ||Y|| ^

H en ce

|C |2 > (2 n )

2 ||Y||_ 1

v2 A 2 .

But

u? A?

49

§6. A BASIC INEQUALITY n2

l|[Y,H]||2 = I £

v.A il?i||2 = 2 vi Ai- Thus I C |2 > (2 n )

2 H Y i r 1 ||[ Y , H ] ||2

|H| = ||^t1 (H)|| = ||H||.

B y d efin itio n ,

C on se q u en tly

1

X

i^l >_ (2n) 4

JL

IY ||2

1Y It ' IIH ||

H

It is o f c o u rse d e s ira b le to free Lem m a 6.3 o f the h y p oth esis that K = G H 0 (n , R ). F o r any

A c c o rd in g ly , w e introduce the fo llo w in g notation.

g

ancf 2

1 d (p, 77(p)) = ||Y||

2 "

W. A j

1

= (2 n )

v2 .

H'

> (2 n ) IHi

L E M M A 6.4.

Let

X

iY iig

be a s im p ly -c o n n ec te d symmetric Riemannian s p a c e

pact or v ecto r normal subgroups).

77: X -» F T

Let

t

n m ig

r and of ne gative curvature ( i . e . , its isometry group has no c o m ­

of rank

let

Moreover

F

Let

be an X

de note the orthogonal projection of

be an

i-flat in onto

F.

X

and let

Let

x-dimensional s u b s p a c e of the tangent s p a c e to

denote the restriction of 77p

T.

to

p e X X

at

ancf p.

Then

_L det

t

< c d (p , F )

where

c

Proof.

N o gen erality is lo st in assu m in g

2

is a constant depending only on the s p a c e

and that the s ta b iliz e r o f 77(p) T h e map

r

is

of the e llip s o id .

is s e lf-a d jo in t in

G L (n , R )

G H 0 (n , R ).

takes the unit b a ll in

up to a constant d epending only on

G

X.

r,

T

into an e llip s o id and

det r

is ,

the product o f the p rin cip al a x e s

Since

n

is a p rojection , the lo n g e st a x is o f the e lli p ­

so id has length at most

1.

On the other hand, the shortest p rin cip al a x is

§6. A BASIC INEQUALITY

51

must be no lon ger than the length of a radius of the e llip s o id a lo n g some regular g e o d e s ic is s u in g from the center of the e llip s o id . We may assu m e that In the c a s e

det r ^ 0,

d(p , 7T(p) ) = d (p, F ) 6.2.

det r ^ 0,

set

w here

o th e rw ise there is nothing to prove.

C = r _ 1 (H ).

We have

exp Y • 7r(p ) = p

and

Y

|C| = 1,

and

is c h o sen as in Lem m a

B y Lem m a 6.3

1

1

T h erefo re

1_ 2

-1

1

1 |H| < ( 8 n) 4 c (H ) d ( p , F )

2 , where

c (H ) - 1 = in f {||[Y/||Y||, H/||H||]||; Y f (p o l F ) 1 n f \ Inasm uch a s in

||2Y|| =

?

H

lie s a lo n g a re gu lar g e o d e s ic of

lie s in the L ie a lg e b ra of

th is, Lem m a 6.4 fo llo w s at once.

pol F .

F,

.

the c e n tra liz e r of H

C on se qu en tly ,

c (H ) < oo.

From

§7. Geometry of Neighboring Flats

In this section we shall determine the intersection of an r-flat F with a tubular neighborhood TV(FQ) of an r-flat F Q. The principal result (Theorem 7.8) states that the intersection is approximately an intersection of half-spaces with singular faces. We continue the notation and assumptions of Section 4. Let F be a flat subspace of X. In Section 5 we have defined pol F as the unique maximal polar subgroup of the stabilizer Gp. Suppose now S denotes either a geodesic ray, or a chamber wall or a chamber in X, and let F denote the unique minimal flat subspace of X containing S. Clearly the stabilizer of S lies in Gp, and we may define pol S as the stabilizer of S in pol F. Let

denote the intersection of all chambers

and chamber walls containing S. Then pol S C pol hd(pkS, pS) > hd(kS, S ) -

hd(pS, S)

h d (p S ,S )

= 00 , u n le s s

kS = S,

(c f. proof o f Lem m a 4.1).

k e G g C Z (p o / S) = Z (p o / ^ S ) C P (S ). tion ( i i ) fo llo w s from the fact that

L E M M A 7 .2 . and only if (ii)

(i)

L Q = gL

Let

lim a - 1 g a = 1 for a -> 00 be rays in

X.

Then

be chamber walls in

g e G

su ch that

g L j = L Q,

mal com pact subgroup of the origin of

^F.

Set

g e U (S ).

h d (L 0 , L ) < o o

X.

"*F *

B y (2 .7 )

L Q w h o se origin is

g " * F * = "*F q ,

with

X.

K

s in c e

h d (L Q, L ) < 00.

G

operates

T h erefo re there is a ray

G = K P (L j)

w e take

g = kp

h d ( ^ S Q, ^ S ) < 00

be a c lo s e d cham ber s im ila rly related to

such that

G;

Then

w here

K

L1

k e K

and

p e P O ^ ).

Then

h d (k L 1 #L ) < h d (k L 1 , k PL 1) + hd(kPL l f L ) < h d (L x , p L j ) + h d (L 0 , L ) < 00 .

in

is any m axi­

to be the isotropy subgro up of

h d ( L j , p L j ) < 00 by Lem m a 7.1. Su ppose now that

if

g e P (^ S ).

= g M S with

T h en there is a

A sse r­

g e P (L ).

^S

L Q and let

im p lies

Lem m a 7.1 ( i ) now fo llo w s .

tra n s itiv e ly on the set o f cham bers o f ^F

kS = S

L e t ^ F q be a c lo s e d cham ber co ntaining

the o rigin o f L.

L

and

with

^ S Q and

if and only if

Proof.

L0

Let

But

T hen

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

54

It fo llo w s at once that

kLx = L

H o w ev er, both

and

L

( v i i ) of ( 2 . 3 ) if

k a k -1 = b

of a p olar subgroup, then L1 = L

and

b e lo n g to the sam e c lo s e d cham ber and by with a = b

k e Z ( p o l L ).

k 6 Z (p ol L ) C P (L ).

a

and

and

Thus

H ence

tion ( i ) of Lem m a 7.2.

s in c e both are rays w ith the sam e origin.

b

in the sam e c lo s e d cham ber

k

lo g

n - 1 T r ( a - 1 v _ a ) * (a - 1 v _ a )

> - l o g n + lo g ^

( v j C j j A r 1^

.

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

60

Inasm uch a s

v__ ^ 1,

there is a fundam ental root a

t]ie L i e a lg e b ra of

pol

su ch that

root s p a c e o f

that is ,

a ( a ) = lo g

—a ;

v_

w hich is p o s itiv e on

h as a non-zero com ponent in the w *th c ij ^ 0.

Thus

d (a _ 1 v _ a p , p ) > ( —lo g n + lo g c 2j ) + 2 a ( a ) .

d (a _ 1 v _ a p , F ) > - 2

B y Lem m a 7.6,

2 d (a _ 1 v _ a p , p ) .

lo g n + (4 n )

P u t­

ting together the in e q u a litie s a b o v e , w e get

d (g F , a p ) = d (F , a - 1 v _ a q ) > d (F , a - 1 v _ a p ) — 2r > c d (a _ 1 v _ a p, p) — c > 2c a (a ) _

w here

c = (4 n )

ap e T y ( g F )

c"

1_ 2,

c " = c ' + c (lo g n — lo g c 2j).

c ' = 2r + 2 lo g n,

im p lies

2 x a (a ) — c " < v;

im p lies a s s e rtio n ( i ) of Lem m a 7.7. is the la rg e s t w a ll of

that is ,

Let

w hich lie s in

R =

a ( a ) < (2 c ) _ 1 (v + c " ) - T h is fl ( g F f l ^ F ).

gF fl

Thus

F.

C le a rly R

B y a p p ly in g a s s e rtio n

( i ) of Lem m a 7.7 rep eated ly , w e find a seq u en ce o f cham ber w a lls S Q, S 1 , . . . , S n w ith

SQ = ^ S , S i

H T y ( g F ) C T v (S p

v > 0,

a w a ll o f for som e

of in c lu s io n s , w e find for any p o s itiv e

S-__ 1 , S n = R ,

and for eac h

v- (i = 1 , . . . , n).

From this chain

v

n T v ( g F ) C T t(R )

for som e finite

T H E O R E M 7.8. s > d (x , F q ).

t.

T h is p ro ves a s s e rtio n (ii ).

Let

F

and

F0

be

t

X,

let

xeF

Then

F0 nx F C W where

r-flats in

is a number depending on

n F C T t( F 0 n x F > F Q, F , x

and

s.

and let

§7. GEOMETRY OF NEIGHBORING FLATS

B y ( i i ) of Lem m a 7.5,

Proof. Let

b e a cham ber or cham ber w a ll in

elem ent of

G

and only if

g f P (^ S )G p .

su ch that

for som e fin ite the in

F Q H x F C T g ( F 0) fl F

^S, F

t

if

g F = F Q.

F

for

o f o rigin

s > d (x , F

x.

B y Lem m a 7.5 (i ) ,

B y Lem m a 7.7, / F Q fl

61

F.

Let

g

q ).

be an

C F Q flx F

if

^ S H T S( F Q) C T t( F Q H x F )

Inasm uch a s

F

is the union o f a ll

and there are only a finite number o f cham bers and cham ber w a lls

of o rigin

x,

w e s e e that there is a fin ite

t su ch that

F fl T g ( F 0)

c Tt(F0nx F). L E M M A 7.9. and let

F

Let

and

T y(F ) D F q

X

be a symmetric s p a c e of constant n e g a tiv e curvature,

F Q be g e o d e s i c lin es in

has length at most

X

2 v / (l — (c o s h d0) - 1 ),

and d

is a su it ably normalized invariant metric.

Proof.

We adopt the notation o f Lem m a 6.2.

pro jection of

X

on

F.

d (F , F Q) > 0.

with

Let

C

onto

T h en for any tangent vector

exp Y ( 77(p ) ) =

F^^y

and

Y

C

to

w here

is orthogonal to

Fq H

F n^

for any

Y e F j (p ).

at

p,

is the pro-

with

p. In a s p a c e o f constant curvature, w e have

and T r (c o s h ad Y ( H ) ) 2 = T y (c o s h 2 a ( Y ) ) ( H ) 2

w here

(ad Y ) 2 (H ) = a ( Y ) 2 H.

N o w in the metric o f

ad joint represen tatio n , o f the group T r (ad Y ) 2 = (dim G / G p ) a ( Y ) 2 .

G

X

in duced from the

o f iso m etries o f

X

d ( p , 77(p ) ) 2 =

T h u s m ultiplyin g the d is ta n c e by

(dim G / G p ) - 1 , w e get an in variant m etric with

|a(Y)| = d (p • ^ (p )) = d (p , F )

and th erefore

Let

of

T V( F )

|C|/|H| > co sh d (p , F ) > co sh d Q.

n F0

and let

t d enote the length o f

s < 2v + t < 2v + s / c o s h d Q.

s

denote the length

n ( T y ( F ) n F 0).

H en ce

s < 2 v / ( l - (c o s h d 0) _ 1 ) .

Then

§8.

D en sity P ro p e rtie s o f D is c re te Subgroups

L E M M A 8.1 (S e lb e rg ).

G

Let

be a locally compact group and let

V

be

a dis crete subgroup. (i)

If

G /r

where ( i i ) If

is compact, then for all Z (y )

G/r

y eT ,

Z (y ) / Z ( y ) H T

is compact,

de note s the centralizer of y . g e G

has finite measure, then for any

hood

U

1

u gnu

n r ^ 0.

of

G,

in

and for any neighb or-

there is a p o s it iv e integer

n

such that

Since the proof is very short, w e present it. (i)

Let

T h en

y £r.

k

C o n sid e r the map

is continuous.

k_ 1 ( k ( 0 )

k (r ) C T,

S ince

is c lo s e d ; that is ,

k : G -» G

T Z (y )

(ii)

k (T )

T\G

k (g ) = g y g - 1 .

is c lo s e d and acc ord in gly ,

is c lo se d in

is a c lo s e d su b se t of the com pact s p a c e Z ( y ) / Z ( y ) PI r

given by

G.

T h erefo re

and is com pact.

find two d istin ct p o s itiv e in tege rs u- 1

H en ce

is com pact.

Since left tran slation is a m easu re-p reservin g map of

H en ce

T \ r Z (y )

g^—

H F / 0.

k and

I

such that

A p p ly in g this remark to

G/F,

w e can

g ^ U T fl g ^ U T ^ 0 . U H U-1 ,

asse rtio n

( i i ) fo llo w s .

L E M M A 8.2. group G

G.

Let

Let

re s p e c t iv e ly .

V

A

be a maximal polar subgroup of the s e m i-s im p le

and Let

W

denote neighborhoods of the identity in A

c > 1 and s et

A q = \a e A ; a ( a ) > c Then there is a neighborhood ae

and

U

for all p o s it iv e of the identity in

A c

62

R -roots G

a\ .

such that for any

63

§8. DENSITY PROPERTIES OF DISCRETE SUBGROUPS

U a U C W[M a V ] g [x ]

where

gxg- 1

denotes

and M

is the maximum com pact subgroup of

Z ( A ).

T h is re su lt is proved in my paper “ On in tersection s of C artan s u b ­ groups with d is c re te s u b g ro u p s / 7 Indian Journal of M athem atics, V o l. 34 (1970), 203-214. Let

r

b e a d is c re te subgroup of the sem i-sim p le group

denote the sym m etric s p a c e a s s o c ia t e d to be an r-fla t in

X.

D E F IN IT IO N.

F

L E M M A 8.3.

Let

is r-c o m p a c t if and only if

V

X

and

Th en the s e t of V - c o m p a c t

Let

Proof.

group of Then

of

1

Gp,

g e Ac

Let d > 1.

F

V

in

A = pol F .

for som e

G

such that

and

d > 1.

Lem m a 5.2 (i i i ) .

We h ave proved: in

y

X

wF

A

1 in

is com pact.

G /T

is compact. r-fla ts of

be the maximum

1

such that

in

G.

A QV C A ^

A.

A.

w ith

S e lec t a neighborhood

U

F in a lly , by S e lb e r g ’s Lem m a,

B y Lem m a 8.2,

is p olar regular. Z (y )

X.

polar s u b ­

be a p olar re gu lar elem ent in

A

F

y e w[M A ^ ]

y e w X (A )w “ 1 e

C le a rly ,

operates tra n s itiv e ly on

as Z (y )/ Z (y ) n F

with

is com pact and

wF

by

Z (y ) = G w p ,

is T -com p act.

G iv en any neighborhood

W o f the identity in

there is a T -com pact r-fla t o f the form

wF

G

and

with

T h u s, Lem m a 8.3 is proved.

REMARK. group

F

g

y e V H U A CU .

Inasm uch

r\TF

X

Let

and som e ordering of the R -ro o ts on

Furtherm ore,

it fo llo w s at once that

any r-flat

Let

U A CU C W [M A CV ].

H en ce

w G p w ""1 = G w p .

w e W.

c > 1

and let

W be a neighborhood of

w e can find an elem ent w €W

in X,

Let

r = rank X.

r-flats is d e n se in the s e t of all

b e an r-fla t i.e .,

and let

be as a b o v e . A s s u m e that

b e a neighborhood of

Let

G

G.

G

Lem m a 8.3 is v a lid for d is c re te su bgro u p s such that

G/T

h as fin ite m easure.

V

o f a sem i-sim p le

T o s e e th is, one needs an

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

64

Z ( y ) / Z ( y ) fl F

adequate condition for

to b e com pact.

Such a condition

h as b e e n given by R aghunathan. R aghunathan d e fin e s an elem ent

g 1and

with

s is t s en tirely of R -h y p er-regu lar elem ents.

Ub^U

y e w [M b ^ V ]

w ith

U

b e a n eigh bo r­ ; t

bJv

S e lec t a neighborhood

a e A Q, U a U C W [M a V ].

y eV

V

for some w e W.

j > 0. H en ce

co n ­ U

of

B y S e lb e r g ’s Lem m a, Since

y

U b^U C

is R -h y p er-regu lar.

Since an R -h y p er-regu lar elem ent is p olar regu lar (c f. [1 3 ], Rem ark 1.2; note that their “ R -r e g u la r ” is our “ p olar re g u la r” ) and s in c e

Z ( y ) / Z ( y ) fl r

is com pact, our proof a p p lie s to the c a s e that

fin ite m easure.

G /r

has

T h u s, modulo the proof of R agh un ath an ’s criterion (w h ich

is proved in [1 3 ]), w e h ave proved

LEMMA 8.3'.

Let

V

X

and

be as a b o v e .

A s s u m e that

G / T has

finite

measure.

Then the s e t of T -c o m p a c t x-flats is de nse in the s e t of all

r-flats of

X.

LEMMA 8.4 (Mautner).

Let

G

be a s em i -s i m p l e analytic group having no

compact normal subgroup of p o s it iv e dim en sio n, and let group of polar regular elem ent s in G normal subgroup of

G/r

has finite

(i)

A

(ii)

TxA

G ).

Let

G -invariant

V

operates ergod ica ll y on is d e n se in

G

be a s e m i­

(w h ic h is contained in no proper

be a c l o s e d subgroup of

measure.

A

Then

G/r.

for almost all

x e G.

G

such that

§8. DENSITY PROPERTIES OF DISCRETE SUBGROUPS

A s s e rtio n ( i ) w a s proved by Mautner for p olar su bgro u p s A s s e rtio n ( i i ) fo llo w s im m ediately from (i ) .

L E M M A 8.5.

G

Let

A

in [1 1 ].

D e t a ils may b e found in [1 2 g ].

be a s e m i -s i m p l e analytic group having no compact P

normal subgroup of p o s it iv e dimension, and let

be a parabolic subgroup.

F be a c l o s e d subgroup such that G/r has finite G-invariant mea­

Let sure.

Then

Proof.

T P = G.

N o g en erality is lo st in assu m in g that

subgroup.

Let

r u ^A -1

is d en se in

^A

b e a cham ber such that G

for alm ost a ll

find s e q u e n c e s o f elem ents

l y n S in

y n u - 1 a ” 1 -> g

1 as

wn

65

1

Set

as

n

and

and

n -» °c.

is a minimal p a ra b o lic

P = P ( ^A) in

G.

(c f. (2 .4 )).

G iv en

l a n S in

g e G,

^A

w e can

such that

T h u s y n = w n g a fl ufl

N = U ( "*A _ 1 ),

with

the unipotent subgro up gen erated by the root ^A.

T h en

u n = v n pn

with

vn e N

and

Thus y P = w g a v P = w g (a „ v a ~ 1 ) P /n n to n n n bVn n n /

We h av e

H ence

oo.

s p a c e s of the n eg ativ e roots on pn e P .

T

u

P

an vn a” 1

1 as

n -> °o.

H en ce

yn P

gP

as

.

n -> .

T h e re ­

fore ~ T P = G.

L E M M A 8.6 (B o r e l).

Let

G b e a s em i -s i m p l e linear analytic group having

no compact normal subgroup of p o s it iv e dimension, and let subgroup such that Z a r i s k i-d e n s e in

G/r

has finite

G -invariant

measure.

V Then

be a c l o s e d V

is

G.

T h is lemma is proved by B o re l in [2 a ] and, g e n e ra liz e s a re su lt first proved by S e lb e r g (c f. [1 6 ]) in the c a s e given in [ 12 f ] .

G = S L (n , R ).

A nother proof is

§9.

Let map.

X

Let

and k

P seu d o-isom etries

X ' be metric s p a c e s and let 0 : X -> X ' be a continuous

and

b

be p o s itiv e numbers.

DEFIN ITIO N.

0

is a

(k ,b )

pseu do-isom etry if and only if

(9 .1 .1 )

d (0 (x ), 0 ( y ) ) < k d ( x , y ) ,

for a ll

x ,y

in

X

and ( 9 . 1. 2 )

d ( 0 ( x ) , 0 ( y ) ) > k_ 1 d (x , y ) ,

T h e map 0

if

d (x , y ) > b .

is c a lle d a pseu do-isom etry if it is a

isom etry for som e

(k, b ).

If, for exam ple,

X

(k, b ) p se u d o -

is com pact and

fie s a L ip s c h it z condition with constant

k,

isom etry w here

T hu s the condition that

b

is the diam eter of

X.

then 0

a pseud o-isom etry is not much of a restriction u n le s s Let

0 : X -> X / be a

(k, b )

tubular neighborhood o f radius (9 .1 .1 )'

is a

s a t i s ­

X

p seud o-iso m etry , and let r,

(k, b )

p se u d o be

is non-com pact. S C X.

F o r any

w e get from (9 .1 .1 )

0 ( T r(S )) C T k f 0 ( S ) .

From (9 .1 .2 ) w e get (9 .1 .2 )'

0 - 1 ( T r( 0 ( S ) ) C T r,(S ), r ' =

A continuous map

sup (kr, b ) .

cf> : X -» X ' s a tis fy in g only (9 .1 .2 ) is c a lle d

(k, b )-

inc ompres s ible. We s h a ll require the fo llo w in g w ell-k n o w n fact from the theory of fib e r bu n dles.

66

§9.

PSEUD O-ISOME TRIES

LEMMA 9.1.

Let

the group

operates f r e e l y ; that is

V

X

X ' be contrac tible topolo gic al s p a c e s on which

and

g = 1 . A s s u m e that

u nles s p lex es.

67

V \X

gx / x

x e X

for any

g eT

and

and F \ X ' are finite s im p licial c o m ­

T -s p a c e morphism 0 : X -> X ' suc/i

Then there is a

d uced map 0 : T \ X -> F \ X '

is simplicial.

M oreo ver

£/?e in-

cf> is a homotopy

eq uivalenc e.

P r o o f may be found in [1 7 ]. T h e next lemma is cen tral for our method.

LEMMA 9.2.

Let

G

G ' be s e m i -s i m p l e analytic groups, let

and

K ' be maximal compact subg roups in X = G/K,

X '= G '/K '.

such that

G /r

and

Let

V

G

K

and

and G ' r e s p e c t iv e ly , and s e t

and T ' be torsion-free dis cre te subgroups

G ' / T ' are compact.

Then there is a p s e u d o -i some try

0 : T -» T ' be an isomorphism.

Let

: X -> X ' such that

0 ( y x ) = d(y)(x) for all

y eV

and

x e X;

that is,

0

is a V - s p a c e morphism and a p s e u d o -

isometry.

O b s e rv e first that

Proof.

the s t a b iliz e r o f a point fore

T H Gx

have

x

o p e rates freely on in

X,

then

is d is c re te and com pact.

T n G x = (1 )

fre e ly on

T

s in c e

T

Gx

F o r if

Gx

is co n ju gate to

Since

is to rsio n -free.

X.

T fl G x

K

F \X

is homeomorphic to

is thus fin ite, w e

Sim ilarly,

T ' o perates

is a d iffe re n tia b le m anifold and is com pact s in c e it T \G/K .

fin ite s im p lic ia l com plex.

T h e re fo re , it can be trian gulated and is

S im ilarly

T 'X X ' is a fin ite s im p lic ia l com plex.

O ne can now ap ply Lem m a 9.1 and ch o o se a F - s p a c e morphism It rem ains to sh o w that 0

T h en

0

and th ere­

X'.

The space

Let

d enotes

cf> : X -> X'.

is a p seud o-isom etry.

0 : T \ X -» T 'X X ' denote the induced map o f is a s im p lic ia l map by our ch o ice.

T\X

We regard

F \X

to

T 'X X '. a s a metric

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

68

s p a c e with metric induced from

X.

Sim ilarly for

c i a l map of a fin ite s im p lic ia l com plex,

Inasm uch a s

X

cf> s a t is fie s a L ip s c h it z condition

k1 : d (0 (x ),0 (y ))
s Q.

inf id (x ', y x ') ; x ' e X'| =

is a g e o d e s ic line s t a b le under

w e are u sin g that every elem ent of

V '.

T '\ X '

is

C o n se q u en tly ,

y ' when w e identify the inf jd (x ', y V ) ; y ' e T ', y V

1,

From this (9 .2 .1 ) fo llo w s .

A s a direct c o n seq u en ce of (9 .2 .1 ), of radius le s s than N e x t w e a sse rt: ( 9 .2 .2 )

s Q.

that is ,

tt'

is in je c tiv e

on

any b a ll in

X'

r'. Let

B ' be a c l o s e d ball in diam 0 _ 1 ( B y)
b,

w e have for a l l

S e Cg,

H n+ 1 ( R n , 2b,

A s s u m e that

and 0 ( B 2^ ) C B 'r^ .

Then

is n > m.

(k, b )

dim X = m Let

and

cl e X,

s e t B f=

in co m p ressib le on

If moreover

n < m,

then

n = m and o . From this it fo llo w s at on ce that

d0*

ddry

n i + dT^°We h av e

f ( s 0) = c t — d (x ( s ) , g x ( s 0) ) + -^ 4 — d ( x ( s Q), g x ( s » ,

s0

s0

f ' ( s ) = s in ( 0 ^ — 90) + sin ( $ 2 — 90) = —c o s

— co s 02

H en ce d 6>

ddn

and thus

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

82

f ' ( s 0) = 0 , w e have

A t a point w here

thus

( d0l

^ ( s 0) = s ^n

d0 2 \ + “3 ^ 7

maximum v a lu e.o n the segm ent f(s ) < a

for a ll

x (s )

betw een

c o s 6^ = —co s

-

T h erefo re

[x y ] x

that is

f(s )

must a c h ie v e its

at one o f the endpoints.

and

y

0 ^ + 6 2 =^

T h a t is ,

[x , y ] C I(g , a).

and thus

T h is

p ro v es (i). (ii)

B y d efin itio n

x (s )

a s ab o v e.

1(g) = I(g ; |g[).

x, gx, gy, y

is

y b e in

f '( s ) = 0

1(g).

for a ll

s

D e fin e and thus

360°.

H en ce the four points lie in a fla t

by (3 .8 ), and a s a matter o f fa c t form a p arallelo gram .

g e o d e s ic lin e p a s s in g through fore lie s in in

and

P ° i n ts- In particu lar, the sum o f the four a n g le s o f the

q u a d rila teral at F

x

From ( i ) it fo llo w s that

^1 + ^2 = 77

space

Let

1(g).

1(g)

by (5 .3 .3 ).

x

and

gx

is s ta b le under

S im ilarly, the line through

H en ce the join of th ese two lin e s lie s in

g e o d e s ic lin e p a s s in g through

x

and

y

lie s in

y

1(g). 1(g).

g

The

and th ere­

and

gy

lie s

In p articu lar, the Thus

1(g)

is a

g e o d e s ic su b s p a c e . (iii)

If

g

is sem i-sim p le and

gate to an elem ent o f assu m e that

c o in c id e s w ith If

g

G H 0 (n , R ).

g e G fl 0 (n, R ).

Z (g ) H G H P (n ,R )

and

fore fo r any

and

g

is c o n ju ­

Without lo s s of gen erality w e can

T h en the orbit of the identity under g

and th is in turn

gy

x

|g| / 0,

then for any

and

is a p a ra lle l tran slate o f the lin e

and by (5 .3 .3 )

gx

x ,y

in

both are s ta b le under

1(g),

pol g.

w e have

T h ere­

x e 1(g ), K g) C Z (p o l g )x

M oreover

pol g = 0

1(g ).

is s e m i-sim p le and

y

then

is c le a rly the fix e d point s e t of

s een that the line through through

|g| =-0 ,

Z (p o l g )x

.

is a g e o d e s ic s u b s p a c e of

X

for

x e 1(g).

We can ap ply the sam e argument o f the fo re g o in g p aragraph to the operation o f

g(p o l g ) ~ 1

on the g e o d e s ic s u b s p a c e

Z (p ol g )x

clu d e that

Kg) = z (pol g)x n z (g (pol g)- 1 )x = Z(g)x .

to con­

§12.

PSEUDO-ISOMETRIC INVARIANCE

T h e “ only i f ” part of ( i v ) fo llo w s from (ii i).

83

T h e “ i f ” part fo llo w s

from (5 .3 .2 ).

LEMMA 12.2. k = gp - 1 .

Let

g

be a s e m i -s i m p l e elem ent in

G.

Set

p = pol g

and

Th en

n i(k ).

(i)

i( g ) = i(p )

(ii)

There is a p o s it iv e constant

e

such that for all

x 6 X,

d (x , 1( g ) ) < e (d (x , I ( p ) ) + d (x , I (k )) < 2 e d (x , 1( g ) )

Proof. We

s e le c t

x Q e 1(g).

.

T h en by Lem m a 12.1 ( i i i )

K g ) = Z ( g ) x 0 , I(p ) = Z ( p ) x 0 , I(k ) = Z ( k ) x Q .

Since

p = pol g,

w e h ave

Z ( g ) = Z ( p ) fl Z ( k )

K g) = Kp)

n i(k )

by (2 .6 ).

C o n se q u en tly ,

.

Let

0 denote the a n g le (in the G -in v a ria n t m etric) betw een

I(p )

and

I(k ),

that i s , the minimum a n g le betw een a g e o d e s ic lin e in

I(p )

orthogo­

1(g )

nal to

at

x Q and a g e o d e s ic lin e in

We can assu m e Let

x€X,

0 > 0 and let

I(k )

orthogonal to 1( g ) at

x Q.

in p ro vin g (ii ). x Q e 1(g)

be s e le c te d so that

Introducing g e o d e s ic co o rd in a tes in

X

at

x Q,

d (x , x Q) = d (x , 1(g)).

w e s e e that in the

(E u c lid e a n ) g e o d e s ic co o rd in a tes d g (x , 1( g ) ) = d g (x , I ( p ) ) c s c O' = d g (x , I ( k ) ) c s c Q " w here I(k ).

0 '

and

C le a rly

c o in c id e .

0 "

are the a n g le s formed by

6 ' + 0 " > 0,

H en ce

the ray

x Qx

with I(p ) and

s in c e E u c lid e a n and Riem annian a n g le s

sup {O', 0" \ > 0 / 2

d (x , 1( g ) ) = d (x , x Q) -

and thus by Lem m a 3.2 ds (x , x Q) = ds (x, 1( g ) )

< (d g (x , I (p )) + d g (x , I ( k ) ) c s c 0 / 2 < e (d (x , I (p )) + d (x , I (k ))

at

xQ

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

84

w ith

e = c s c 6/2.

at once from

1(g) C I (p )

LEMMA 12.3.

and

1(g) C I(k ).

Then there is a function I(g ,a )C

Proof.

Let

group o f

c such that for a ll a,

T c ( a ) (I(s )) .

t denote a s em i-sim p le elem ent in the s m a lle s t a lg e b r a ic

G L (n , R )

group co n tain in g Let

T h e right h a lf fo llo w s

g e G and let s denote the s e m i -s i m p l e Jordan c o m ­

Let

g.

ponent of

T h is p ro ves the left h a lf o f (ii ).

Z (s )^

co ntaining s,

g.

T hen

and th erefore

t

lie s in the s m a lle s t a lg e b r a ic

Z ( s ) C Z (t ).

denote the cen tra lize r o f

s

in

C o n se q u en tly G L (n , C ),

I(t) 3 I (s ).

and s et

C =

! z g z - 1 i w here

z

va rie s over the connected component o f the identity

of

C

is an irred u c ib le a lg e b r a ic variety.

Z (s )^ .

Then

the s e m i-sim p le Jordan component o f ea c h

g ' e C.

to any a lg e b r a ic group co n tain in g an elem ent Set s

and

p = pol s

R e p la c in g

Let set

k = sp-1 .

g

X

w ith a s u b s e t of

g ' e C.

p = pol g ' for a ll

1 e I(s )

s in c e

n : X -> I ( s )

orthogonal to

g e C ; hence C.

in the u s u a l w a y (c f. (2 .1 1 )).

is tran sitiv e on

Y

F o r any elem ent p o s itiv e re a l number

X onto

I(s )

c o n s is t s o f the one param eter su bgro u p s in

Furtherm ore,

f e G, a,

assum ­

X.

denote the orthogonal p ro jection o f

T h en

I (s ).

G

com ponent o f the identity in

set

pact and bounded s u b s e t of g jj

P (n , R )

is

t b e lo n g s

by a c o n ju ga te if n e c e s sa ry , no gen erality is lo s t in

Y = 77 1 (1 ).

L et

T hen

T h erefo re

s

k b e lo n g to any a lg e b r a ic group co n tain in g an elem ent o f

We identify

ing that

and

C le a rly

X = ZY

w here

Z

1

d (f) = d ( l , f) = Tr (lo g f *"f) 2

C Q = i g 'e C ; d ( g ') < a|. C

lie s in

T h en

C Q for som e fin ite

denote the co m p lex -v alu ed function on

and sim ila rly d efin e the function

c o e ffic ie n t of

t-

by

and for any Ca

is com­

a.

G L (n , C ) x C

by gij(h> gO = (i, j )

X

is the connected

Z (s ). set

and

h g 'h

d efin ed

§12.

t i j(h , g O = (i, j )

for a l l on

h e G L (n , C ).

G L (n , C )

1 < k < j, Cn

For each

c o e ffic ie n t o f

(i, j)

with

h t h -1

j < £ < nJ b e c a u s e

gen erated

n—j +1

standard b a s e vecto rs if

by Igy?; 1 < k < j, j < ^ < nj.

g k£ ^

t-

p o s itiv e in teger

h g h- 1

d o e s.

b e lo n g s to the ra d ic a l o f

J

- k < j ’j -

b e lo n g s to the ra d ic a l o f the id e a l

G L (n , C )

that is

^

^

Aj

•

and therefore there is a

q such that

tjq = ^

w here e a c h

t-

Set

- k < h

tj = ^

tj

the s e t of z e ro s o f

of every w h ere re gu lar ration al function on

gj = 2

T h en

i < j

k e ep s invariant the s u b s p a c e of

B y the Plilbert N u lls t e lle n s a t z th erefore, A.

1

h th

co n tain s the s e t o f common z e ro s of the fu n ction s

spanned by the la s t

the id e a l

85

PSEUDO-ISOMETRIC INVARIANCE

ay? g j^ (1 < k < j, j < £ < n)

a^j? is an every w h ere re g u la r ra tio n a l function on

G L (n ,C )x C .

Set C j(a ) = supi |ak£(h, g ')| ; h 6 0 (n , R ), g ' 6 C a , 1 < k < j, j < I < n!

T hen

Cj(a)
0

A s su m e for d e fin ite n e s s that

= d (x , 1) sin 0

< d (x , I (p )),

T h erefo re , for any

x e Y

d ( x , g 'x ) > c x + c 2 d ( x S , p x S ) ,

and

and thus

0 „ > 0 / 2. p — d (x , I (p )) >

g' e C a

by (1 2 .3 .5 )

> c x + c 2 c 4 d ( x S , I ( p ) ) 2,

by (1 2 .3 .6 0

1_ > c t + m—1 sin 0

c 2 c 4 d (x , I(p ) ) 2

by ( 12 . 3 . 7 ') 1_

> c 1 + m- 1 c 2 c 4 s in 2 0 / 2 d (x , I ( s ) ) 2

w here c

c2, c4

are p o s itiv e constants d ep e n d in g on

a ls o d epend s on

a.

From (1 2 .3 .8 ) w e in fer for a ll

1_ (1 2 .3 .9 )

a,

d (x , g 'x ) > c 5 (a ) d (x , I ( s ) ) 2

and the constant g 'e C Q

STRONG RIGIDITY OF LOCALLY SYMMETRIC SPACES

90

for a ll

x £Y

such that

d (x , I ( s ) )

p o s itiv e constant d epen d in g on We h av e seen z e Z

is s u ffic ie n tly la rg e w here

z e Z

X can be e x p re s s e d a s

> d ( l , g 1) w here

g' = z ~ 1 g z ,

e s d is ta n c e s . a > d ( g ')

z

•x

with

and x c Y ,

d (z x , g z x ) = d (x , z —1 g z x ) >

(1 2 .3 .1 0 )

is a

a.

that any point in

and x e Y . F o r any

c 5( a )

d(/7(x ), g ' 77( x ) )

=

d (g ')

77: X ^ I ( s )

s in c e the orthogonal projection

From (1 2 .3 .1 0 ) w e infer that if

a > d (z x , g z x ),

d im in ish­

then

and 1_

( 12 . 3 . 11 )

Since

d (z x , g z x ) > c 5 ( a ) d (x , I ( s ) ) 2 .

I(s )

is s ta b le under eac h

and co n seq u en tly for a ll

z e Z,

d (x ,g x )< a .

c ( a ) = (a / c 5 ( a ) ) 2 .

Set

d (z x , I ( s ) ) = d (x , I ( s ) )

x e X

d (x , I ( s ) ) 2
d ((e x p th )u (exp — th)x, x ) — |s| w here 91 e n.

|s| = d (s x , x). H en ce

Now

(e x p th )u (exp — th) = exp n (t)

d ((e x p n (t)x , x ) -> oo as

d oes not b e lo n g

to

I(g , a )

T b (I (g , a ) ) C K g, a + 2 b )

for

t -» oo

and c o n seq u en tly

a < d ((e x p n (t))x , x ) — |s|.

for any p o s itiv e

d ((e x p — th)x, I(g , a )) ->

a

and b,

com es from c o n s id e rin g b a lls around as

t

(e x p th)x

oo

n (t) = > (e x p — th)x

Inasm uch as

it fo llo w s

as t ->

OO

From this the firs t a s s e rtio n o f the lemma fo llo w s .

(ex p — th )u (ex p th) -> 1

where

that

.

T h e seco n d a s s e rtio n

and u s in g the fac t that

oo.

We can now g iv e a criterion for se m i-sim p lic ity and unipotence o f e le ­ ments o f

G

that is p reserved by p se u d o -iso m etric F-m orph ism s o f

P R O P O S I T I O N 12.5. X

G

be a s em i -s i m p l e analytic linear group, let

b e the a s s o c ia t e d symmetric space, let

d (x , g x ) < a}. (i)

Let

g

g e G

is s e m i -s i m p l e if and only if for ev ery

( i i ) A s s u m e that G a > 0,

for e v e r y

Let

I

I C T c (I (g , a ) ) I(g , a ),

for

and let

I(g , a ) = {x e X;

Then a, I(g , a )

Hausdorff dis tance of a g e o d e s i c s u b s p a c e of

Proof.

X.

has no center. I(g , a )

Then

g

X;

is unipotent if and only if

contains s om e arbitrarily large balls of

be a g e o d e s ic s u b s p a c e such that and

lie s within finite

I(g , a ) C T C(I )

d (x , g x ) = d (g x , g • g x ).

for som e finite H en ce

h d (I(g , a ), I) < oo. c.

C le a rly

g n I(g , a ) = I(g , a )

X.

T hen

g I(g , a ) = for every

n,

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

92

and

g nI C T c ( g n I(g , a ) C T c (I(g , a )).

B y Lem m a 5.4,

tran slate of a g e o d e s ic s u b s p a c e o f

I,

late of

Thus

w here

I

— for d im ension re a so n s.

and hence g l = hi

hi = g l C T 2 c (I )

for a ll

h e Z (p o / I)

x e I.

Z (p o / I).

C o n se q u en tly ,

I

with

and

I

h e Z ( p o l I), gen erated by the In p articu lar,

g l = hi

and

x e I,

B y Lem m a 5.4,

h- 1 g h- 1 g 6

p a ra lle l to it s e lf and therefore

g = h • h - 1 g e ( pol I).

contains a point

Z (p o / I)

Gj

so that

We can assum e without lo s s o f gen erality that group and that

is a p a ra lle l tran s­

It fo llo w s that for a ll

d (x , h - 1 g x ) < d (x , g x ) + d (g x , h _1 g x ) < a + 4c. m oves each g e o d e s ic line in

is a p a ra lle l

g nI C T c (I(g , a )) C T 2 c (I).

We have

and w e can s e le c t

d (x , h x ) = d (I, h i) < 2c

Gj

gl

pol I d enotes the subgroup o f the s t a b iliz e r

p olar subgroup s of G j.

gl

G

x Q fix e d under

is a s e lf-a d jo in t

G fl 0 (n , R ).

Then

are s e lf-a d jo in t groups and w e have by (5 .4 .1 ) I = (p o l i n P ( n , R ) ) • ( G j f l 0 (n , R )) pol I = (p o l i n P ( n , R ) ) • (p o l i n 0 ( n , R ) ) .

Set

Z

= Z ( p o l I).

Since

Z ( p o l I)

is s e lf-a d jo in t, w e have

Z = ( Z n P ( n, R ) ) • ( Z n 0 (n, R ) ) . F o r any Since

p e pol Iand

zfZn O (n ,R )

I= pol I x Q, w e s e e that

fix e d each point in T h e orbit

la te s of

C le a rly I, and

z e Z.T h e

Z

convex

co n vex s et s in c e g

Z n 0 ( n , R ) C G j,

p x Q.

in fact Z n 0 (n , R )

Z x Q is a g e o d e s ic s u b s p a c e by (3 .4 .1 ) and pol I

5.5 that the g e o d e s ic lin e s in D.

zpxQ= pzxQ=

I.

mutes elem en tw ise with

space

w e have

D

s in c e

pol Z x Q C Z .

Z x Q orthogonal to

pol Z x Q com­

It fo llo w s by Lem m a I

form a g e o d e s ic s u b ­

is a s et o f re p resen ta tiv e s for the p a ra lle l tran s­

operates on hull o f

D

v ia

y

z ( p o l I)y n D for

i g nI; n = 0, ±1, ...j

g nI C T 2 c ( I )

for e a c h

n.

g

in

D,

a bounded

T h erefo re in its action on

keeps in variant a c lo s e d bounded co n vex set.

B rouw er fix e d point theorem ) that

in terse cts D

y e

D,

It fo llo w s (from the

k e ep s fix e d a point in

D

under the

§12.

PSEUDO-ISOMETRIC INVARIANCE

ab o v e action; that is , c a n o n ic a l action. z l;

i.e .,

( n , R ) ).

g

93

s t a b iliz e s a p a ra lle l tran slate of z e Z

T h e re fo re there is an elem ent

z-^ gze G jH z .

We h ave

From the e x p re s s io n for

G j,

I

under the

su ch that

g z l =■

G j fl Z = (G j f l Z f l P (n , R ) ) (G j H Z f l w e s e e that each elem ent in

G j n Z fl P (n , R ) commutes w ith each elem ent in G j fl Z fl 0 (n , R ). Gj H Z

seq u en tly , every elem ent in

Con­

is sem i-sim p le and in p articu lar,

g

is sem i-sim p le. We have thus proved:

If

I(g , a )

o f a g e o d e s ic s u b s p a c e , then

g

lie s w ithin fin ite H a u sd o rff d is ta n c e

is sem i-sim p le.

T h e c o n v e rse fo llo w s

im m ediately from Lem m a 12.3 w hich im p lies that g

h d (I(g , a ), 1 (g)) < oo if

is a s em i-sim p le elem ent. If

g

is a unipotent elem ent, then its sem i-sim p le part is the identity

elem ent and therefore

I(g , a )

co n tain s arbitrarily large b a l ls in

Lem m a 12.4.

T h is p roves the “ only i f ” a s s e rtio n of ( i i ).

s u p p o s e that

I(g , a )

co n tain s arbitrarily large b a lls of

the s em i-sim p le part of T c (I(s )) that

s

for som e fin ite is cen tral in

g.

G.

It fo llo w s at once

X.

B y our h y p o th e sis on

I ( s ) = X.

by

C o n v e rs e ly ,

T h en arbitrarily large b a lls of

c.

X

Let X

s

denote

lie within

T h is im p lies

G , s = 1;

that is ,

g

is

unipotent.

P R O P O S I T I O N 12.6. groups, let

F

and

Let

T'

G

and

G ' be s e m i-s im p le analytic linear

be subgroups of

G

and

G',

and let

de note the symmetric Riemannian s p a c e s a s s o c ia t e d to s p e c t i v e ly . (£>': X '-> X

Let

0

be an isomorphism, let

and

and

X'

G ' re -

cf> : X -> X ' and

be p s eu d o -i s o m e tr ie s equivariant with r e s p e c t to 0

r e s p e c t iv e ly . (i)

0 : T -> V '

G

X

and

1

Then

sends s em i -s i m p l e elem ents to s e m i -s i m p l e elem ents and v i c e -

vers a (ii)

if

G ' has no center, then

el em ents.

0

s en d s unipotent el ements to unipotent

94

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

Let

Proof.

g / e

be an elem ent in

p ositio n into sem i-sim p le part and let are

s

T,

let

g = su

and unipotent part

s ' u ' be the Jordan d ecom position of g'.

(k, b )

be its Jordan decom ­ u,

set

g '= 0 (g ),

A ssu m e that 0

and

cfV

and

0'

p se u d o -iso m etries.

We b e g in with the sim ple o b se rv a tio n s that T c (I(g , a )) C I(g , a + 2 c) 0 ( I ( g , a ) ) C Kg', ka) for any

c

and

are s u rje c tiv e .

a.

B y Lem m a 10. V, dim

It fo llo w s that for

X = dim X ' and both

0

a > b,

n,

w e co n clu d e

m = n.

dim I ( s ) > dim I ( s ') .

Sin ce

it is su rje c tiv e by Lem m a 10.1'.

C T c (0 (I (g )),

s in c e

iJj : I ( s ) -> I ( s ')

C o n se q u en tly ,

X and

0

F ' is

X,

A

let

0

w hich

~ 1 re­

denote

Then there is a unique

Y '-c o m p a c t, and A ' is the

A

F

in

G

and s e t

is a m axim al p olar subgroup o f F,

and

G

is the norm alizer of Gp

A

is sem i-sim p le.

B y 2.1,

je x p t lo g u; t e R i

A = pol F .

G,

A in

operates G.

F o r let

b e the Jordan product d ecom position w ith

unipotent.

containing

r-flat in

A ' = 0 (A ).

and s e t

We claim that every elem ent in

u

F in a lly , let

0 : X -> X ' and

Y -c o m p a c t

denote the s ta b iliz e r of

sim ply tra n sitiv e ly on

and

G ' re sp e c tiv e ly such

in T'.

T h en by Lem m a 5.1,

and let

and

is s u rje c tiv e by Lem m a 10. V.

X ' stable under A ',

sta bilize r of

Let

F

Let

the sta bilizer of

Proof.

G

G '/ T ' have fin ite m easure. A ssu m e there e x is t maps

denote the

G', K', X ' be a sim ilar triple.

p se u d o -iso m e trie s eq u ivarian t with re sp ect to

s p e c tiv e ly .

x-flat

Let

T ' be d isc re te subgro up s o f

isom orphism . are

G /K .

X

s

g e Gp

sem i-sim p le

lie s in any a lg e b r a ic group

and therefore lie s in the connected component of the identity A

is a m axim al polar subgroup of

that the c e n tra liz e r of

A

u e Z (A ).

Z (A ) = M x A

B y (2.6. iv ),

G,

w e get by (2 .3 .iv )

is of finite index in the norm alizer of (d ire c t) with

96

M com pact.

A . H ence It fo llo w s

§13. THE BASIC APPROXIMATION

at once that every elem ent o f Thus

g = s

Since

Z (A )

and every elem ent is

A fl Z ( A )

97

is sem i-sim p le and in p articu lar Gp

u = 1.

is sem i-sim p le.

is of fin ite in dex in

A

and

is fin ite , w e can A 1 C Z (A )

s e le c t a subgroup of

Aj

of fin ite in dex in

A j H M = (1 ).

T h en

Aj

is an a b e lia n subgroup s in c e it in je c ts into the

a b e lia n group

M \ Z (A ).

B y h y p o th esis

r\rF A

= A\F

F

and

is F -co m p ac t. A j\ A

A X\ F

-

M A1 H A\M

M \ M A 1 = M A j n A,

TNFF

is com pact.

A ^ F

w e have

M A X\ M A

w e get

fl A \ A

S ince

Since

and hence

.

Aj = M H A j\ A 1 = M X M A ^

rank A j = dim A = R -rank G = r. sen d s s em i-sim p le elem ents of

0

and

From

is com pact.

F = M \M A

=

T ' and v ic e -v e r s a .

R -ran k G = R -rank G'. elem ents of

F,

is com pact.

B y P ro p o s itio n 12.6, sim p le elem ents o f

H en ce

such that

fin ite w e get that

operates sim ply tra n sitiv e on

T h erefo re

A

A fl M

Thus

0 (A j)

F

to sem i­

T h erefo re , by Lem m a (1 1 .3 .iii)',

is an a b e lia n subgroup o f sem i-sim p le

T ' of rank e q u a l to R -ran k G ' and thus co n tain s an

R -h y p erreg u lar elem ent

y'

of

G'.

B y R ag h u n ath an ’s criterion

Z ( y ' ) fl T ' is com pact.

M oreover, the elem ent

Z ( y ') /

y ' is p o la r-re g u la r s in c e

any R -h y p erreg u lar is p o lar-re gu lar. B y Lem m a 5.2 ( i i ) , 5.2 (i i i ) ,

Z (y ')

Z (y ') H r ' \ F '

y'

s t a b iliz e s a unique r-flat

s t a b iliz e s

F ' and acts tra n sitiv e ly on

is a quotient of

F ' is a r - c o m p a c t r-flat of Let

Tp/

Z (y ') H r ' \ Z ( y ' )

A,

r.

T p'

0 (A j)

r,

w e s e e that

is of finite in dex in

A,

F'

w e s e e that

group of fin ite index of rank of rank

F '.

X'.

B y Lem m a

H en ce

and is com pact; that is ,

X'.

denote the s t a b iliz e r o f

resu lt proved a b o v e for

F ' in

Since

in Tp/

^ (A j)

F'

A p p ly in g to

Fp/

the

co n tain s an a b e lia n s u b ­ is an a b e lia n subgroup of

is o f fin ite index in

s o a ls o is the subgroup

n

Tp/. g

Since 1.

Aj

Thus

geA without lo s s o f gen erality w e may assu m e that A j 0 (A j )

is normal in

0 (A ).

F o r any

g ' e d (& ),

is normal in

A

and

0 ( A 1) g ' F ' = g '0 ( A 1) F ' = g 'F ';

98

STRONG RIGIDITY OF LO CALLY SYMMETRIC SPACES

that is ,

s t a b iliz e s the r-flat

unique r-flat, w e get

g 'F '.

g ' F ' = F ' for a ll

Since

00^)

s t a b iliz e s a

g e 0 (A ).

Thus

# (A ) C T p / .

A p p ly in g the sam e re su lt to

0 ~ l , we

get

REMARK.

G /T

G '/ T ' are com pact, there is no

In the c a s e that

and

0- 1 ( T 'p ') C A .

Thus

(9(A)^ T p ^ .

need ab o v e to em ploy R agh un ath an ’s criterion, s in c e in this c a s e , a ll the elem ents of

F

and

L E M M A 13.2. the unique

F

Let

r-flat in

r-flat

be a T -c o m p a c t

F

b

r-flat in F 0 (F p ).

F ' s ta b i liz e d by k and

depending only on r -compa ct

are sem i-sim p le (c f. Section 11).

V'

and let

F'

denote

Then there is a constant v

but not on the particular c h o i c e of the

such that h d ( 0 ( F ) , F /) < v .

Let

tt

onto

F

Proof. of

X

y 6Tp

and

: X -> F

and

and o f

x e X,

X ' -» F ' denote the orthogonal p ro jection s

X ' onto

and

We claim first that

tt':

F '.

We note that

n ' ( y ' x ) = y ' n ' ( x ) for a ll n'(cf) ( F ) ) = F'.

F

onto

F '.

and

y 'e T

for a ll x e X'.

We can d edu ce this from the re su lts

o f Section 12, u sin g Lemma 12.3 to sh o w that of

n (y x ) = y n (x )

tt'

° 0

is a p seud o-isom etry

In ste ad , ho w ever, w e p resent here a sim p le to p o lo g ic a l

argument. Let

A

b e a free a b e lia n subgroup of finite index

denote the restriction of A -b u n d le map of

F

duced map of A \ F in

X

to

tt'

° cf) Set

F '.

to A '\ F '.

to

A ' = 0 (A )

is com pact, the subgroup A fl G x

is a p rin cip al

space,

F

A -b u n d le .

is a u n iv e rsa l

if/

may be

and let

is fin ite for

H en ce

A

Sim ilarly

and let

F

i[i

regarded a s a

a ll

of any point x e X

operates fre e ly on

Furtherm ore, sin c e

A -b u n d le .

Tp,

denote the in­

Since the s t a b iliz e r G x

contains only the identity elem ent. F

F . T hen

in

x

and thus F

and

is a co n tractible

F ' is a u n iv e rsa l

A '-

bun dle.

A s is w ell-k n o w n from the elem entary theory o f u n iv e rsa l b u n d les,

the map

if/^ : A \ F -> A r\ F r is a homotopy e q u iv a le n c e .

A\F

is an r-dim en sio n al torus,

ifj^

Inasm uch as

in du ces an isom orphism o f the

§13. THE BASIC APPROXIMATION

hom ology group

H r( A \ F )

onto

H r( A ' \ F ' ) -

99

Inasm uch as

A ' \ F ' is a

com pact m anifold, any top d im en sio n al c y c le has the entire s p a c e a s its support. tt' ( 0

(f » =

f

H en ce

t/ ^ (A \ F ) = A '\ F '.

A '\ F '

i/r(F) = F ' and

:

N e x t w e claim that b > 0.

C o n se q u e n tly

Let

77(T|3( 0 ~ 1( F r) ) = F .

: A \ X -> A ' \ X ' ,

be the m aps induced by

0,

We can assu m e

77A : A \ X ^ A \ F , and

n,

and

tt^

k > 1 and \ A " \ X '-> A ' \ F '

We can assu m e, upon re p la c in g

n\

A

by a subgroup of fin ite in dex if n e c e s s a ry that the c a n o n ic a l p ro je ction s

of

X

onto

radius

2b

b a lls in Let

X

F ' onto

of radiu s

b/k,

p