Crash Course on Kleinian Groups

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

400 A Crash Course on Kleinian Groups Lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

Edited by Lipman Bers and Irwin Kra

Springer-Verlag Berlin. Heidelberg. New York 1974

Lipman Bers Columbia University, Morningside Heights, New York, NY/USA Irwin Kra SUNY at Stony Brook, Stony Brook. New York, NY/USA

Library of Congress Cataloging in Publication Data

American Mathematical Society. A crash co~rse on Kleinian groups, San Francisco, 1974. (Lecture notes in mathematics, 400) i. Kleinian groups° I. Bers, Lipman, ed. II. Bira, Irwin, ed. Ill. Title. IVo Series: Lecture notes in mathematics (Berlin, 400) QA3.I28 no. 400 [QA331] 510'.8s [512'.55] 74-13853

AMS Subject Classifications (1970): Primary: 30-02, 32G15 Secondary: 30A46, 30A58, 30A60

ISBN 3-540-06840-6 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-06840-6 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

To Lars

V. A h l f o r s

PREFACE

It has r e c e n t l y b e c o m e c u s t o m a r y to h a v e sessions" at m e e t i n g s of the AMS, lectures,

"special

c o n s i s t i n g of short invited

and intended for groups of specialists.

A n n u a l W i n t e r M e e t i n g at San Francisco,

At the

we tried to h a v e a

s p e c i a l s e s s i o n a d d r e s s e d to non-specialists.

The lecturers

w e r e asked to p r e p a r e in advance texts of their talks, these w e r e d i s t r i b u t e d at the meeting. revised,

are c o l l e c t e d in the present

These texts, fascicule.

and

slightly

(We also

included an a b s t r a c t of a f o r t h c o m i n g paper b y H. Masur.) The p r e s e n t

"crash course" does not intend to do

m o r e than to give a reader an i n t r o d u c t o r y s u r v e y of some topics w h i c h b e c a m e important K l e i n i a n groups. means complete,

in the m o d e r n theory of

The references

to literature,

though b y no

should enable anyone interested in more de-

tailed i n f o r m a t i o n to o b t a i n same. Lars Ahlfors,

w h o p l a y e d a d e c i s i v e part in the recent

r e v i v a l of K l e i n i a n groups, Francisco.

could not b e p r e s e n t at San

It is fitting to d e d i c a t e this modest effort to

him. LoB.

I.K.

CONTENTS

Chapter i What is a Kleinian group? by Lipman Bers Chapter 2 Quaslconformal mappings by C. J. Earle

1

and uniformization

15

Chapter 3 Automorphic forms and Eichler cohomology by Frederick P. Gardiner

24

Chapter 4 Deformation spaces by Irwin Kra

48

Chapter 5 Metrics on Teichm~ller by H. L. Royden

space

71

Chapter 6 Moduli of Riemann surfaces by William Abikoff

79

Chapter 7 Good and bad Klelnlan groups by Bernard Maskit

94

Chapter 8 Kleinlan groups and 3-dimensional by Albert Marden Researcn Announcement The curvature of Telchm~ller by Howard Masur Some Unsolved Problems Compiled by William Abikoff

topology 108

space 122

124

i.

WHAT

IS A K L E I N I A N

GROUP?

L i p m a n Bers Columbia University

This is hoped,

is the

will

the present

first of a series

give a picture,

(they are, discrete

by

can be

the way,

subgroups

or as a tool

course,

the two points The theory

for r e p r e s e n t i n g

for their

Riemann

was

of

own sake

studied

class

of

of infinite

surfaces.

of v i e w cannot be neatly groups

one,

groups.

the only e x t e n s i v e l y

of K l e i n i a n

and Klein

dormant,

of F u c h s i a n decade

in the

except, groups.

is based,

eonformal

Of

separated.

founded b y

Schottky,

groups

The b u r s t or

groups

with

[i]), b u t a p p l i c a t i o n s algebraic

curves

indirectly, tool

1965 p a p e r

such groups.

to compact

(and to h i g h e r

the

it case

last

on the use of quasi-

[3]

function

finitely

and our

Infinitely

and p r e s e n t

special

during

in c o m p l e x

of attention,

are also of interest

important

of a c t i v i t y

as a w o r k i n g

are at the center

exclusively

For many years

of c o u r s ~ for the

directly

mappings

19th century.

Since Ahlfors' seminal

deal

either

it

/

Polncare was

an i n c o m p l e t e

of K l e i n i a n

studied

which,

of a Lie group w i t h q u o t i e n t s

volume)

•

albeit

state of the theory

Such groups

of lectures

generated

lectures

surfaces

dimensional

will

generated

new p h e n o m e n a

Riemann

theory.

and

algebraic

(Abikoff

varieties, generated

cf.

Griffiths

groups.

formations is as yet

in

[i0])

involve

Discontinuous

R n, n > 3,

in infancy

primarily

groups

finitely

of M6bius

are not discussed.

trans-

Their

theory

and seems

to h a v e

no f u n c t i o n

theoretical

is the first

lecture

it contains

mostly

interest. Since

this

definitions

and examples.

A group of t o p o l o g i c a l is c a l l e d

(properly)

self-mappings

discontinuous

infinitely

m a n y of its translates.

a subgroup

of the

discontinuously C = C U [~}.

(complex)

on some

that

multiplication

by

(-1).

projective

line

az+b z~--9 cz+d

thus M 6 b can be

isomorphisms

of

2 by

A Kleinian group

M~b

set m e e t s

group

[ 7 ] , [9],

2 matrices,

= ~

is

sphere

[ii].) is the group of

determined

(ab ~ cd )

G

, w h i c h acts

subset of the R i e m a n n

The e l e m e n t

up to

acts on the

b y the rule

;

identified

~

if no c o m p a c t

M~b = S L ( 2 , ~ ) / { ~ I}

all c o m p l e x u n i m o d u l a r

(i)

open

(General references:

Recall

complex

M~bius

of a space

with

the group

of all h o l o m o r p h i c

.

The real u n i m o d u l a r

matrices

±(ab) cu

form the real

3

M~bius upper

group

c M~b.

half-plane

Recall U

M~R

that

into

U =

Thus

motions

element

may be viewed

of

U

and also

as

of

E ~, y >

M6b R 0}

maps

onto

the

itself.

12/y 2

ds 2 = I dz

of the n o n - E u c l i d e a n

M~b

self-mappings

element

[z = x + iy

• , P o l n c a r e line

the

a model

plane.

Every

makes

(Bolyai-Lobatchevski)

as the g r o u p the g r o u p

of a l l

conformal

of a l l n o n - E u c l i d e a n

in the plane. The

complex

tation.

The u p p e r

with

Poincare

the

metric

a point

number (z,t)

A complex

(2)

group

half-space

of the n o n - E u c l i d e a n a complex

M~bius

R3 + =

ds2

space.

x + iy

E IR~

unimodular

=

(I dz 12 + d t 2 ) / t 2

+-(cd ab )

of all

=

interpret >

0]

is a m o d e l

and

identify

x + iy + j0 + k0,

z + jt = x + iy + jt + k0.

the q u a t e r n i o n

matrix

the g r o u p

quaternions

the q u a t e r n i o n

(z + jr) ~-9 (z' + jt')

Now M6b becomes

a similar

{(z,t)l z E C, t E R,

We use

with

with

admits

acts

3+

on

by

the

rule

[a(z + jr)+ b] [c(z + jr)+

non-Euclidean

motions

d]

in

space. A discrete uously

on

3

+

(see M a r d e n ' s not act

and

subgroup

G c M~b

the q u o t i e n t

~/G

lecture).

discontinuously

it is c a l l e d set on w h i c h

Kleinian, G

acts

On

the o t h e r

o n an o p e n as n o t e d

always

is a l w a y s hand,

subset

above,

discontinuously

acts

and

G

of the

a 3-manifold may

~.

discontin-

or m a y

If it does

largest

is d e n o t e d

by

open

~ = Q(G)

-i

and

is c a l l e d

A = A(G) the

the

region

= C\~(G)

is c a l l e d

set of a c c u m u l a t i o n

closure

of the

hyperbolic)

The consists

set of

and

of

infinite.

limit 0,

fixed

set

1 or

structure

holomorphic. Riemann

Thus

locally

z

order of

point

of order

tached

to

of

points,

and

the

(including

G.

finite G

(in this

is c a l l e d

is a p e r f e c t

case

it

elementary) nowhere

.)

Thus one

or

set of

.

at p o i n t s G

z

points

~ ~ Q/G

z E Q

with

is c a l l e d of

to b e

G is

non-trivial

group

near

a

of

(One says:

v-to-i

a component equipped

union

is a c y c l i c

is

is g i v e n

~ ~ Q/G

is a d i s j o i n t

The p r o j e c t i o n

case

i.e.,

and

the p r o j e c t i o n

the p r o j e c t i o n

but

of points,

is a 2 - m a n i f o l d

requiring

except

~.

It is a l s o

of some

z, a n d the

a ramification

Q/G

is n o t just

with

a discrete

with

integers

a

set o f v >

1

at-

them. If

A

Q/G

In this

under

surface,

ramification

A

v, the p r o j e c t i o n

image

Riemann

z

.

case

G.

capacity.

---

one-to-one, G

and

complement

loxodromic

~/G = S 1 + S 2 + "'"

S I, S 2,

stabilizer

of

may be

the q u o t i e n t

surfaces,

represents

finite

by

points

The

set of

of o r b i t s

2 points

The q u o t i e n t complex

limit

elements A

latter

logarithmic

the

points

parabolic

In the

positive

o__[fd i s c o n t i n u i t y .

is a g a i n

A

is a c o m p o n e n t

a Kleinian

group,

of

and

~, the A/G A

stabilizer

is a R i e m a n n

GA

of

surface

(with r a m i f i c a t i o n points). called conjugate if A I, A 2 . . . . of

G

A 2 = g(Al)

~(G)),

n/G = AI/GAI

The g r o u p component genus

p

for some

A1

A if

+

G

A/G A

£2' are

If components

then

A2/GA2

+

-o°

is said to be of finite type over a is o b t a i n e d from a compact surface of

b y r e m o v i n g finitely many,

(punctures)

and

g 6 G.

is a complete list of n o n - c o n j u g a t e

(i.e., of

(3)

Two components,

say

n

and if there are finitely many,

r a m i f i c a t i o n points on

A/G£, of orders

~ 0, points say

n O ~ 0,

vI ~ v2 ~

--- ~ n O"

Set

n = nO + n .

The pair

(p,n)

is called the type of

A/G£, and the sequence

(4)

(p, n;

~i'

---, v

no

, ~

= n

is called the siqnature. The group

G

) times

(One may w r i t e

(p) instead of (p,0).)

is called of finite type if it has

only finitely m a n y n o n - c o n j u g a t e

components and is of finite

type over all of them. E l e m e n t a r y K l e i n i a n groups are easily enumerated. The m o s t i m p o r t a n t are the cyclic and elliptic groups. the cyclic group

G = {gk, k = 0, ±i,

g = id

is e l l i p t i c of order

or if

g

---},

A

For

is empty if

k, A consists of ! p o i n t

6

if g is p a r a b o l i c , natures

of 2 p o i n t s

of ~/G are

respectively.

(0),

6 Z 2, Im w w > 0} has one G

G

is l o x o d r o m i c .

(0, 2; k, k),

An elliptic

If

if

group G =

(0, 2; ~, ~) a n d

[z~9 z + n w + mw',

limit point

is n o n - e l e m e n t a r y ,

~; ~/G h a s

component

A

of

~

metric

of c u r v a t u r e

latter

can b e t r a n s p l a n t e d

finite

type o v e r a c o m p o n e n t

if

(i).

from n o w on,

/

(-i).

complete

Since

G

~/G.

to A

conformal respects

Riemannian

the m e t r i c ,

The g r o u p

the

is of

G

if a n d o n l y if

k(z) 2 d x d y <

~ ;

Area

(A/G A) = ~

Area

n 1 (A/G A) = 21~{2p - 2 + E (i - ~-~)} 1 3

so

(6)

where !/~

(n,m)

carries a Polncare metric

ds 2 = X ( z ) ~ dz i2, the u n i q u e

(5)

(i),

signature

as w e a s s u m e •

every

The sig-

=

(p, n;

of

A/GA

and

0. A basic

groups

is the s i g n a t u r e

v I ..... Vn )

result

is the f i n i t e n e s s

in the r e c e n t theorem

that a finitely generated (The c o n v e r s e a r e the a r e a

theorems

[5]:

(Ahlfors

Kleinian

n e e d n o t b e true.)

t h e o r y of K l e i n i a n

group

[3]) w h i c h

states

is of f i n i t e type.

A quantitative

refinement

6

(7)

if

Area

G

has

(8)

(n/G) i 4n(N - i)

N

generators

Area

(~/G) ~ 2 Area

for a n o n - e l e m e n t a r y component

and is not elementary,

40

(A0/G)

(cf. Gardiner's

for all

GA0 = G~

finitely generated group with an invariant lecture).

An important open ~ u e s t i o n mes A = 0

if

(Ahlfors)

finitely generated groups

is w h e t h e r (cf. Maskit's

lecture). The inequality

(7) is sharp.

example of a Schottky group follows. curves,

Let

CI, Cl, C 2, l

such that C. 3

Then G.

gj

and let

"''' gp

The limit set

-, C

~/G

A(G)

is of type

Schottky before

187~

Koebe in 1907) asserts p > 0

gl'

be

i

P

p

constructed as

2p

disjoint Jordan

in the u n b o u n d e d

"'', gp

be M6bius

component transformations

component of the complement

component of the complement

are the free generators

so that

retrosection

Area

of

C~. ]

of a Kleinian group

is totally disconnected,

(p,0)

The classical

genus

o

maps the u n b o u n d e d

onto the b o u n d e d gl'

and

•

of genus

each containing all others

of its complements

of

G

This is shown by the

mes A = 0,

(Q/G) = 4~(p - i).

theorem

(suspected b y

stated by Klein in 1882, proved by that every compact Riemann surface of

can be represented by a Schottky group.

8

(A f u n d a m e n t a l is a s e t points

w c

of

w

equivalent

such that

are

G

mes

of

for t h e S c h o t t k y by

the

w

2p

Jordan

The b e s t

studied

of a Kleinian

(Cl(w)\w)

equivalent,

to s o m e p o i n t

Example: bounded

Q

region

group

= 0, n o

and every

cl(w).

Clearly,

C I,

two

z E Q

constructed

curves

group

---,

G

interior

is

G

Q / G = w/G. above

C' P

the r e g i o n

is a f u n d a m e n t a l

region.)

Fuchsian of

MOb

called A of

groups.

o f the

first

is t h e w h o l e A R.

A Fuchsian

(or c o n j u g a t e

R

lower half-plane

G

case,

L, b o t h

second kind

U [~} ~

groups

in

according dense

two c o m p o n e n t s ,

invariant.

In t h e

It is to w h e t h e r subset U

second

of

subgroup

M6b).

or a n o w h e r e has

is t h a t

is a d i s c r e t e

to s u c h a s u b g r o u p

A R = R

first

of Kleinian

group

or o f t h e

of

In the

class

and

case,

the Q

is c o n n e c t e d . The m o s t

first kind its

are

subgroups

finite

classical •

Klein

and by

Koebe

in 1907)

every

given

represented

of Fuchsian

modular

group

groups

of the

SL(2,Z)/[±I~

and

index. limit

circle

theorem

(conjectured by

/

Polncare

in 1882,

asserts

Riemann as

examples

the e l l i p t i c of

The

famous

U/G,

that,

surface G

proved

but with

by

Poincare

and by

for a f e w e x c e p t i o n a l ramification

a Fuchsian

group.

points

cases,

can be

If the

G

is a finitely

first kind,

mirror same

images

the R i e m a n n

and

its

image

in

Jordan

of the

if

uniformization Riemann

group

polygon

C

is a K l e i n i a n on

~

The t h e o r e m

among other

group.

fixed.

the

G in

can U

group G

is c a l l e d

on s i m u l t a n e o u s

things,

signature

that any

two

can be r e p r e s e n t e d

The only k n o w n

on the t h e o r y of q u a s i c o n f o r m a l

p r o o f of this mappings

(cf.

lecture). A finitely

a b-group 40 .

G

of the same

b y a given q u a s i - F u c h s i a n

Earle's

curve

[4] asserts,

theorem depends

are

they have

for a F u c h s i a n

~roup

A = C.

surfaces

L/G

L.)

leaving a d i r e c t e d first kind,

group of

(8) is sharp.

chosen as a convex n o n - E u c l i d e a n

mirror

and

In particular,

region

A quasi-Fuehsian

given

Fuchsian

U/G

Thus the i n e q u a l i t y

(A f u n d a m e n t a l be always

surfaces

of each other.

signatures.

generated

generated

if it h a s a s i m p l y

If so,

there

a Fuchsian

group

a boundary

group

bijections

W.: 3

Kleinian

connected

group

invariant

is a c o n f o r m a l b i j e c t i o n

P

such that

if there L - W

(L) 3

G = W F W -I

is a s e q ue n c e such

G

W:

G

is c a l l e d component

L ~ 40

and

is c a l l e d

of c o n f o r m a l

10

that

lim W

= W

uniformly

on c o m p a c t

subsets

of

L, a n d a l l

3 groups

W . F W ~ 1 = G. 3 3 3

that all b-groups

[8]).

ramification which

Indeed

the

that

that

of

right

~/G = S

"almost

finite

side

the name.

Riemann

surface

and with

totally

But not a single

groups

are

(Bers a n d

(6) p o s i t i v e ,

(in M~b)

[6].

hence

40 = ~

type,

of

It is c o n j e c t u r e d

a l l '~ b o u n d a r y

is s a t i s f y

non-conjugate

with

groups,

for a n y g i v e n

points,

makes

ably many G

out

degenerate,

Maskit

quasi-Fuchsian.

are boundary

It turns totally

are

S

with

a signature

there

are u n c o u n t -

degenerate such

group

b-groups has b e e n

constructed. A b-group with

equality.

[14]),

All

a n d a l l of

to a p p e a r ) .

surface

S

to a p o i n t

contracting

exceptional two

case

ramification An

deqenerate inequality.

on

each

Such

S\[set

groups

(Marden,

C. 3

points

of o r d e r

~ ~ 40 can b e

S I, S 2, by

image

curves

type

represents ---,

"drawing {

of

of r a m i f i c a t i o n

if a

groups

constructed

obtained

occurs

with

been

surfaces

of t h o s e

if it s a t i s f i e s

b-group

on the m i r r o r

intermediate

group,

have

A regular

to h a v e b e e n

-- -, C r

Cl,

homotopic then

groups

a n d one or m o r e

thought

curves

such

regular

them are b o u n d a r y

Abikoff,

may be

is c a l l e d

(8) (Maskit

Harvey, a Riemann S~

which

Jordan S, n o t points},

and

into a p u n c t u r e . "

bounds

a disc

(An

containing

2.)

of a b-group and

is a p a r t i a l l y

(8) h o l d i n g

constructed

with

a strict

[14] a s s u m i n g

the e x i s t e n c e

of totally

degenerated

The c o n s t r u c t i o n s

b-groups.

mentioned

a b o v e use

combination

t h e o r e m a n d its e x t e n s i o n s

[15].

is a n o t h e r a p p l i c a t i o n .

Here

G 1 , • -- , G r

Let discs

AI,--',A r

with

with

n. ~ i, 2 < 3

~ <

~

and

nl+

by

far apart,

then

is K l e i n i a n ,

components

A0,

has

signature

follows

GI, .... G r.

from Klein's

into

if one i d e n t i f i e s SO + ~0'

~0

surface with

a n d if (unique)

k

k

pairs

i = 1 ..... k,

G.3(i)

and

-i (trace gi )2 = 2 + s. + s.1 '

norm and

let

in A.3(i)and i n s = G

be

are s u f f i c i e n t l y

non-conjugate and

S O = A0/G

p = pl+...+pr.

(This

points

on

(PI,P{)--- (Pr'P~) of a pair, ~0

let

r

P. l

G£(i)

loxodromic M~bius

Let

G

Q/G

s u c h that, becomes

is a s o - c a l l e d

"Riemann

nodes."

s i C C, [ si I p o s i t i v e

fixed points

r + 1

~ ..... ~)

Let

Aj

2k r a m i f i c a t i o n

This

[13],

theorem.)

the two p o i n t s

connected.

For

subgroups of

the

= 2k.

r

invariant,

with

combination

Now we divide AI/G 1 .... , A./G.33

A0

v ..... ~)

(pj, nj;

If the

has

A 1 .... ,Ar, w i t h (p, k;

... +n

[12],

group acting on

of s i g n a t u r e

the g r o u p g e n e r a t e d G

by Maskit

be a Fuchsian

Aj/Gj

Klein's

and

FI l

corresponding a n d small,

let

transformations, with

A~(i).

(s I ..... s k) b e the g r o u p

be elliptic

E Ck

to

P.1

and

gi

be the

Pl,1

with

-i giFigi = Fi, a n d w i t h

If

s.1 = O, l e t

be a vector with

generated by

GO

and

gi = id. small gl .... 'gk"

12

Then

G

is a K l e i n i a n g r o u p

s

second c o m b i n a t i o n theorem). i d e n t i f i e d to

PI l

for

(this follows b y M a s k i t ' s Also,

s. = 0, is l

a Riemann surface w i t h

k - k(s)

SO + E nodes,

n u m b e r of n o n v a n i s h i n g c o m p o n e n t s The g r o u p s b-groups,

~(Gs)/Gs, w i t h

G

s. l

Pi

where

s

~

is

s

k(s) b e i n g the

of

s.

can be u s e d ~ i n s t e a d of regular

s

in the theory of m o d u l i of R i e m a n n surfaces d e g e n e r a t -

ing to a surface w i t h nodes

(cf. A b i k o f f ' s

lecture).

An important class of K l e i n i a n groups are web groups (Abikoff

[2]), that is f i n i t e l y g e n e r a t e d K l e i n i a n groups

such that the s t a b i l i z e r of each c o m p o n e n t is q u a s i - F u c h s i a n of the first kind. We give only one, h i g h l y pathological, Let

G

be a F u c h s i a n group leaving the unit disc fixed and

r e p r e s e n t i n g two compact surfaces of type be the group o b t a i n e d from

G

a large p o s i t i v e number. and

G'

types

(p,O) and

be real numbers,

by

z~

Gt

Let

The g r o u p

GO

G'

z~

Iz,

generated by

G

is K l e i n i a n and represents three compact surfaces of (p,O),

by

(p,0).

b y c o n j u g a t i n g it b y

(2p,O),

respectively.

lows from Klein's c o m b i n a t i o n theorem.)

G

example.

and

let

(i + t)e G'.

Gt

all

~, there is a n u m b e r

for

0 ~ t ~ s,

Gt

~

and

t ~ 0

be the group o b t a i n e d b y c o n j u g a t i n g

z, and let

Maskit

Let

(This fol-

Gt

(to appear) s > 0

be the group g e n e r a t e d showed that,

such that

Gt

for a l m o s t is K l e i n i a n

represents three surfaces of types

(p,O),

13

(P,0) and

(2p,0)

representing

for

0 ~ t ~ s, and

only two surfaces,

In this example,

Gs

is a web group

b o t h of type

(p,0).

just like in the case of p a r t i a l l y

and totally degenerate b-groups,

a Riemann surface

Its debris is, in some sense, hidden

"disappeared."

in the limit set.

Could

it be that such a limit set has positive m e a s u r e ?

REFERENCES

[i]

Abikoff,

W.• Some remarks on Kleinian groups,

i__nnthe theory of Riemann 66(1971), [2]

[3]

of Math.

Studies,

limit sets of Kleinian groups, Acta

13___~0(1973), 127-144.

Ahlfors•

L. V., Finitely generated K l e i n i a n groups,

J. Math.• [4]

Ann.

1-5. , Residual

Math.,

surfaces,

Advances

Bers,

8_66(1964), 413-429;

L., Simultaneous

Amer.

8_~7(1965), 759.

uniformization,

Bull. Amer. Math.

So e., 6_~6(1960), 94-97. [5]

. . . .

Inequalities

groups,

J. Analyse Math.,

[6]

• On b o u n d a r i e s Kleinian groups

[71 [8]

for finitely g e n e r a t e d Kleinian

I, Ann.

of TeiehmOller

London Math.

Bets,

L. and B. Maskit,

in Contemporary Functions,

moduli,

Soc., 4(1972),

and Kleinian

Moscow,

groups,

257-300.

On a class of Kleinian

Problems

Nauka,

spaces and on

of Math., 9_~i(1970), 570-600.

• Uniformization, Bull.

i_88(1967), 23-41.

groups,

in the Theory o__fffAnalytic (1966), 44-47.

14

[9]

Ford,

L. R., A u t o m o r p h i c Functions,

New York, [I0]

2nd Ed.

(Chelsea,

1951).

Griffiths,

P. A.,

Complex a n a l y t i c p r o p e r t i e s of certain

Zariski open sets on a l g e b r a i c varieties, Ann.

of Math.,

9_~4(1971), 21-51. [ii]

Kra,

I., A u t o m o r p h i c Forms and K l e i n i a n Groups, W. A.

Benjamin, [12]

Maskit, Math.

Reading,

Massachusetts

B., On Klein's c o m b i n a t i o n theorem I, Trans. Amer.

Soc.,

[13]

120(1965),

499-509.

, On Klein's c o m b i n a t i o n theorem II, Trans. Amer. Math.

[14]

Soc.,

131(1968),

32-39.

, On b o u n d a r i e s of T e i c h m ~ l l e r spaces and on K l e i n i a n groups II, Ann.

[15]

(1972).

Maskit,

of Math.,

9_~i(1970), 608-638.

B., On Klein's c o m b i n a t i o n theorem III, A d v a n c e s

in the theory of R i e m a n n surfaces, 66(1971),

297-316.

Ann. of Math.

Studies,

2.

Q U A S I C O N F O R M A L M A P P I N G S AND U N I F O R M I Z A T I O N C.J. Earle* Corne!l U n i v e r s i t y

Since the p i o n e e r i n g work of Teichm~ller, maps have p l a y e d a aignificant

role in t h e

quasiconformal

study of R i e m a n n

surfaces and F u c h s i a n and K l e i n i a n groups.

In this brief talk

I want to survey several aspects of the theory and indicate some a p p l i c a t i o n s

to K l e i n i a n groups.

§I. i.i

Let

D

and

D'

Q U A S I C O N F O R M A L MAPS be domains in

p r e s e r v i n g homeomorphism.

For each

@

and

z

in

f:D ~ D' a senseD

set

H(x) = lim sup r ~ 0 where L(z,r)

= max{If(~

z,r)

we say that H(z)

= min~If(~)

f

- ~(z)I;I~

is q u a s i c o n f o r m a l

is a b o u n d e d f u n c t i o n on

for every (K-qc)

) - f(z)I;l~

in

z

in D

D).

The

qc

if and only if

D

(qc)

- z I =

r)

- zl

=

r}°

in

D

if and only if

(a fortiorl, H(z) mapping f

H(z) ~ K

is

is finite

K-quasiconformal

for almost all z

in

f

The author thanks the N a t i o n a l Science F o u n d a t i o n for f i n a n c i a l

support through Grant GP-28251.

16

D.

Obviously any

1.2

Suppose

morphism.

qc

mapping is

f: D ~ D'

Let

fz

and

is a sense-preserving f~

= ~-~

f~

= ~

C1

K.

diffeo-

be the complex derivatives

li~f

fz

Since the Jacobian

K-qc for some (finite)

-

i/~f

i~f]

~-'

.Sf

+ ~f).

ifz 12 - if[l2

of

f

is positive, we

see that L(z,r) = r(Ifz(Z)l + If~(z)I) + o(r),

~(z,r) = r(i~z(~)l H(z)

Therefore the

(1)

ifz(Z)i + rf~(z)P Ifz(z)1 -If~(z)I

=

CI

diffeomorphism

f

is

K-qc

if and only if

rf~(~)li~-~f fz(Z)i

for all 1.3

- if~(z)i)'+ o(r),

z

in

D.

We want to extend the criterion (I) for quasiconformality

to the general case. generalized

We need to recall the definition of

(distribution) derivatives.

We say that D

if and only if

D

satisfying

f

has generalized derivatives fz

and

f~

are locally

L2

fz' f~

in

functions in

17

0 : 5(

fz + f z)dXdy : SS( f

for all smooth functions

$

+

)dxdy

with compact support in

D.

The

analytic, definition for quasiconformal maps states that the homeomorphism

f: D ~ D'

is

generalized derivatives in where.

K-qc D

if and only if

satisfying

1.4

has

(1) almost every-

The equivalence of this definition with our first one

is proved, for instance, in Lehto - Virtanen 4.

f

Notice that

1-qc

maps are conformal, by Weyl's lemma.

We list some useful properties of

Proposition

[5], §4 of Chapter

(see [i], [2], or [5]).

qc

Let

mappings. f: D ~ D'

be

Then (a)

f

is differentiable a.e.

(b)

Ifzl > 0 a.e.

(c)

mes(f(E)) = ~ E ( I f z 12 - If~]2)dxdy for all measurable sets

(d)

f-l: D' ~ D K'-qc,

then

§2. 2.1

If

f

is

qc

in

is

K-qc.

gof

is

If

g:D' ~ D"

KK'-qc

in

is

D.

BELTRAMI EQUATIONS D,

then

equation (~)

E c D.

f~ = ~fz

f

solves the Beltrami

K-qc.

18

in

D,

where L~

whose such

~

~(z) = f~ (z)/fz(Z)

norm in

D

is a measurable

is less than one.

there is a qc

map in

D

function

Conversely,

which solves

(2).

for any Further,

this map can be chosen to depend nicely on the p a r a m e t e r To be more explicit, the Banach space

let

M(~)

~.

be the open unit ball in

L~(~,C).

Theorem

(Ahlfors-Bers

unique

qc

ma~

w~

[2]):

For each

o_~f C

and solves the equation

~

i_~n M(~)

there is a

onto itself that fixes zero and one

(2).

For any fixed

~

i__nn C

the map

~-....-.>~(~) is a holomor~hic

,,~(¢) where

o(II~II )

function on

= ¢ + ~(¢)

M(~).

Ex~licitely we have

+ o(ll~ll),

~o,

is u n i f o r m on compact

subsets

of

¢

and

P~

i_As giv e____~n b_z

2.2

For any domain

which vanish a.e. is a of

qc

self-map

D,

let

M(D)

in the complement of

{

be the set of of

D.

If

which is conformal

Z

in

~ ~ M(D),

M(@) w~

in the exterior

D. If

~ ¢ M(U)

we denote by

of the upper half-plane solves defined

(2) in by

U.

Notice

U

w

the unique

that fixes that

w

= w v,

0,i,

and

where

qc

self-map ~

and

v ~ M(C)

is

19

(3)

~(z) = ~(z),

~(~)= ~(z)

Further,

w ~ o ( w ) -1

is the conformal

that fixes

0,I,

and

§3. 3.1

Qc

Let

(Wn)

EXTREMAL

SELF-MAFS

maps have the following be any sequence

Wn = w n.

on compact

z ~ U.

map of

U

onto

w~(U)

~.

in

i

qc

map

compactness

Wn. 0 and

w ~,

property.

with for all

Then a subsequence

sets to a

OF U

important

M(C)

<_ k<

II ~nll

Let

for all

n.

converges

uniformly

II ~II <_ lim infll ~n.ll. J-~ 0

3.2

We call the

II ~II ~

II ~II

real axis.

qc

map

for all

Given any

that there

in terms

~

if

introduce in ~(z)

N.

w

on

U

M(U)

extremal

if and only if

with

= w

w

annihilator such that

property

to describe

coefficients

[4] gave an analytic

space

on the implies

with the same boundary values.

importance

is extremal.

w

the compactness

of their Beltrami

the Banach

Its

:U ~ U

in

is an extremal

R.S. Hamilton by

~ w ,

It is of considerable w

w

A(U) -L

of

~.

condition

To formulate A(U)

the extremal maps

LI

consists

satisfied

this condition

we

holomorphic

functions

of the

functions

L~

20

0 = ~u~(Z)0(z)dxdy

in

M(U

satisfies Hamilton's

(4)

Reich and Strebel

tremal if and only if We call

w

~

~ ~ A(U).

condition if and only if

li~ll ! l l ~ + o l l

Recently,

3.3

for all

for all

o ~A(U)-

[6] have proved that

satisfies Hamilton's

a Teichm~ller mapping and

w

L

is ex-

condition.

~

a Teichm~ller

differential if and only if

=

where

0 ~ k < i and

k~/l~l

~

is a holomorphic function in U.

follows from Reich and Strebel (Strebel

[7]) that

w

It

[6], or by direct arguments

is extremal if

~ e A(U).

It is an

important unsolved problem to give necessary and sufficient conditions on

§4. 4.1

~

to make

w

BELTRAMI COEFFICIENTS FOR A GROUP

For any Kleinian group

a~

=

G

~G(~)-I

is a group of homeomorphisms ~(O)(G)).

extremal.

In order that

~

and any

= [~

of

A C,

~,

the group

o g o (~)-i

g c G}

acting discontinuously

on

be a Kleinian group, it is necessary

21

and sufficient G.

in

to have

This happens

(5)

~

,go(w~) - I' *--

Z

M(G,Z)

M(G,Z)

support in

Z.

the set of

For each such

holomorphically qc

If

G

the set of

deformations

of

the group

G.

~.

of

~

be Fushsian.

~

in

However, w~(~)

quasi-Fuchsian

is Kleinian.

of

G~

depend

G~

are

M(G,~), depends

we denote by If

so

M(G)

~ ¢ M(G),

then

G~ = w~G(w~) -I

real analytically

depend real analytically

obtain the family of For most

~

with

G.

(5).

Since

G~

G

(5)-

They play a central role in

satisfying to

G.

for

The groups

W

M(U)

of

satisfying

coefficient

U,

the elements

4.3

~,

M(Z)

is a Fuchsian group on

a Fuchsian group.

and

in

of the group

defined by (3) belongs

~(U)

W

on the parameter

in

g

g ¢ G.

easily that the elements

the study of variations 4.2

for all

is called a Beltrami

Theorem 2.2 implies

called

= ~

be any invariant union of components

We denote by in

for all

if and only if

(~°g)~'/g' Let

conformal

on

~,

on

is ~,

and we

qc Fushsian deformations

G

of

M(G)

G~

will not

the Kleinian group

every element of

onto themselves.

G~

G.

will map the sets

These groups

G~

are the

groups.

This construction

of quasi-Fuchsis~a groups,

due to Bers,

leads easily to a proof of the theorem on simultaneous ization mentioned

in the first lecture.

uniform-

22

Let g ~ 2.

SI

and

S2

be closed R e i m a n n surfaces

Choose a F u c h s i a n group

mirror image of

S I.

so that

U/H

m o r p h i s m of

onto

w

G~

is a q u a s i - F u c h s i a n

is

G.

Let

S 2,

f.

wu(L)/@

is

by

S 2.

of

and

Then

L

is

S~

qc,

all

properties

f

for

G

and let

w: U ~ U

~ = w~/w z group.

G

belongs

Obviously

them. ~

G

on

U.

w

= wv

on

Hamilton satisfies

finitely generated

w

G.

If

G

is a quadratic

holomorphic

in

U

and

is

S 2,

~ ~ M(G)

II ~II ~ Again,

II vll

for

compactness

exist, and it is important

an appropriate

U/G

we

w

analog of (4).

by Strebel

is compact theorem

is extremal The

[8] for

(or has finite

[3] tells us that

if and only if

= ~T/I~I, ~

diffeo-

M(G),

[4] p r o v e d that if

area), Teichm~ller's

is extremal for

where

to

If

~.

converse has recently been established

noneuclidean

of

be a lifting

w~(U)/G ~

if and only if

such that

then

the

is the quotient

be a sense-preserving

ensure that extremal maps

to describe

S~,

EXTREMAL MAPS F O R A GROUP

extremal for

~ ~ M(G)

is

Choose another Fuchsian group

Consider a F u c h s i a n group w

SI

U/G

S 1.

§5-

call

so that

Then of course,

the lower half-plane H

G

of genus

o k k<

l,

differential

for

G.

That is,

all

g ~ G

~

is

and satisfies

~(g(z))g.(z)2=~(z),

an~

z E U.

23

REFERENCES

[i]

L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand-Reinhold, Princeton, New Jersey, 1966.

[2]

L.V. Ahlfors and L. Bers, Riemann's mapping theorem for variable matrices, Ann. of Math., 72(1960), 385-404.

[3]

L. Bers, Quasiconformal mappings and Teichmuller's theorem, in Analytic Functions, pp. 89-119, Princeton University Press, Princeton, New Jersey, 1960.

[4]

R.S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc., 138(1969), 399-406.

[5]

0. Lehto and K.i. Virtanen, Quasikonforme Abbildungen, Springer-Verlag, Berling, 1965.

[6]

E. Reich and K. Strebel, Extremal quasiconformal mappings with given boundary values, to appear.

[7]

K. Strebel, Zur Frange der Eindentigkeit extremaler quasiknoformer Abbildungen des Einheitskreises II, Comment. Math. Helv., 39(1964), 77-89.

[8]

K. Strebel, On the trajectory structure of quadratic differentials, in Discontinuous Groups and Reimann Surfaces, pp. 419-438, Princeton University Press, Princeton, New Jersey, 1974.

3.

AUTOMORPHIC

FORMS

AND

EICHLER

COHOMOLOGY

Frederick P. Gardiner Boston College

One method of obtaining information about a Kleinian group is to study the p r o p e r t i e s of its automorphic forms and E i c h l e r cohomology groups.

This chapter shows how this method leads to

Ahlfors' finiteness t h e o r e m and B e r s ' a r e a t h e o r e m s . A good listing of the a r t i c l e s and r e s e a r c h done in r e l a t e d a r e a s can be found in the bibliography to [6].

§1.

AUTOMORPHIC FORMS AND POINCARE THETA SERIES

In t h i s s e c t i o n we a s s u m e

that

I" is a n o n e l e m e n t a r y

Kleinian group acting discontinuously on a connected open subset A of

C.

In a p p l i c a t i o n s ,

h w i l l b e a n i n v a r i a n t c o m p o n e n t of D(F),

of discontinuity of F.

Under t h e s e conditions,

the set

& is a hyperbolic

Riemann s u r f a c e and has a Poincar~ m e t r i c ~ = XA. If f is a holomorphic mapping from D1 to D2 and ~0 is a function defined on D2, defined on D1.

then fp, * qq~(z) = ¢p(f(z))f'(z)Pf'~q is a function

In this definition p and q must be h a l f - i n t e g e r s and

p + q an integer.

We also use the convention that f = f p p, o

W i t h t h i s n o t a t i o n t h e i n v a r i a n c e p r o p e r t y of k expressed

by

can be

f l / 2 , 1/2Xf(&) = X& f o r any c o n f o r m a l m a p p i n g

f.

25

A measurable function

automorphic

~ with the property

two Banach

spaces

measurable

functions

that

of such forms U on

form of weight

yqU = U f o r a l l for any integer

such that

A

(-2q)

y 6 F. q > 2.

Yq~l =U a n d

in

A is a

We define L®(A, F) = a l l q

II~ll~ < ~

where

II~II~= sup [x-q(z)l~(z)13. Z

L 1 (A,F) = all m e a s u r a b l e q

II~ll r < ~

functions

v

on

A

s u c h that

y *v = v q

and

where

ll~llr=ff~2-q I-I Idz^~l oo

and

•

is any fundamental

domain for

F

in

A.

We shall require that all definitions of spaces that we make , be invariant under f whenever f is a conformal mapping. q In particular,

i f ~ ~ A,

to be the closed subspaces Lq(A,F),

respectively.

is holomorphic as

z~

Remark.

of holomorphic

When

~ E A,

co t r a n s l a t e s

B (A,F) q

functions in

and

A (A,F) q

L~(A, F) q

and

the condition that a function

into the condition that

%0

~ ( z ) = 0( Iz 1-2q)

~.

In t h e c a s e t h a t

A (A) i n s t e a d o f q A2(U)

at

we define

B (A,F) q

F

is the trivial group we write

and

coincides with the space

A (A,F). q A(U)

and

A =U,

the space

by Earle

in Lecture

Ifwelet

introduced

B (h) q

2. T h e d e f i n i t i o n s of t h e s e s p a c e s

generalize

to the ease where

26

A

is replaced

is defined

q

(F,F)

XA.

the Petersson

the spaces

is a pairing

to any component

LI(E,F), q

L~{E,F) q

and

A A

q

between

I.

antilinear

(z, r),

between

LI(E,F) q

and

L~(E,F) q

given by

domain

F

in E.

that this pairing

the dual of

will need

for

standard

melhods

establishes

LI(E, F) q

the following

By

and

theorems

an antilinear

L~(Z, F). q about automorphic

(Bers

[5])

isomorphism

The

Petersson

between

scalar

13 (E,F) q

product

establishes

and the dual space to

A (~, F).

q

Definition.

Let

theta series

is defined by

~p be a holomorphic

®~(z)=

~

function

~(¥(z))¥,(z) q,

in E.

The

Poincar6

zEE

yEF

compact

of

and theta series.

Theorem

whenever

kZ

product:

one can show

We

of Q(F).

just as before.

scalar

isomorphism

union of components

that its restriction

tu is a fundamental

in analysis

forms

an invariant

Then

are defined There

where

Z,

by stipulating

be equal to B

by

the right side converges subsets

of E.

absolutely

and uniformly

on

an

27

Theorem

2.

® : A ( E ) - - > A (E, F) q q

linear mapping. such that

To e a c h

®~ = ~

Definition.

Let

~ E A (E,F) q 2q- 1

there corresponds

S = A/F b e of f i n i t e t y p e .

parabolic punctures restored. a puncture,

surjective a ~ E A (E) q

i ~ - ~ - _ 1 I1¢11r.

IMI

and

is a n o r m d e c r e a s i n g ,

the o r d e r of p

For each

if p

A S be

Let

A p E S,

let

S with the

~(p) = ~

is a ramification point, and

if p

is

1

otherwise.

The

space

of cusp

forms

automorphic

q-forms

differential

A S,

(Here,

on

as usual,

convention

that

(Ahlfors

and

Theorem and

p

3.

such

a pole

[i], page

at

as

the set Im

If A/F

--

the dimension

that, p

~p is viewed

of order

Notice

that

< x

and

~(p)

we

= I

as a

q [q - --~].

at most

A If p C S - S,

416)

z~

[z

U [Im

use

the

implies

~

is

that

z > c}

and

is of finite type, then

of these

spaces

A

q

there

exists

U-Iyu(z)

= z + 1

such

7r(U(z))

that

(A,P) = B

is

A pES A g = g e n u s (S).

such

then

~.

(2q-1)(g-1) + E

where

of holomorphic

when

integer

transformation

contains

approaches

is the space

p. )

a M~bius

U-I(A)

has

A

[q - q] = q - I.

Lemma

and

(A, F)

[x] = the largest

at

Y E F

q

¢p in

~

holomorphic

I.

C

[q - ~--~p}l

q

(A,F) = C (A,F) q

28

Lemma

2.

Suppose

A/F

is of finite type.

on a h o l o m o r p h i c a u t o m o r p h i c

q-form

1)

sup { l - q ( z ) ] ~(z)l} < ~ , zEA

2)

f f ~ . 2 - q l ~ o [ [dzA dz[ < ~ , w

3)

If l i m z 124~

= { where z

n

n

The

following

conditions

in A are equivalent:

is contained in cusped r e g i o n

A

b e l o n g i n g to a p u n c t u r e on S a n d ~ E A t h e n l i m q0(zn) = 0. rl-.i,~ C o m p l e t e p r o o f s to t h e s e r e s u l t s a r e c o n t a i n e d in K r a [61.

§2.

Let J 2q-2.

If F

-~2q-2

-~2q-2

COHOMOLOGY

be the vector

is a group

of MSbius

on the right by defining

v E T~2q_2

and every

ad-bc

then

= i,

see that more,

EICHLER

v.

y

y E F.

v • (YlY2)

= (V.Yl) Under

cohomology

first eohomology mapping

P : F-->

P: F-->

then

If y(z) = (az+b)(cz+d) -I and,

F

acts on

for every where

using this fact, it is easy to v

is in -~2q-2"

it is easy to check

circumstances,

Further-

that

HJ(F,-~2q_2 group

and now

-~2q-2

). We

one can form

T[2q_2

the group

will only be concerned

give an explicit description

is called a

l-coeyele

PY1Y2 = PY1 ' ¥2 + PY2 f o r a l l ¥1 a n d ¥2" mapping

of degree

• Y2"

these

groups

transformations,

whenever

by use of the chain rule,

of polynomials

v . y = v(y(z))y'(z) l-q

y'(z) = (cz+d) -2 is in -~2q-2

space

of the f o r m

A

with the of it.

if

1-eoboundary is a

P¥ = v . y - v.

One e a s i l y

A

29

checks

that any

1-coboundary is a

1-cocycle.

The first cohomology

group

H I ( F , ] - [ 2 q _ 2 ) i s t h e v e c t o r s p a c e of 1 - c o c y c l e s f a c t o r e d b y t h e

v e c t o r s p a c e of 1 - e o b o u n d a r i e s . Let

B be a M~bius transformation

and A F = B _ 1FB"

c o n j u g a t i o n b y B -1

induces an isomorphism

1n and H (F,-~2q_2).

T h e m a p p i n g is d e t e r m i n e d by s e n d i n g the c o e y c l e

P

PA w h e r e

into the eoeycle

¥ 6 F.

eoboundaries.

HI(F,-~2q_2 )

~ (B- 1¥B) = P ( y ) • B = B ~ _ q P ( y )

It i s e a s y to s e e t h a t t h e m a p p i n g

preserves

between

Then

for all

A P - - > P is i n v e r t i b l e a n d

It i s i m p o r t a n t to r e a l i z e t h a t if F 1 a n d 1"2

a r e K l e i n i a n g r o u p s a n d g : F 1 - - > 1"2 i s a n a l g e b r a i c i s o m o r p h i s m , t h e r e w i l l not, i n g e n e r a l , b e a n y r e l a t i o n s h i p b e t w e e n a n d H l ( F 2 , - ~ 2 q _ 2 ).

HI(F1,-~2q_2 )

The s t r u c t u r e of H I ( F , - ~ 2 q _ 2 ) d e p e n d s on t h e

g e o m e t r i c m a n n e r i n w h i c h F is a s u b g r o u p of the full M~Sbius g r o u p . H o w e v e r , one c a n find a b o u n d on d i m H I ( F , ] - [ 2 q _ 2 ) t e r m s of the n u m b e r of g e n e r a t o r s

L e m m a 3.

Suppose

Let

vE T[2q_2

would mean that

of F.

q >_ 2 a n d I" is a n o n e l e m e n t a r y K l e i n i a n g r o u p

generated by N elements.

Proof:

in

Then

dim HI(F,]~2q_2)<_ (2q-1)(N-1).

and assume that

v ( y ( z ) ) y ' ( z ) 1 - q = v(z)

5 v = v . y - v = 0.

for e v e r y

y i n F.

v(z 0) = 0 t h e n v(Y(Z0)) = 0 for a l l y a n d t h i s i m p l i e s infinitely many zeroes since

F is n o n e l e m e n t a r y .

T h u s if

v has

Therefore,

i n j e c t i v e a n d t h e s p a c e of c o b o u n d a r i e s h a s d i m e n s i o n t h e d i m e n s i o n of - ~ 2 q _ 2 ).

This

2q-1

5 is

(which is

S i n c e a e o c y c l e is u n i q u e l y d e t e r m i n e d by

i t s v a l u e s on g e n e r a t o r s of F,

t h e d i m e n s i o n of t h e s p a c e of c o c y c l e s

30

is < (2q-1)N.

We c o n c l u d e t h a t

Next, HI(F,-~2q_2 A(F)

As

introduce usual,

is an infinite set).

invariant for

).

we

F

under

with

support

I ((?-~) = 0,

coefficients

assume

Let

in

~

E F.

we

mean

that

k2-q(z).

(For

A continuous

function

~F

(2.1)

~

By

= U in %he sense

elements

coefficient

function

for all these

of

Beltrami

a measurable

y E F

of

(so that

of components

a generalized

and

~i,

with

such

that

are the ordinary

Beltrami

2. ) F(z)

coefficient

to obtain

is nonelementary

q = 2,

in Lecture

Beltrami

F

Iu = ~

discussed

generalized

way

be a union

of

# ¥1-q,

such

I~(z) I <_ (const.)

an analytic

we

the action

d i m H I ( F , - ~ 2 q _ 2 ) <_ ( 2 q - 1 ) ( N - 1 ) .

u

will be called

a potential

for the

if

of distribution

theory

5z

and if

(2.2)

F(z) = 0(Izl2q-2),

If F

is a potential

potential then

~.

~

and

Conversely,

if v E-~2q_2, if F 1

and

then F 2

F + v

is also a

are potentials

for

~,

F 1 - F 2 E TV2q _2.

Remark. ]Beltrami A

for

for

z-* ¢o.

* = Al_q,

A * F = AI_qF

If A

is a M~Sbius

coefficient i~

for

F,

is a generalized is a potential

for

transformation, and

F

a potential

]Beltrami A Li.

~

a generalized for

coefficient

~, for

then A

IFA

and

31 Lemma

4.

If ~

nonelementary

is a generalized

Kleinian group

a I, a2, .... a2q_l

(2.3)

F(z)

=

is a potential

Beltrami

F with s u p p o r t in Z,

are distinct points

in

A(F),

for the

a n d if

then

~t(~)d~ A d~ (~-z)({-a I) ... (C-a2q_l)

( z - a 1) ,.. ( z - a 2 q _ l ) f f 2wi Z

for

coefficient

U. /k

Proof:

Let

A b e a MiSbius t r a n s f o r m a t i o n

and l e t

~I-"

One

F = AI_qF.

f i n d s by c h a n g e of v a r i a b l e t h a t

A

F(z)

where

=

A~a(¢)d~ A d-~

A-I(~)

A A (~-z)(~-a I) ... (~-a2q_l)

a n d Aa.j = A - l ( a j ) '

a n d the r e m a r k

generality that

ff

2M

A u = A l *_ q ~

observation

A (z-a2q_l)

(z-Aa1) . . .

~ E f2 a n d

j = 1. . . . .

conclude that in ( 2 . 3 ) is

A is b o u n d e d .

z ~ a..j

F(a.) = 0, J

F(z) F We

satisfies

~(z) = 0 ( I z I 2 q - 4 ) ,

Moreover,

therefore,

It i s s i m p l e c o n s e q u e n c e

1/X (z) = 0(lz t2),

H e n c e , f r o m the f a c t t h a t

0([C [-4)

z -* ~.

(see Kra

ha I < ( e o n s t . ) x 2-q,

we

It f o l l o w s t h a t the i n t e g r a n d

and the i n t e g r a l c o n v e r g e s a b s o l u t e l y f o r a l l the i n t e g r a l is

0([z-ajl

becomes must

Usingthis

a b o v e , we m a y a s s u m e w i t h o u t l o s s of

of S c h w a r z ' s I e m m a t h a t if = E f~, t h e n [6], p a g e 168).

2q-1.

log[z-ajl

continuous show

(2. I) and (2.2).

0(llogIz-ajl

that By

F

at

) as

) as

z-~ a.. J

z-* a.j and,

Thus

by letting

a..

is continuous

elementary

everywhere

estimates

and

one shows

32

F(z) = 0(]z]2q-2)

as

variable formula,

one c a n s h o w that,

constant

C(R)

z-b ~ and, by r o u t i n e u s e of t h e c h a n g e of for every

R > 0 t h e r e is a

such that

I F ( z ) - F ( w ) [ < C(R) [ z - w I l o g l z - w l [

whenever

lzl

and

twt < R.

It remains tions.

Let

support. every

(See K r a [61 , p a g e 137-141.)

to show

that

8F/Sz

~0 be a test function, We

must

show

test function

~p.

Let

If F~o~ dz A d---z -

=

that is, a

- ff

that

= ~

F~

in the sense C°

function

dz A d-'~ = ff

~

p(z) ff

~ - ~

~(~)

(~-z)p(~)

with

compact

U~ dz A d---z for

p(z) = (z-a l) • • ' (z-a2q_l).

-

of distribu-

Then

d~ A d--~dz A d-'~

1 ff ~(~)/f ~(~(z)p(z)) dzA d-7 dC ^ d-~ = 2--~ p~ (z-~)

by F u b i n i ' s t h e o r e m .

By the general Cauchy integral formula,

i n s i d e i n t e g r a l in t h i s l a s t e x p r e s s i o n y i e l d s lemma

2vi ~0(~)p(g),

and t h e

is p r o v e d , Let

M (E,F) q

b e t h e s p a c e of a l l g e n e r a l i z e d B e l t r a m i

c o e f f i c i e n t s w i t h s u p p o r t in E.

T h e o r e m 4.

There is a canonical linear mapping

: MQ(Z.r)

> Hl(r,-ff2q_2).

the

33 Proof:

Given any

~/ 6 M ( E , F ) , q

form a potential

y 6 I" let P (z) = F(y(z))y'(z) l - q - F(z). Y

F

From

for

~.

For each

F(z) = 0(Iz12q-2),

one easily sees that P (z) = 0(Iz12q-2). Using the facts that ¥ A * 8F Yl-q, i.U = U and - ~ = ~/, one sees that : ~ P (z) = 0 for all z 6 C. 8z 8z Y Therefore

Py 6 T[2q-2"

cocycle relation.

it is easy to s h o w that Py

T h e choice of a potential

but obviously the cohomology

c l a s s of

It i s e q u a l l y c l e a r t h a t t h e m a p p i n g

P

F

satisfies the

for ~/ is not unique,

is uniquely determined

~(U) = ( t h e c o h o m o l o g y

by

U.

c l a s s of

P)

is linear. The next lemma when the cohomology

Lemma

5.

Let

gives us a useful condition for deciding

class associated

U 6 M (Z, F ) . q

to

~ is zero.

The following conditions are equivalent:

(i) ~(U) = 0.

(ii) ~ has a potential

F

# such that Y l _ q F

(iii) ~/ has a potential

F

such that F I A = 0.

= F

for all y 6 I'.

P r o o f : If (i) h o l d s , t h e n U h a s a p o t e n t i a l F 0 s u c h t h a t # # Yl_qF0 - F 0 = Yl_qV0 - V 0 for some v 0 E ~-[2q-2 and for every ¥ 6 F.

Thus

F = F 0 - v 0 has property

a potential

F

hyperbolic)

fixed point

and if y ¥'(~) ~ I.

such that

F(C)y'(~)l-q ~ 6 h.

is the loxodromic

(ii).

If (ii) h o l d s ,

= F(~)

But such fixed points are dense in

transformation

(iii) holds, then each polynomial A,

U has

for every loxodromic

with fixed point

O n e concludes, by the continuity of F,

vanishes on

then

~,

an infinite set, and therefore

P

(z) = 0. Y

A

then

that F I h = 0.

P (z) = F(y(z))y'(z) l-q - F(z) ¥

(or

If

34

functions that

Recall

that the elements

~

support

with

k-q(z)I~(z)l

obvious

that

in

E

< (const.).

of

B

such

(F,E)

that

Therefore,

ILl(Z) I < (const.)

q

are

)(y(z))y'(z) q = ~(z) if we

k2-q(z).

holomorphic

let

~

=k 2-

Furthermore,

and

2%,

one

such it is

can

easily

#

show

that

Theorem

Yl-q,

5.

1 ~ = U,

Consider

and

so

u E M

where

i(~) = k2-q~ - and

Proof:

What

generalized of

i(~)

must

A

q

and

z

/~ > H I ( F , - ~ 2 q _ 2 )

is defined

as before.

is that the cohomology

coefficient ~ E B

distinct points in

~

show

Beltrami

for s o m e

(E,F).

the mappings

Bq(E,F) i > Mq(E,F)

we

q

is the same

im(/3oi)

class

of any

as the cohomology

= im

~.

class

Note that if [a I ..... a2q_l } are

(E,F).

2q- 1

then

== E fL

1

Then

(z-al) " " "

• " (C-a2q_ 1 ) is an element

of

2, we can f o r m

Aq(f~) z

~p (C)

for each

z E A - [a I, .... a2q _I}.

= ®qZC)"

F(z)

=~/

theorem

L e t ~a be a g e n e r a l i z e d B e l t r a m i

c o e f f i c i e n t for F with s u p p o r t in E.

(2.4)

By

A potentiai

@z({)u(~)dg

F f o r ~a is given by

A d-~

C

= f f @z(c)~(C)d { A d--~ E By

using

the invariance

properties

of

U

and

the fact that

®

q

@

Z

Z

= ~0 ,

35 one c a n s h o w t h a t

F(z) = S [ ~oz(~)~C)d~ A-~.

(2.5)

z/r Since the restriction A (Z,I"), q

by the duality between

E B 0",F) q

=

F

coefficient

and completes

i(~) We

mapping?

now

raise

potential function (~,F).

F

k2-2q~

question:

is

is uniquely determined

a2q_l},

Suppose

for

we know there

exists

Beltrami

/~ o i an injective

to asking whether the cohomology

~ (~),

spana

as

c l a s s of t h e

by the automorphic

giving an answer, Z

If t h e f u n c t i o n s

z varies

dense subspace

/3 o i(~) = 0.

- such

that

F(~)

we prove

over the set

of Aq(5"),

then

F

ff

Then

by lemma

I A = 0.

z

In fact,

4, there F

is a potential

is given

by

/2-2q~-

which

2-2q

Z

since

both

the integral

vanish at the F(z) = 0

form

H I ( F , TT2q_2 ) i s i n j e c t i v e .

0 i : Bq(E,I')-->

F

B , q

of

t h e p r o o f of t h e t h e o r e m .

a crucial

As a step towards

6.

A - { a 1. . . . .

Proof:

and

q

is also a potential for the generalized

This amounts

Theorem

to E is an element

Sf ~z({)k2-2q(~)~-(C)d ¢ A d-~. z/r

This shows that

q

A

Z

such that

F(z)

E B

of ~

2q-I

and

points,

for all z E A,

F

are

potentials

a 1 ..... a2q_l.

the fact that , = 0

for

Since by hypothesis follows f r o m the

36

assumption that the functions

z

from the fact that the Petersson pairing between

Remarks and

1.

A (Z) a n d q

known

scalar product is a nondegenerate

we will show F

/3 o i i s i n j e c t i v e w h e n

finitely generated and

/3 o i is injective

whether

and

B (E). q

Ultimately,

/3 o i i s i n j e c t i v e f o r

s p a n a d e n s e s u b s p a c e o f A (~) q

when

F

q >_ 2.

is infinitely

q = 2

It i s n o t

generated

and

q>2. 2. shown

that

By

HI(F,T~2q_2

that it is the direct of Eichler Kra's

introducing

integrals.

papers

the space

) is the middle

sum

of the space For

§ 3.

term B

an exposition

on this subject,

see

of Eichler

q

(Z,F)

integrals,

of an exact

of this theory

and

has

sequence

and an appropriate

and space

a listing of

[6].

AN APPROXIMATION THEOREM

In t h i s s e c t i o n w e w i l l c o n c e n t r a t e o n t h e c a s e before,

Kra

q = 2.

As

let

1 (z-al)(z-a2)(z-a3) z(.~) 27r---i(C-z)(C-al)(~-a2)(~-a3) =

where

a I,

a 2,

varies

over

the set We

when

q = 2,

a3

have

are three A - [a I, seen

it suffices

distinct fixed points

a 2,a 3}

A

and

z

g

that in order to prove

in

to prove

the following

that

~ o i is injective

theorem.

37

Theorem

7.

( B e r s [2])

Kleinian group

I',

Let

fl b__~eth__eelimi_____tse___ttof a n o n e l e m e n t a r V

(I" m a y b e i n f i n i t e l y g e n e r a t e d ) .

2,a3]

~z(C) w h e r e

z 6 h - [a 1,a

Remarks

O b v i o u s l y the t h e o r e m

1.

a r e d e n s e in A2(E) w h e r e b e c a u s e in t h i s c a s e 2.

T h e n the f u n c t i o n s

s p a n a d e n s e s u b s p a c e of A2(fi). i m p l i e s t h a t the f u n c t i o n s

tz(~)

E i s any u n i o n of c o m p o n e n t s of ~,

A2(E) c A2(i~ ).

For generalizations

of t h i s t h e o r e m

s e e B e r s [2] o r K r a

[6].

Proof:

Suppose £ is bounded l i n e a r functional on A2(fi).

To prove

the theorem, we m u s t show that, if £(~z) = 0 for each z 6 A - {a 1,a 2 , a 3 ] ,

then £(~0) = 0 for all ~0 6 A2(D). Of course, one

can find a bounded m e a s u r a b l e function ~ with support ~ such that

(3.1)

£(~) = f f ~(~)~(~)d~ ^ d-~ for all ~0 6 A2(Q).

Now we let F be a potential for the function ~.

(3.2)

F(z) = f f

In p a r t i c u l a r , let

~z (~)~(~)d~ A -d~.

The hypothesis t e l l s us that F(z) = £(~z) = 0 for all z 6 A - {a 1,a 2 , a 3 ] . ~F F is continuous and -~_ = ~ in the sense of 5z

Recall that by l e m m a 4,

distributions. To show that ~(~) = 0,

(3.3)

~(~) - - I f 5F = -ff

we t r y to argue as follows:

dC ^ - ~ = f f ~(F~)dz^~

g(F~0 dz)

=f

Fop dz

=

0.

38 If w e k n e w fact that

~

to be c o n t i n u o u s a n d

F = 0 on

these hypotheses

~

would make this argument

late all the variables

B ( ~ ) = a 3.

B.

difficulties,

in t h e p r o b l e m We choose

B

A F = B-1FB

By letting

to the statement

valid.

the

However,

are not satisfied.

To get around these

transformation

~Q t o b e s u f f i c i e n t l y s m o o t h ,

o u r f i r s t s t e p is to t r a n s -

by t h e a c t i o n of a M S b i u s so that

B ( 0 ) = a 1,

and ~ = B

1 ,

B ( 1 ) = a 2,

the theorem

and

reduces

that the functions

1 z(~)

z(z-l)

2~i (C-z)(~)(~-i)

span a dense subspaee of A2(Q) where 0, 1,~ 6 A and z 6 A - {0, i , ~ ] . The function F(z) is now given by

F(z)

(3.4)

=

ff

~z(~)U(C)d C A d-~

= z(z-l) [~ U(~)d~ A d~ 2wi %~ (~-z)({)({-l) "

Lemma

6.

F(z) i)

F

defined by (3.4) has the following properties: is c o n t i n u o u s on

C,

~F

ii) --_ = ~,

%z iii) F(z) = 0(Iz I logIzl), iv) for every

R > 0,

z ~ ~,

there exist

and C(R)

JF(z)- F(w)[ <_C(R) Jz-wl logrz-wJ This Lemma

4.

lemma

is proved

by the same

such that

for all

methods

f zr, rw l < R .

used

to prove

39

Let distance

5 (z) b e t h e s m a l l e r

from

z to

5ft.

the above lemmatells Iz I < R . on

Since

us that

of t h e n u m b e r

F(w)=

0 for

e

w in

and the

5~,

Note that this inequality tells us not only that

is this fact which makes

it possible

F

j(t)

j(t) = 0 for

integer

and

n

for

t<

z E f?,

5(z) < e

1 and

set

-2

,

F

t>

2.

n

1 ")" log log 5(z)

so that the

log iog

since

For each

(Note that we

term

in w

n

Since j ' ( t )

I6(z) - 6(w) l < [ z - w l ,

and we can compute

~z

n

a7

would be

(log l o g 5

1 -1)2 '

5__5_5a r e b o u n d e d , 5g

log 5

way.

1

~5

-------~ . 5 ~

' 5z

we c o n c l u d e

i

n

< (eonst.)

--

5 has generalized

in the ordinary

n = J'( " )

and

(log log 5-1) 2

0

log 6 -I

1

5(z)

Also we may assume

1 < n ( l o g l o g 6 - 1 ) -1 < 2 s i n c e o t h e r w i s e

vanishes.

we a r r i v e

(3.5)

Therefore

in (3.3)

n

Moreover,

derivatives

It

0 <__j(t) <_ 1 f o r a l l

j(t) = 1 for

Wn(Z) = j (

~D.

mollifier.

--

defined.)

vanishes

t o p a t c h up t h e a r g u m e n t

be a smooth function with

t E IR a n d

I for

for points near

b y t h e u s e of a n i n g e n i o u s d e v i c e k n o w n a s A h l f o r s ' Let

part (iv)of

I F ( z ) [ <_C(R) 5 ( z ) l l o g S ( z )

5f? b u t a l s o p u t s a b o u n d o n t h e s i z e of

arranged

-2

5Wn

~-

j'

at the inequality

<_( c o n s t

)1

1

n 5 log 6-1"

It follows f r o m the definition of ~n

that ~n(Z) = 0 w h e n e v e r

40 n

6(z)<_exp(-e ). Let D ( R ) = ~ N [ z

F(R) = a fl {z I IzJ = R}.

(3.6)

f

D(R)

ff

Wn%oFd z -

I ff D(R) Since •

n

%oF

d~

~

A dz I < (const.) f f

I,

we s e e that

l%o(z)-q----

D(R)

1 in b o u n d e d p o i n t w i s e c o n v e r g e n c e ,

t a k e the l i m i t in (3.6) as n-* ®,

(3.7)

%oF ~Wndz ^ d---z.

IF(z)[ < C(R)5 ( z ) I l o g 5(z)

--

approaches

~z

D(R)

8z

we find

wn G ( % o F ) d z A d--z

= ~

F(R) B y (3.5) and the fact that

and

Applying Stoke's theorem,

f f ~n%o~/ d z A ~ D(R)

=-

I [z/
I ff

if we

we find that

%oudz^d--{zl <_ I f

D(R)

%0(z)F(z)dzl.

F(R)

By part (iii) of lemma 6, the right hand integral is less than a constant

time s

(3.s)

Rloga

f

I%ol Idol.

r(R) Notice that ~0 { f

I%oI I dz I}dR = ~

r(R)

n

I:PI Idz ^ ~zl 2 <®"

If (3.8) is a l w a y s l a r g e r t h a n a p o s i t i v e n u m b e r

ff

I%o dz ^ --d~I >- --J9 ~ ~R log R dR = ~. T h e r e f o r e ,

6,

then

we c o n c l u d e that

n

l i m inf f

IF%o[ Idzl = 0

R-*~ F(R) f r o m which it follows that f f %OUdz A d-~ = 0 w h e n e v e r f~

%oE A2(f~).

41 This completes

the proof.

Remark.

It i s u n k n o w n w h e t h e r t h e a n a l o g o u s t h e o r e m

valid for

q > 2.

8.

AHLFORS'

( A h l f o r s [1])

Kleinian group,

7 is

S e e K r a [6].

§4.

Theorem

to theorem

then

FINITENESS

If r

fl(r)/r

THEOREM

is a finitely generated

nonelementary

is a finite union of Riemann

surfaces

of

f i n i t e type.

Remark.

This theorem

components.

does not say that

In m o s t c a s e s t h e n u m b e r

~(F) h a s f i n i t e l y m a n y

of components

of fi(F) i s

infinite.

T h e p r o o f of t h e f i n i t e n e s s classical

Lemma

theorem

depends on the following

lemma.

7.

Let

A be a component of ~(F) and let F 1 be the sub-

group of F which leaves

& invariant.

Then

A2(&,F I) has finite

dimension if and only if &/F 1 is of finite type.

Proof:

By classical

can construct

function theory

an infinite dimensional

abelian differentials

on

( s e e K r a [6], p a g e 3 2 4 - 3 2 8 ) o n e s p a c e of s q u a r e

integrable

S = A/F 1 if S has infinite genus or if S has

infinitely many elliptic points or punctures. these abelian differentials

T h e p r o d u c t of a n y t w o o f

will be in A2(&,F1).

i s of f i n i t e t y p e t h e n t h e c l a s s i c a l

Riemann-Roch

Conversely, theorem

if ~/r 1

asserts

that

42 dim

A2(A,F

terms

I) < =

and gives

of the signature

Remark.

of

It is possible

an explicit formula S

for

A/F 1 is a thrice punctured

Corollary.

If F

component

Proof: single

E

B2(A,F

F

which

Let

by lemma

invariant.

By theorems

6 and 7,

Since by lemma

(2, 3, 7).

group,

of Q

then each

which

cover

component Then

of E

a and

of theorem

components.

following

theorem.

Theorem

g.

B2(E,

HI(F,-~2)

F)

has finite

and these

equals

>

spaces

E/F),

have

is of finite

Let

of q/F

F

8, we must

To accomplish

be a nonelementary

is of finite type.

thai

Q(F)/F

this, we prove

Kleinian Then

show

group

~ o i :B

q

the

such that each >

(Q,F)

) is an injection. R

Proof:

Let

w

be a fundamental

region

in Q

for

F.

w=U

wi i=l

where

R

F1

obviously

/3 o i :

3,

(which

when

7.

has finitely many

HI(F,-~2q_2

A/F 1

example,

of finite type.

B2(A ,FI) = A2(5 ,FI)

To finish the proof

component

surface

A

Therefore

For

Kleinian

5 be a single

leaves

we conclude

finite dimension. type,

I) = 0.

with signature

is a Riemann

) is injective.

dimension,

sphere

of Q/F.

I) = B2(E,F).

HI(F,-~2

A2(A,F

be the union of all components

component

the subgroup

dim

in

3).

is a finitely generated,

of Q (F)/F

Let

(cf. Theorem

for this dimension

is a positive

integer

or

R =

and each

7r(wi) yields

43 precisely

one of the components

/3 o i(~) = 0.

By lemma

of fi/F.

X2-2q~-- h a s a p o t e n t i a l

¥ E F.

_Oi(z) = t~(z) if z E F~u. a n d

show that

F

such that

1

.t~i = 0 f o r e a c h

ff

~ E B

4, w e k n o w t h a t t h e g e n e r a l i z e d

coefficient Let

Suppose

q

(fi,F)

and

Beltrami

¥1_qF = P

for each

0.(z) = 0 o t h e r w i s e .

We must

1

i.

[,ilak2-2q[dz

A dz I = f f * i 3 - ~ - Idz ^

¢i

goi

=ff ~-g (F,i)[dz A Tzl gO. 1

Notice that punctures

on 7r(mi),

F ~ i dz

is i n v a r i a n t .

If t h e r e a r e no p a r a b o l i c

one can apply Stokes' theorem

and this last

integral becomes

(4.1)

___1 2 f

F ~ i dz = 0.

If ~r(wi) h a s a p a r a b o l i c p u n c t u r e , c a n a s s u m e t h e r e is a p a r a b o l i c e l e m e n t y(z) = z + 1 anda Fw i_~ ~z Stokes'

constant

[ 0_< Re z < i and theorem

argument

c

t h e n by l e m m a

1, one

¥ E F of t h e f o r m

such that

Im z> to show

c}. that

One f

can use a similar

F~i dz = 0 if one shows

that

~). 1

01 l i m J0 ~(x + i b ) F ( x + ib)dx = 0. b4~

(4. 2)

But by (2.2) we know cusp

form

we know

F(x + ib) = 0((x 2 + b2) q-l) that

and because

I~o(x + ib) I < (const.) e -27rb.

Hence

~0 is a the limit

44 in (4.2) is zero.

Corollary

i.

dim Bq(~,F)

C o r o l l a r y 2.

_< dim

HI(F, Tr2q_2 ) .

If F is g e n e r a t e d by N e l e m e n t s

R

{(2q-1)(gj-1) + E

[ q - v-~p}]] < (2q-1)(N-1).

A pES. J

j=l

H e r e the s u m is t a k e n o v e r all e l l i p t i c o r p a r a b o l i c points p of fl(F)/F and g: is the genus of the c o m p o n e n t S. of Q/F. J J C o r o l l a r y 2 follows f r o m c o r o l l a r y 1, l e m m a 3 and t h e o r e m 3.

Corollary

3.

components

Proof:

If F

is generated

of ;}(F)/F

By elementary

is R,

methods

by

then

N

elements

2 for

q = 4 and

one shows

completes

the proof of Ahlfors'

§5.

Theorem I0.

THE

q = 6,

that the inequality

the result follows.

finiteness

AREA

of

R < 18(N-I).

dim A4(Aj, F) + dim A6(Aj, F) > 1 for all j. Adding corollary

and the number

in

This corollary

theorem.

THEOREMS

( B e t s ' i s t a r e a t h e o r e m [4]) Area (Q/F)<_ 47r(N-l)

where the a r e a is Poincar4 a r e a and N = the number of g e n e r a t o r s

of F.

Proof: -

Merely multiply the inequality in corollary 2 to t h e o r e m 9 by

27r and let q-~ ~. q -

The left hand side approaches the Poincar~ a r e a

and the right hand side approaches 4Tr(N-1).

45 Theorem

11.

( B e r s ' 2nd a r e a t h e o r e m

n o n e m p t y o p e n s u b s e t s of fi, suppose

~1 U f12 = fl and

[4])

Suppose

fll

and

~2

e a c h of w h i c h i s i n v a r i a n t u n d e r

fll N ~2 : ~" I f fil

is c o n n e c t e d ,

are

F,

and

then

area(fl2/F) < area(fll/l~ ). Proof: mapping

O u r m e t h o d w i l l b e to e s t a b l i s h t h e e x i s t e n c e of an i n j e c t i v e L : Bq(fl 2, F) - - > Bq(fl 1, F).

J u s t a s in t h e p r o o f of t h e f i r s t

area theorem,

the inequality for areas will follow multiplying the

corresponding

inequality for dimensions

and letting

q-* ~. To construct the mapping,

~(z) = k 2 - 2 q ( z ) ~ ( z )

for

z E ~2

and

be a potential for the generalized L(~)(z)

2w of c o h o m o l o g y g r o u p s by - q

d2q-1 - F(z) d z 2 q -1

holomorphic

for

which potential

By differentiating

F

that

is c h o s e n ,

~(z)= 0 for

Beltrami

z E ~1"

f u n c t i o n in f12'

l e t ¢p E Bq(fl 2, F) a n d l e t

coefficient

It i s c l e a r t h a t L(~)

and that

the integral expression

z E C-

Q2"

Let

F

~ and define

L(~)

will be a

d e p e n d s o n l y on ~ a n d not on L

is an a n t i l i n e a r m a p p i n g .

for F

(2q-l) times,

one

finds that

L(cp)(z)

(2q-l)! ff k2-2q(c)~(C ) d~ A d--~. 2wi fil (~ -z)2q

Using this fact and the formula e a s y to s h o w t h a t = L(~).

L(~)

To s h o w t h a t

(AC-Az) 2 = (~-z)2A'(,()A'(z),

i s a d i f f e r e n t i a l f o r m of w e i g h t

(-2q).

it i s Let

, E Bq(fl 2) we m u s t s h o w t h a t if a p u n c t u r e is

r e a l i z e d by t h e p a r a b o l i c t r a n s f o r m a t i o n

y(z) = z + 1 with

46

~2 ~ [ z [ I m z > e ] ,

t h e n ~(z) = 0(e-2~'Y).

But ~(z) = ~

a e n

27rinz

for

-eD

Im z > e.

d2q-1 ~(z) = ~ F ( z ) dz2q -1

By the fact that

F(z) = ~

a

(27rin)1-2qe 27rinz n

and hence ,

~ a e 2rrinz n=l

=

'

+ a0z2q-1 + v

0( [z12q-2).

v E "]-[2q-2" But F(z) =

one finds that

=

for

Im z>

Therefore,

0(e -2Try)

as

y-~

0 where

an = 0 for n<_ 0

~.

n

T h e o r e m 12.

If ~1 is connected, then L : Bq(~ 2 , F ) - >

Bq(fl 1,F) i_.ss

injective.

Proof:

Suppose

L{~) = 0.

F 0 is a polynomial

Then

of degree

<__ 2q-2

polynomial and let F = F 0 - v 0. vanishes on

f~l" Since

continuity, that theorem

9,

F[A

k2-2q~

in f~l" Let

Hence

[~I is connected

= 0.

Hence,

has a potential

F

F0

such that

v 0 be this

is a potential w h i c h

A = ~flI.

by l e m m a

5,

Thus,

w e have, by

/3 o i(~0) = 0.

By

~ = 0.

REFERENCES

[I]

L.V.

Ahlfors,

Math., [2]

L, Bets,

Finitely generated

8__~6(1964),

413-429

An approximation

Kleinian

and 8__77(1965), theorem,

groups,

Amer.

J.

759.

J. Analyse

Math.,

14 (1965),

I-4. [3]

L. Bets, (1967),

[4]

L. Bets, Analyse

OnAhlfors'

finiteness theorem,

Amer.

J. Math.,

89

113-134. Inequalities Math.,

for finitely generated

18 (1967),

23-41.

Kleinian

groups,

J.

47

[5]

L. Bets,

Automorphic

generated [6]

I. Kra, (1972).

forms

Fuchsian

groups,

Automorphic

Forms

and Poincar6 Amer.

series for infinitely

J. Math.,

87 (1965),

and Kleinian Groups,

W.A.

196-214. Benjamin

4.

DEFORMATION

SPACES*

Irwin Kra SUNY at Stony Brook Let G be a (rich-elementary) group.

finitely generated

In this chapter we show how the set ~(G)

"marked" Kleinian groups quasiconformally forms a finite dimensional the set ~(G) conformally

This deformation completely surfaces

(of finite

equivalent

to G and

of Kleinian groups quasi-

to G forms a normal complex space.

theory contains

dependent

of all

complex analytic manifold,

of conjugacy classes equivalent

Kleinian

as a special case and is

on Teichm~ller

space

type) or equivalently

space theory for finitely generated

theory for Riemann on Teichm~ller

Fuchsian groups of the

first kind. We fix once and for all a finitely generated elementary)

(non-

Kleinian group G, and let Z be an invariant

of components

for G.

be denoted by ~.

The region of discontinuity

of G will

All Kleinian ~rou~s under consideration

be assumed finitely ~enerated.

§i. QUASICONFORMAL

DEFORMATIONS

Let f be a quasiconformal

will

Almost all Fuchsian ~roups con-

sidered will be finitely ~enerated~

~Work partially

union

of the first kind.

OF KLEINIAN GROUPS

automorphism

of the complex

supported by NSF grant GP-19572.

49

sphere ~. Beltrami

We say that f is compatible with G provided coefficient

~ of f vanishes

on the limit set

A = A(G) of G, and fGf -1 is again a Kleinian group. condition

is equivalent

the

This

to ~ being a Beltrami coefficient

for G; that is,

i) ii)

iii) WARNING.

~(g~)g'-rg~V/g, (~) : ~(z),

all g ~ G, a.e.

z ~

~,

~IA = O, and

li~ll : ess sup

i~l < i.

All quasiconformal

mappings

be considered will be assumed set (of G).

(compatible with G) to

to be conformal

It is not known whether

satisfied by all finitely generated

on the limit

this is automatically groups.

(See Bets'

lecture.) We denote

(as in Earle's

(open unit ball in a closed Beltrami coefficients

lecture) by M(G,Z)

linear subspace

the set

of L (~)) of

for G with support in Z.

is an open subset of a complex Banach space,

Since M(G,E)

it has a natural

topology and complex structure. To every ~ E M(G,Z) a unique normalized

there corresponds

(Ahlfors-Bers

(fixing 0,1,~) ~-conformal

[4])

automorphism

of

denoted by w ~. DEFINITION AND LEMMA 1.1.

A Beltrami coefficient

is called

trivial if it satisfies

followin$

conditions:

a)

~ogo(w~)-i

: g, all g ~ G,

one (hence both)

W E M(G,Z) of the

5O

b)

w~(z) = z, all z E A, the limit set of G. A trivial Beltrami coefficient

G as a consequence LEMMA 1.2.

of the following

fixes each component elementary

A__nnorientation preserving

maps every component

o_~f the complement

of

topological

automorphism o f ~ of its fixed point

set onto itself. The set of trivial Beltrami eoefficients support in Z is denoted by Mo(G~Z). holomorphic

mappings

for G with

It acts as a group of

known as right translations

on M(G,~)

by

where -i

w~

: w~o(w~) -1

The %uasiconformal

deformation

space~ o_~f G w i ~

support

in Z, is

~(a,z)

= M(a,z>/Mo(a,z)

endowed with the quotient structure

topology and quotient complex

(of a ringed space).

An element ~ E M(G,Z) vided w u is homotopic of Z modulo

to the identity

the ideal boundary

that the set of strongly ~o(G,Z),

is called strongly

of A.

deformation

on each component It is easy to show

trivial Beltrami

forms a normal subgroup

trivial pro-

coefficients,

of Mo(G,Z).

The strong

space, o_~f G with support i_~nZ, is defined as

51

9(a,z) = M(a,Z)~o(a,z). If we set

~(G,z) = Mo(a,z),,,'fo(a,z), then it is clear that

If E = ~, the entire

~(G) = ~ ( G , ~ ) ,

region of d i s c o n t i n u i t y

~(G) = ~ ( G , ~ ) ,

and Z(G) = $ ( G , ~ ) .

§2. T}{E D E C O M P O S I T I O N Let ~ l , ~ 2 , . . . , ~ r components in G.

lecture) of finite

THEOREM

be a complete

list of non-conjugate

of E and Gj = G~ , the stability

j

By Ahlfors'

Finiteness

Theorem

[3]

subgroup

Furthermore,

of 4.

J

(see Gardiner's

this is indeed a finite list and ~j/Gj type.

we write

is a surface

each Gj is finitely generated.

The study of deformation

spaces of Kleinian groups

reduces

to

the study of deformation

spaces of f u n c t i o n groups

(Kleinian

groups w i t h an invariant

component)

as a result of the

following THEOREM 2.1

(Kra [22]).

~(~,~) ~ ~(GI,~ l)

We have

X...x

~(Gr,&r) ,

and

5(a,z) ~ 5(al,~ l) x...x %(Gr,~r). The first isomorphism

is trivial.

The second

is a conse-

52

quence of Lemma 1.2 and another elementary LEMMA 2.2.

Let D be an open subset of ~ and f ~ topological

homeomorphism of~ fI~kD = id. coefficient

that is

quasiconformal

Then f i_~s q u a s i c o n f o r m a l is supported

o__n_nD and such that

and its Beltrami

in D.

Lemma 2.2 together with some arguments Kleinian groups THEOREM 2.8

leads

(Maskit

to the following

[25]).

quasiconformal

§3°

(conformal)

REDUCTION

universal

Then f is the A automorphism F of C

all g £ G.

Further,

if f i__ss.

of Z.

Let h : U ~ ~ be a holo-

covering map.

Let F be the F u c h s i a n model

of G over 4; that is, the group of conformal ¥ of U such that there is a X(Y) We then have an exact sequence [i}

> H in~ > F

X

with H (the covering group) of finite Riemann

type

F i_~s

TO THE F U C H S I A N CASE

Let A be a component morphic

automor-

all g 6 G.

topological

with FISkZ = id and F6g = goF,

to

exteqsion

Let f be a topological

p h i s m o f Z such that fog = g-f, restriction o f ~ global

specific

automorphisms

£ G~ with ho~ = X(y)oh.

of Kleinian

> GA

> [i}

and F Fuchsian~

(as a m a t t e r of fact~

groups

and F and GA

U/~ ~ A/G A as

surfaces with ramification points).

53

REMARK.

The Fuchsian model of G over A used above differs

from the one defined by Maskit lead to equivalent

results

For ~ E M(r) = M(r,u) (h*~)oh = ~h'/~',

(3.1) is

[25].

However,

both models

(Theorem 7.1). we define h*~ ~ M(G~,~) by

and verify almost trivially

that

h~ : M(r) ~ M ( % , ~ )

a (linear,

whenever possible)

isomorphism. PROPOSITION

Furthermore, 3.1.

norm p r e s e r v i n g

(surjective)

we have the following

The map h* of (3.1) induces

holomorphic

surjections

T(r) = ~ ( r , ~ )

~ ~(%,~)

T(F) ~ ~ ( G a , ~ ) (with

the

REMARK. for

s e c o n d map b e i n ~ The s p a c e T ( F )

the Fuchsian

a_~n i s o m o r p h i s m ) .

is called

the Teichm~ller

space

group F.

B e f o r e we c a n c o n t i n u e m a p p i n g s , we m u s t t u r n

the

investigation

t o a more c a r e f u l

of the above

examination

of

§4. THE FUCHSIAN CASE For r Fuchsian,

we denote by B(r) = B2(F,L)

space of bounded automorphie

forms of weight

support in the lower half plane, For ~ E M(F),

let ~

the Banach

-4 for F with

L (see Gardiner's

be the Schwarzian

derivative

lecture). of

54 f = w~IL (the definition makes sense since ~

is holomorphic

and univalent on L), f,, ,

llf"~2

A result of Nehari [26] shows that ~

E B(F).

Lemma i.i

shows that ~W depends only on the equivalence class of in T(~).

Furthermore,

the induced mapping (from T(r)

into B(r)) is easily seen to be injeetive. ~ ~

We may view

as a holomorphic mapping : M(F)

~

B(F).

The derivative of this mapping (at the origin) can be computed and seen (Bets [8]) to be surjective.

Since

B(F) is a finite dimensional space, it can be shown using the implicit function theorem that ~(M(F)) = ~(T(F)) is open in B(F).

We have the following

THEOREM 4.1 (Bers [8]).

The Teichm~ller space T(F) has a

unique complex structure so that : M(F)

~

T(F)

i___ssholomorphic with local holomorphic sections.

We can

realize T(F) as a bounded domain (of holomorphy)

i__%B(r).

Further T(r) i_~stopolo~ically a cell. The fact that T(r) is a domain of holomorphy was proven by Bers-Ehrenpreiss

[12].

(See also Royden's lecture.)

Teichm~ller's theorem (see for example Ahlfors [2] or

Bers

[6]) implies

a unique

that every ~ £ M(F)

Teichm~ller

coefficient

coefficient;

is equivalent

to

that is, a Beltrami

~ with

~(z) = k ~(~)/I~(~)I , z ~ U, k ~ ~,

0 ~ k < l,

It follows

0 /~

easily from this result

(as topological

depends

been obtained simple proof

[13] show that T(F), only on the type of r.

independently see Earle-Kra

as the Teichm~ller

space

by Marden [17].)

An automorphism

quasiconformal

(This result has

[23].

For a rather

We thus define

T(g,n) (g,n).

GROUPS

e of the Kleinian

with respect

as a complex

T(E) for some group r of type

§5. MODULAR

geometric,

that T(r) ~ B(F)

manifolds).

Bers-Greenberg manifold,

~ B(r).

group G is called

to Z, if and only if there is a

automorphism

f of ~ compatible

with G such

that fZ = Z and

e(V) = f~yof -I, all y ~ G. The map f induces

a biholomorphic

by sending ~ E M(G~Z) w~of-IIz. equivalence mapping

into the Beltrami

It is easy to check classes

automorphism

and hence

that

th~

induces

of M(G,Z)

coefficient mapping

of

preserves

a biholomorphic

self-

e* : %(G,z)

- %(G,z)

w h i c h depends o n l y on the automorphism B, i n f a c t the conjugacy class We thus d e f i n e

o f 8 modulo i n n e r

automorphisms o f G.

the m o d u l a r group Mod(G,Z) as the q u o t i e n t

o f the group o f g e o m e t r i c

(with

respect

t o E) automorphisms

by t h e subgroup o f i n n e r a u t o m o r p h i s m s . Mod(G~E) on ~ ( G , E ) REMARK.

(The a c t i o n

of

is not always effective.)

The above d e f i n i t i o n

definition

o n l y on

used by Bets

o f Mod(G,E) d i f f e r s

f r o m the

[ii].

The moduli space is defined by

~(a,z) = %(~,~)/Mod(o,~). As usual we set

~(o) : and observe

~(am),

that it denotes

the set of cnnjugacy

of Kleinian groups q u a s i c o n f o r m a l l y If F is a Fuchsian group Mod F = Mod(F~U). nature of F. signature

Further,

is a subgroup

of finite

The group Mod(g,n) (a classical

(operating

result,

to G.

on U), we set

Mod F depends

We define Mod(g,n)

(g,n;~,...,~).

equivalent

classes

only on the sig-

to be Mod V for a group of

If F has type

(g,n)

then Mod F

index in Mod(g,n). acts d i s c o n t i n u o u s l y

on T(g,n)

for the proof see, for instance,

[i0]) and thus the Riemann space

Bets

(for F)

R(r) = ~(r,~) is a normal complex

space

(as a result of a theorem of Cartan

57

[14]).

For more about the group Mod(g,n)

see Royden's

lecture. REMARK. classes

If F is Fuchsian,

of conjugacy

of Fuchsian groups with the same signature

This assertion follows i)

then R(F) consists

as r.

from the following:

Two Fuchsian groups are quasiconformally

equivalent

if

and only if they have the same signature. 2)

For each ~ E M(~)

there exists a unique normalized

~-conformal

automorphism

Bets

This automorphism

[4]).

of U denoted by w conjugates

(Ahlfors-

F onto another

Fuchsian group. 3)

For ~,~ E M(F), ~ and v are equivalent

(in T(r))

if and

only if wwI~ = w~I~. 4)

The modular group Mod F can be defined by looking at quasiconformal

automorphisms

global compatible

extensions)

f of U (all such f have that conjugate F onto

itself. Note also that T(g,n) surfaces

represents

"marked" Riemann

of type (g,n), while R(g,n) = T(g,n)/Mod(g,n)

represents

the conformal

equivalence

classes of surfaces

of type (g,n). §6. THE FUNCTION GROUP CASE We return to the situation quaslconformal

automorphism

treated in §3.

of U with

If w is a

58

(6.1)

mHw -I = H, wFw -I = F,

then m induces a q u a s i c o n f o r m a l

automorphism

f of ~ such

that fob = ho~ and

(6.2)

fGAf -I = G~ Further,

satisfies

every quasiconformal

(6.2)

is so induced

automorphism

(w is unique

f of & that

except

that it may

be replaced by ¥ow, ¥ E H). If f and w are as above, (6.3)

then

fogof -I = g, all g E G~

if and only if

(6.4)

~oyo~-~¥-!

It follows

from Theorem 2.3

is the restriction satisfies

E H, all y E F. (and Lemma 2.2)

that such an f

to ~ of a global compatible

(6.2)) and whose Beltrami

coefficient

f (that is supported

on G~ = [g(~);g E G]. Let ModH(r)

denote

by all w satisfying

the set of elements

(6.1), and let MOdH(r)

induced by all ~ satisfying Further,

MOdH(F ) is a normal

P R O P O S I T I O N 6.1.

(6.4).

Furthermore,

T(r)/Modi(r) ~ ~(G~,~),

the subset of

Both of these are groups.

subgroup

The group MOdH(F)

in Mod r induced

of ModH(F).

acts freely on T(r).

59

and thus $(GA,A)

is a finite dimensional complex analytic

manifold with ho!omorphic

§~

universal coverin~ space T(r).

THE GENERAL CASE

We now consider the situation described in §2. Fj(j=l,...,r)

Let

be the Fuchsian model of G over ~j defined

by the holomorphic universal covering map hj : U ~ 4.j with covering group Hj c Fj. HI Mod*(G,E) = Mod

We define

H F I x...x Mod rF r

Mod,(G,E) = MOdHIF I x...x MOdHrF r. Proposition 6.1 implies almost immediately the following THEOREM 7.1 (Maskit [25]).

We have

~(~,z) ~ T(r l) x...x T(r ), and is hence simply connected.

Further,

~(G,Z) ~ (T(r 1) x...x T(rr))/Mod.(a,Z). COROLLARY i.

The deformation space ~(G,Z)

is a complex

analytic manifold with universal covering space ~(G,Z). COROLLARY 2.

If each component of Z is_ simply connected,

then

9(a,z) ~ %(o,~) ~ T(r l) ×...x T(rr). We can also obtain THEOREM 7.2 (Bers [ii]). holomorphic

The group Mod(G,E) is a group of

automorphisms o f ~ ( G , E ) .

It acts properly

60

discontinuously o_n_n~(G,E), and ~(G,E) is thus a normal complex space. The

group Mod(G,Z) is induced by quasiconformal auto-

morphisms f of ~ that conjugate G into itself and fix E. There is thus a normal subgroup MOdo(G,E) of finite index in Mod (G,E) that is induced by quasiconformal automorphisms of A C that fix each E~~ = [g(Aj);g E G]. Let f be such an autoA morphism of $. For each j, there is a g~ ~ G such that fj = gjlof fixes Aj and fjGjfj I = Gj.

By the introductory

remarks of §6, there is an automorphism w. of U that J satisfies fjohj = h j

j

and

®jHj®~ 1 = Hj, ~jrjw] 1 = rj. Further, wj determines a well defined element of Mod

Hi

If f induces the identity automorphism of G~ then the Hi element of Mod (Fi) determined by the above procedure actually lands in MOdHi(ri). PROPOSITION 7.3.

We thus obtain

There is a normal subgroup of finite

index o_~fMod(G,E) which is isomorphic to a subgroup of

Mod*(a,Z)/Mod.(G,Z). Proposition 7.3 easily implies Theorem 7.2.

(ri).

61

§8. THE COMPLEX STRUCTURE We have shown that of genus g(~2),

T(g,O),

IS C A N O N I C A L

the space of marked Riemann surfaces is a complex analytic manifold.

what sense is the complex

structure canonical?

Let X be a compact Riemann surface Let

(al,...,ag,bl,...,bg)

i aj

of genus g ~ 2.

be a canonical h o m o l o g y basis for

X; that is, the intersection

~ i'aJ

In

matrix

ai'bJ]

i,j = l,...,g

bi'bj/

(where a-b denotes

the intersection

number of the curves

a and b) is of the form 0 ( -I

I 0)

(where 0 is the g×g zero matrix, matrix).

Let ~l,...,Wg be a canonical basis for abelian

differentials basis;

and I is the g×g identity

of the first kind dual to the given homology

that is, 0 if i J j

Sa i ~j= Consider

lifi=

j.

the g×g period matrix : (vij),

i,j = I, .... g,

where

~ij = Sb.~j" 1 A classical

result

(due to Riemann)

asserts

that ~ is a point

in

62

Sg, the Siesel upper half 9pace of senus g; that is, the set of g×g (complex) symmetric matrices whose imaginary part is positive definite. We choose a model T(F) for T(g,O) with F a Fuchsian group of type (g,O).

The curves al,.. ,ag,bl,...,bg determine

elements AI,...,Ag,BI,...,Bg

in F (every homology basis lifts

to a homotopy basis and wI(U/F) ~ F).

It is not hard to

see that given any point x £ T(F), with x the equivalence class of ~ E M(F), then the elements A~,. • ., A~g, ~i''" ., B~g in F

= w~Fw~ I (where A~ = w oAiow~ I, B~j = ~ o B j o ~ - i

i,j = l,...,g) project to a canonical homology basis on U/F .

In this way one obtains a canonical mapping

(once a

canonical homology basis has been chosen on U/T) (8.1)

s

: T(g,0)

~

g"

Now T(g,O) is a 3g-8 (= dim B(F) by Riemann-Roch) manifold and ~g is a (i/2)g(g+l)

dimensional

dimensional complex manifold

and we have the following THEOREM 8.1 (Rauch [27], Bers [7]). (8.1) i_~sholomorphic. locally injective) hyperelliptic

The mapping S of

It is of maximal rank (therefore,

except at ~ point x E T(g,o) representing

Riemaran surface.

COROLLARY i (Rauch [27]).

In a neighborhood

elliptic Riemann surface, 3g-3 periods

of a non-hyper-

(that is, entries in

63

the matrix ~) can be used as local coordinates COROLLARY periods

2 (Bets

However,

Thus,

in a very real sense,

is natural.

Of course,

classical

the image S(T(g,O)) important)

results

Farkas-Rauch

structure

we have just touched

on

on the

of Schofitky of determining

For partial

(but deep and

see the work of Andreotti-Mayer

[2] and

[18].

is the complex

generated

structure

here is easier

~l,...,yr

for all non-

the complex

problem

in gg.

Let G be a finitely

answer

set of 3g-3

surfaces.

very difficult

sense

no fixed

can b_~e used as local coordinates

hyperelli~tic

T(g,O)

[7])-

for T(g,0).

on ~(G)

group.

canonical?

than in the classical

be a set of generators

X from G into

Kleinian

the MSbius

group,

for G. MSb,

case.

In what The Let

Every homomorphism

determines

a point

(x(¥l),...,×(Y r) ~ MSbr. Let Hom denote into MSb

the set of parabolic

(that is, those homomorphisms

for every parabolic on 2/G). affine

homomorphisms

element y E G determined

It is quite easy to see

algebraic

X with

variety

(Bers

and that there

[9])

of G

trace2x(y)

= 4

by a puncture that Hom is an

is a holomorphic

mapping

(8.2) where

L : ~(a)~ L(~(~))

~om,

= (w~oYlO(W~)-l,...,w~OYrO(W~)-l),

E M(G).

64

Is L(~(G)) a submanifold of Hom? open.

Partial results have been obtained

Marden [24]). L( 0)?

The question is completely (Bers [9],

Is L($(G)) a manifold in a neighborhood of

Even here only partial results are known (Abikoff

[i], Bers [9], Gardiner-Kra

[19], Marden [24]).

The Fuchsian case is much simpler.

Let r be a

Fuchslan group and Hom~, the affine algebraic variety (over ~) of parabolic homomorphisms of r into the real MSbius group, MSb R.

In analogy to (8.2), there is a real analytic

diffeomorphism L : T(r)

~ HO~R,

with L(T(r)) a submanifold of Hom~ (see, for instance, Gardiner-Kra

[19]).

§9. A FUNCTORIAL APPROACH Let r be a finitely generated Fuchsian group of the first kind operating on the upper half plane U.

For each

E M(r), the domain w~(U) depends on ~(~), the equivalence class of ~ in T(r).

We may therefore form the Bets fiber

space

F(r) = [(~(~),z) E T(r) x ~; ~ E M(F), z E w~(U)], and the punctured fiber space Fo(r) w h e r e Ur

is

-- [ ( ~ ( ~ ) , z ) the

set

E F(r);

of p o i n t s

any elliptic element of r.

~ e M(r),

z E w~(Ur)},

in U which are not fixed

by

65 The group F acts discontinuously on F(F) as a group of biholomorphic mappings by y(~(~),z) = (~(~), y~z) where ~ E M(~), z E w~(U), y E r, and

y~ow~ = w~o¥. The quotient space v(r) = F(r)/r is a complex manifold equipped with a projection ~ onto T(F).

The punctured fiber space Fo(F ) is invariant under

F; in fact, Fo(F ) is the largest subset of F(r) on which F acts freely.

We form the punctured quotient space

V(r)' = Fo(r)/fl. For each point ~(~) E T(r), the fiber w-l(~(~)) is the Riemann surface w ~ ( U r ) / ~ r ( w ~ ) - l .

in V(r)'

The complex

manifold V(r)' (the n-punctured Teichm~ller curve) depends only on the type of F, so we define V(g,n)'

to be V(F)'

for some group r of type (g,n). THEOREM 9.1 (Bers [i0]).

If 2g + n ~ 2, then T(g,n+l) is

the holomorphic universal coverin~ space o f V(g,n)'. If we take r to be of type (g,n) without parabolic elements, then we may define a

another fiber space V(g,n)

(= the (unpunctured) Teicb_m~ller curve = ~ r ~ D

for F as above)

that depends only on the type of the group r chcsen. each "puncture" on V(g,n)' provides

Since

a holomorphic section

66

ej

: T(g,n) - V(g,n)

of the p r o j e c t i o n ~, we have constructed ed n-pointed

(see Earle

family of closed Riemann surfaces

(Let g,n be a pair of integers By an n-pointed

satisfying

[ 1 5 ] ) a mark-

of genus g.

2g - 2+n > O.

family of closed Riemann surfaces

of genus

g, we mean a pair Of complex manifolds V and B, a holomorphic map p : V - B, n holomorphic

sections

ej : B - V

(that is, poe. : id) such that J i) ii)

p is a proper submersion, p-l(t)

is diffeomorphic

to a closed

surface

X of genus g, for all t ~ B, and iii)

the sections

el,...,e n are disjoint

(that is,

ej(t) J ek(t) for all t E B if j J k). If we set n V' = V \ U e~(B), j=l ~ then p : V' - B is a smooth fiber bundle with fiber X n (= X\[n distinct

"distinguished"

p o i n t s ~ and structure group

Diff+(Xn ) (= group of orientation p r e s e r v i n g of X keeping structure

the n-"distinguished"

group of this bundle

the path component

bundlesJ

= V~

fixed).

If the

to Diff~(Xn),

in Diff+(Xn)~

A map of marked n-pointed

is a pair of holomorphic such that F(VI')

is reduced

of the identity

is said to be marked.

points

diffeomorphisms

the family families

maps F : V I ~ V 2 and f : B I - B 2

and

(FIVe,f)

is a map of Diff~(Xn)-

67

The fact that the complex structure canonical

is also contained

THEOREM 9.2

(Grothendieck

e_~d n-pointed

of T(g,n)

is

in [20], Earle

[15]).

Given

family of closed Riemann surfaces

any mark-

of ~enus g

p : V ~ B, there is a unique ma D of marked families

v

F

~ v(g,n)

B

f

~ T(g,n)

The u n i v e r s a l uniquely

property

determines

We c o n c l u d e development

both V(g,n)

this

due to Earle-Eells

sections

chapter

(2)

two r e m a r k s :

to Teichm[ller

The p r o b l e m of describing : V(g,n) ~ T(g,n)

[21] and Earle-Kra

The

space

the theory

in these

all holomorphic has b e e n solved

[17], for special cases,

in a forthcoming paper by the last two authors general case).

(1)

in common with

[16], which is not covered

of the bundle ~

(in Hubbard

with

approach

in the above theorem

and T ( g , n ) .

of this section has points

differential-geometric

lectures.

described

in the

and

REFERENCES

[i]

W. Abikoff, Constructability

and Bers stability of

Kleinian groups (to appear).

[2]

L. V. Ahlfors,

On quasiconformal mappings, ~. Analyse

Math., ~ (1953/4), 1-58.

[3]

L. V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math., 86 (1964), 413-429 and 8_~_7(1965), 759.

[4]

L. V. Ahlfors,

and L. Bers, Riemann's mapping theorem

for variable metrics, Ann. of Math., 72 (1960), 385-404.

[5]

A. Andreotti and A. L. Mayer, On period relations for ab@lian integrals on algebraic curves, Annali Sc. Nor. Sup. Pisa, 21 (1967), 189-238.

[6]

L. Bers, Quasiconformal mappings and Teichm~!!er's theorem in Analytic Functions, pp. 89-119, Princeton University Press, Princeton,

[?]

L. Bers, Holomorphic

New Jersey,

differentials

1960.

as functions of

moduli, Bull. Amer. Math. Soc., 6__7 (1961), 206-210.

[8]

L. Bers, A non-standard

integral equation with appli-

cations to quasiconformal mappings, Acta Math., 116 (1966), 113-134. [9]

L. Bers, On boundaries of Teichm~ller spaces and on Kleinian groups:

[io]

I, Ann. of Math., 91 (1970), 570-600.

L. Bets, Fiber spaces over Teichm~ller spaces, Acta Math., 130 (1973), 89-126.

[ii]

L. Bets, On moduli of Kleinian groups

(to appear).

89

[12]

L. Bers and L. Ehrenpreiss,

Holomorphic convexity of

Teichm~ller spaces, Bull. Amer. Math. Soc., 7__0 (1964), 761-764.

[18]

L. Bers and L. Greenberg, Isomorphisms between Teichm~ller spaces, Advances in the theory of Riemann surfaces, Ann. of Math. Studies, 6 6 (1971), 58-79.

[14]

H. Caftan, Quotient d'un espace analytique par un group d'automorphismes,

in Al~ebraic Geometry and Topology,

pp. 90-102, Princeton University Press, Princeton, New Jersey, 1957.

[15]

C. J. Earle, On holomorphic families of pointed Riemann surfaces, Bull. Amer. Math. Soc., 7 9 (1978), 163-166.

[16]

C. J. Earle and J. Eells, A fibre bundle description of Teichm~ller theory, J. Diff. Geom., ~ (1969), 19-48.

[i? ]

C. J. Earle and I. Kra, On holomorphic mappings between Teichm~ller spaces (to appear).

[18]

H. M. Farkas and H. E. Rauch, Period relations of Schottky type on Riemann surfaces, Ann. of Math., 92

(1970), ~34-~61. [19]

F. Gardiner and I. Kra, Quasiconformal stability of Kleinian groups, Indiana University Math. J., 2_.1 (1972), 1087-1059.

[2o]

A. Grothendieck, analytique,

3).

f

Sem. Cartan 60/61, expose 17 (with appendix

by J. P. Serre, Rigiditi' na

f

Techniques de construction en geometrie

de foncteur de Jacobi d'echelon

70

[21]

J. Hubbard, Sur la non-existence

de sections analytiques

a la courbe universelle de Teichm~ller,

C. R. Acad.

Sc. Paris, 274 (1972), A 978-A979. [22]

I. Kra, On spaces of Kleinian groups, Comment. Math. Helv., 47 (1972), 53-69.

[23]

A. Marden, On homotopic mappings of Riemann surfaces, Ann. of Math., 9__~0(1969), 1-8.

[24]

A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math.,

[25]

(1974) (to appear).

B. Maskit, Self-maps of Kleinian groups, Amer. J. Math., 9_~3 (1971), 840-856.

[26]

Z. Nehari, Schwarzian derivatives and schlicht function~ Bull. Amer. Math. Soc.,55

[27]

H.E.

(1949), 545-551.

Rauch, A transcendental view of the space of

algebraic Riemann surfaces, Bull. Amer. Math. Soc., 7_~i (1965), 1-39.

METRICS

5.

ON

TEICHMULLER

H. L. Stanford

Royden University

In this chapter we discuss invariant metrics of a Fuchsian group no fixed points in surface of genus

~ U

TEICHM~LLER

M(F) = M(W)

U, whose elements have

W = U/I~

~ E M(~)

Then

U/F

we let

coefficient

of Beltrami

is a compact Riemann

w

of

Two elements

~

and

w

= wv~ Y = ( w ) - i

M(~)

(see

Denote by

~

W

y E F.

of

the Fuchsian group Wu

which is

by taking as a conformal of open sets of

are said to be e~uivalent

for each

F

~.

which are homeomorphisms of

for

be the normalized homeomorphism

is again a compact Riemann surface

the restrictions

~

coefficients

w

obtained by defining a new complex structure on

yo ( w ) - I

T(~)

SPACE AND ITS TANGENT SPACE

be the collection

onto itself with Beltrami F = (wu)-l.

space

space

g.

Kra, §l), and for each

w~

on the TeichmUller

on the upper half plane

and whose quotient

§i. We let

operating

SPACE*

W

into

atlas ¢.

if

This is the same as requiring

U w

= w

and

on the real axis.

~

are equivalent

: W --->W U

It is not difficult

which is homotopic

= T(F) = T(W)

W.

A Beltrami

trivial if it is equivalent

to

0.

T

M(~).

If we use the group structure introduced

We have described

of

to the identity map on

space

coefficients

[3]) that

if and only if there is a conformal homeomorphism

is said to be (globally) g

to show (see Ahlfors

F

(or

W)

in Chapter 4, then the TeichmUller

which serves as a base point in

This research is supported grant GP 33942 XI.

T . g

The TeichmUller

is the space of equivalence

on the space

M0(F)

coefficient

classes in

of trivial Beltrami

T(P) = M(F)/M0(F).

space However~

Tg

in terms of a fixed group

r = F0

we could just as well have started

in part by the National

Science Foundation under

72

with any other Fuchsian group

~'

with

U/I~'

a compact surface of genus

we would find a natural biholomorphic equivalence between if we wish to consider properties of x E Tg,

T

T(~)

and

g,

T(~').

and Thus

in the neighborhood of a given point,

g

we may always choose our base group

r

so that

x

is the equivalence

class of trivial Beltrami coefficients. In order to introduce metrics in differential-geometric terms we need to characterize the tangent space of

Tg.

Since

will be the quotient of the tangent space of which are tangent to the subspace to

M(F)

M0(r ).

p~

a family

~(t)

In our present context the tangent space

t = 0

U

onto itself.

is a vector field

h(z) = h(?(z))/y'(z) t = 0

is

differentials

W.

A curve through the origin in

M0(F)

of Beltrami coefficients belonging to a continuous family

of homeomorphisms of

~,

modulo those tangent vectors

or, equivalently, the space of bounded measurable Beltrami

differentials on the Riemann surface

at

M(~)

its tangent space

is the Banach space of bounded measurable Beltrami differentials which

are compatible with

at

Tg = M(~)/M0(~) ,

for each

~h/~z. ~

h

The derivative with respect to

compatible with y E ~.

~ = ~h/~

i.e.~ an

The derivative of

Thus the tangent space at

such that

F,

0

to

h

~(t) M0

w (t) of

w (t)

such that with respect to

t

is the set of Beltrami

for some vector field

or, equivalently, of some vector field on

t

is

h

compatible with

Wo

Such a Beltrami coefficient is called infinitesimally trivial. difficult to show (see Ahlfors [3]) that a Beltrami differential

~

It is not is infinite-

simally trivial if and only if ential

Q

space

on

B2(F )

W.

The space

7 ~Q = 0 for each holomorphic quadratic differW ~(W) of quadratic differentials on W is just the

consisting of automorphic forms of weight

It has dimension

3g - 3,

and we see again that

T

g

-4,

defined in Chapter 3.

has dimension

3g - 3.

Since we have a natural pairing

[~,Q] =

~ ~Q w

between ~(W)

is

M(W) 0,

and

~(W),

we see that

whose kernel in ~(W)

M(W)

is

M0(W )

and whose kernel in

is just the cotangent space to

Tg

at

~ = O.

73

§2.

DIFFERENTIAL

In order to define a differential ll~IIx = F(x,~) negative

on the vectors

~

METRICS

ON T

metric in

T

g we must introduce a norm

g

in the tangent space at

real function defined on the tangent bundle of

smooth curve in

x. T . g

Thus If

Tg , then the length of that curve is given by

F

is a non-

x(t)

gives a

~F[x(t),i(t)]dt,

and we may define the distance between two points as the infimum of the lengths of curves joining them. on

T

Thus different

choices of

F

give rise to different metrics

g One can define a norm on the tangent space to

pseudo-norm ferentials

on the tangent space to on

W,

which annihilates

Another procedure and obtain

F

M(W),

T

at

g

W

by defining a

i.e., on the space of Beltrami

the infinitesimally

is to define a norm

G(~)

dif-

trivial ones.

in the cotangent

space

~(W)

by duality:

F(~) = sup [ ~ , ~ ]

as

~

runs over all elements

convenient,

since

~(W)

of

~(W)

is a concrete

§3. If the norms

F

or

G

with

G(~) = i.

This method is particularly

finite dimensional

space.

HERMITIAN METRICS

come from a positive definite Hermitian

we get a Hermitian metric for

Tg.

On

we specify a conformal metric

%Idzl

~(W) on

W

inner product,

we get such an inner product whenever by setting

(QI,Q2) = wTQI(Z)Q2-~-~ %-2(z)dx dy .

If we take

%

to be the Poincar~ metric given on

induced Hermitian metric on

T

g

is the Weil-Petersson

If we are given an inner product orthonormal

basis

QI,...,Q n

for

U

~

( , )

on

~(W)

by

% = i/2y ,

then the

metric. we can construct an

and define a kernel of the Bergman type by

setting

K(z,~) = ~ Q i ( z ) ~

Then

K

is a holomorphic

quadratic differential

.

in the variable

z

and the con-

74

jugate of a holomorphic

quadratic differential

in

~.

It is not difficult

to show

(see Royden [i0]) that the dual inner product in the tangent space is given by

(*)

(~,,~) = f

f~(z)K(z,~)~--(~y

W

Thus every Hermitian metric on Hermitian z

symmetrie

and defining

form

K

form

(U~)

T

may be obtained by specifying a suitable

g

K(z,~) by

dx dy d~ d~ .

W

which is a holomorphic

(*).

In the case of the Weil-Petersson

is given (on a fundamental K(z,~) = 3

region in ~

U)

~2dz2

- ~ yEl~

basis

the Bergman kernel

Wl,...~Wg form

~2

Hermitian kernel which is a holomorphic

on the tangent space of pseudo-metric) Q(W).

on

k(z,~) = Z wi(z)wi--~-~ .

T

g

of

at

k W,

T

is obtained by taking an

g

on

quadratic differential

that

(*)

W

Then the form

and constructing K = k2

in

z.

is a It follows

defines a Hermitian pseudo-metric

and this will be a metric (rather than a

if and only if the products of the Abelian differentials

Thus this procedure gives us a Hermitian metric on

elliptic

metric the

by

for the Abelian differentials

from the positive definiteness

in

[ z-y--~] 4

A naturally defined Hermitian pseudo-metric orthonormal

quadratic differential

locus, where the metric fails to be definite.

[I0]) that this metric is the pull-back

to

T

T

g

W

span

except on the hyper-

It turns out (see Royden

of the Bergman-Siegel

g

on

Siegel upper half plane under the Riemann mapping which maps each

W

metric in the in

T

to

g

its Riemann period matrix.

§4. There is a natural a metric considered

THE TEICHM~LLER METRIC

Finslerian metric on

by TeichmUller.

defining the distance between

W

T

g

TeichmUller and

W

which is the differential defined a global metric on

form of T

g

to be U

1 l+k dT(W,w~)- = ~ log l-k '

where

k = inf I[~II as

~

ranges over all

~

equivalent

to

~.

The infinites-

imal form of this metric comes from the norm on the tangent space to obtained by

T

g

at

W

by

75

IhllT where

I[ II is the

e~

norm and

Beltrami differentials. Tg

inherits from the

~

the

differential,

~

~

II - 11 ,

inf

ranges over all infinitesimally

trivial

Thus the TeichmUller metric is the quotient metric which L~

metric on

The Ahlfors-Bers-TeichmHller a given

=

M(W)

by considering

as

M(W)/Mo(W).

theorem (Ahlfors [2], Bers [4]) asserts that for

equivalent to

~

which minimizes

i.e., a Beltrami differential

iI~l] is always a TeichmUller

of the form

and that conversely such a differential minimizes Beltrami differentials.

Tg

~ = k~I~I

where

~ ~ ~(W),

IIwll among all equivalent

From this it follows that the TeichmUller metric is the

integrated form of the differential metric introduced in the previous paragraph and that the curve given by dT(Wo,W ~)

joining

W0

~(t) = ~/I~[, and

W

v

with

Since each open geodesic arc of

with

t E [O,k],

is a geodesic of length

~ = k~/I~ ]. Tg

is of the form

~/[~I

it can be completed to a closed arc by adding the endpoint from the Hopf-Rinow-Myers

theorem that

T

with

k~/lW [.

t E [0,k),

It follows

is complete in the TeichmHller metric.

g

This metric can also be defined by specifying the dual norm in the cotangent space to be

IIQII =

IQIW

One c a n a n a l y z e

of

the smoothness of this

infinitesimal

metric

on t h e c o t a n g e n t

Tg , and it turns out (see Royden [81) that it is of class

CI

but not

bundle

C 2.

In fact, a careful analysis of the smoothness properties of this metric shows that the unit balls in the cotangent spaces at

W

and

only if there is a eonformal mapping between Thus the only isometrics of take each equivalence class

[W ]

T

g

H

and

h-i ~ ~

[W]

consisting of

W homeomorphisms

or equivalently of all quasiconformal homeomorphisms hoyo

W

into an equivalence class

be the group of all quasiconformal

onto itself such that

are affinely equivalent

V for the TeichmHller metric are those which

Riemann surfaces conformally equivalent to Let

W

W

for each

y E F,

h

of

W

onto itself,

of the upper half plane and

H0

the subgroup of

those homeomorphisms which are homotopic to the identity on

W,

or, equivalently,

U

76

of

Then the TeiehmUller

modular group or mapping class group

of

W

onto itself for which

is defined to be the quotient

we have a homeomorphism T

U

h o ?= h-i = ~

those homeomorphisms

of

Mod(W)

group

a quasiconformal taking

h*(~)

only on

8

mapping

@

in the coset

to be the Beltrami

of

T

onto itself.

g

@

coefficient

and on the equivalence

self-mapping statement

homeomorphism

Then,

For each element

as in §5 of Chapter 4,

class of

@ 6 Mod(W),

and define of

~

w in

This mapping

@

Then Thus

g

that,

let

h*(~) h*

h 6 H

be

> M(W)

by

of

depends

induces a

is not only a biholomorphic

but is an isometry under the TeichmUller metric.

in the preceding paragraph

self-mappings

h* : M(W)

o h "I. T

? 6 ~.

Mod(W) = Mod(F) = M(g,O)

into the group of biholomorphic

onto itself defined as follows:

g

H/H 0 .

for all

conversely,

It follows from the

each isometry of

Tg

with

the TeichmUller metric is given by such a self-mapping which comes from the action of an element of the modular group.

§5. Every complex manifold variant under biholomorphic one complex manifold

M

THE KOBAYASHI METRIC has a natural pseudo-metric

self-mappings

and such that each holomorphic map from

to another is distance-decreasing

metric was first defined and studied by Kobayashi in

cn

this pseudo-metric

metric is complete morphy.

¢n,

in this pseudo-metric.

[6],

[7].

This

For a bounded domain

and Wu [ii] has shown that,

if this

then the domain must be a domain of holo-

It can be shown (Royden [9]) that this metric can be defined by a differ-

ential metric as follows: with

is always a metric,

for a domain in

on it which is in-

~

Let

<x,~>

be an element in the tangent bundle to

a tangent vector at the point

x E M.

M

Define

F(x,~) = inf R -I ,

where

R

ranges over the set of radii of disks in

mapped into

M

by a holomorphic map

unit tangent vector of the disk to The map E ~(W)

f

f

which takes

¢

centered at 0

x

which can be

and which takes the

~.

of the unit disk into

T

g

given by

~,

is totally geodesic in the TeichmUller metric,

TeichmUller metric

to

0

lies in the image of such a map.

~ ~/[~I

for a given

and every geodesic in the

Thus the pull-back of the

77

TeichmUller metric under such a map is the Poincar~ metric for the disk and has Gaussian curvature

-4

everywhere.

This is the analogue for differential metrics

of the condition for K~hler metrics of having holomorphic sectional curvature everywhere equal to

-4.

(Royden [8]) that

Just as for K~hler metrics with this property, we can show CI

differential metrics having this property also have the

property that the pull-back of the metric by any holomorphic map of the disk into the manifold has Gaussian curvature at most

-4.

The Ahlfors version (Ahlfors [i])

of the Schwarz-Pick lermna then asserts that any holomorphic map of the disk into Tg

is distance-decreasing

metric on

Tg.

from the Poincar~ metric on the disk to the TeichmUller

Since the mappings considered at the beginning of the paragraph are

isometric between the Poincar~ and TeichmUller metric, and there is such a map in the direction of each tangent vector to for

T

g

T

, it follows that the TeiehmUller metric

g

is also the Kobayashi metric for

T

For further details see Royden [8].

g

Since the Kobayashi metric is invariant under biholomorphic self-maps, it follows that each biholomorphic self-map of metric.

T

g

is an isometry in the TeichmUller

This shows that the only biholomorphic self-maps of

arise through the action of the TeichmUller modular group The result of Wu, together with the fact that domain in

cn 9 gives another proof of the fact that

T

T

g

are those which

Mod(W).

is equivalent to a bounded

g T

g

is a domain of holomorphy.

This was first shown be Bets and Ehrenpreis ~5] using other methods.

78 REFERENCES

i.

Ahlfors,

L. V., An extension of Schwarz's

lermna, Trans. Amer. Math. Soc.,

43 (1938), 359-364. 2.

, On quasiconformal mappings,

3.

, Lectures on quasiconformal mappings,

i0, Van Nostrand,

Princeton,

4.

J. Analyse Math.,

3 (1953-4),

1-58.

Van Nostrand Math. Studies

N. J., 1966.

Bets, L., Quasiconformal mappings and TeichmUller's

theorem, in Analytic

Functions by R. Nevanlinna et al., Princeton University Press, Princeton, N. J., 1960, 89-119. 5.

- -

and L. Ehrenpreis,

Amer. Math. Soc., 70 (1964), 6.

Kobayashi,

mappings,

S., Invariant distances on complex manifolds and holomorphic 19 (1967), 460-480.

, Hyperbolic manifolds and holomorphic mappings, M. Dekker,

New York, 8.

spaces, Bull.

761-764.

Jo Math. Soc. Japan,

7o

Holomorphic convexity of TeichmUller

1970.

Royden, H. L~, Automorphisms

and isometries of TeichmUller

space, Ann. of Math.

Studies, 66 (1971), 369-383. 9.

, Remarks on the Kobayashi metric, Springer Lecture Notes in Math.,

185 (1971),

125-137.

I0.

, Invariant metrics on TeichmUller

ii.

Wu, H., Normal families of holomorphic mappings, Acta Math°,

193-223o

space, to appear. 119 (1967),

6.

MODULI OF RIEMANN SURFACES william Abikoff Columbia U n i v e r s i t y

A finitely g e n e r a t e d K l e i n i a n group

G

represents a

finite union of h y p e r b o l i c Riemann surfaces of finite type. theory of m o d u l i is Riemann surfaces ones);

concerned with parametrizing

(of finite type,

The

families of

in fact, p r i m a r i l y compact

one can use K l e i n i a n groups as a tool in this study.

The subject has roots in statements of Riemann and Poincare, among others.

The first s y s t e m a t i c study was c o n d u c t e d b y Fricke

who c o n s t r u c t e d the space of m a r k e d R i e m a n n surfaces. results w e r e o b t a i n e d using either a l g e b r a i c a b e l i a n differentials)

Later

(i.e. periods of

or d i f f e r e n t i a l g e o m e t r i c techniques.

A m o n g the p r a c t i t i o n e r s of the former m e t h o d are Torelli, Satake,

Baily

[3], Mumford,

The latter technique, the p e n e t r a t i n g

Mayer,

in its m o d e r n

and Deligne and M u m f o r d form,

insights of T e i c h m O l l e r

i n t e r p r e t a t i o n and t r a n s f o r m a t i o n latter w o r k is due to Ahlfors,

Siegel, [8].

is b a s e d p r i m a r i l y on

[12] and their s u b s e q u e n t

into a c o h e r e n t theory.

Bers et al.

This

Our d i s c u s s i o n w i l l

focus on the d i f f e r e n t i a l g e o m e t r i c approach.

In one r e s p e c t

it is m o r e g e n e r a l than the a l g e b r a i c a p p r o a c h since we may consider a surface w i t h r a m i f i c a t i o n points. Let

S

be a m a r k e d Riemann surface of finite type w i t h

r a m i f i c a t i o n points,

together w i t h a corresponding,

possibly

80

ramified, map

covering

~.

We assume

signature greater

than

one

that

the

or the

is r a m i f i e d

latter

case

surface where

We may

type

the

has ~i

are

(p,n)

either

~. l

at some

z

at

The g r o u p

is a p u n c t u r e to the

generated

perform

of c o n f o r m a l point

covering

Fuchsian

many

z . l

of the

spaces."

with metric structure

the

S

is d e n o t e d

group

"moduli

case on

l

and

integers

of d e g r e e

2-manifold

At each

S

former

as a R i e m a n n i a n

metric.

U, a n d p r o j e c t i o n

In the

construct

changes

plane

~.

associated

is a f i n i t e l y

half

symbol

there

transformations and

the u p p e r

(p,n;v ! ..... ~n )

covering the

by

and

in

of c o v e r by

first

G kind.

If w e v i e w

S

ds 2 = a(z) I dz 12, w e m a y by perturbating

z 6 S, a p e r t u r b e d

metric

the

may be written

as

ds~ = I ~l(Z) I I dz + U(z)di I2 Since

w e are

factor open

only

~l(Z)

unit ball

the g i v e n Beltrami

ficient

G

for

lower h a l f equation

in

L

(S) The

coefficients

of

also on

lift

U,

plane. ~Wz,

and

space

structure,

functions

taken

constructed

and

The u n i q u e normalized

set

of

solution so that

w

in the

space

for

of

talk).

U

it e q u a l

scale

of moduli

is the

to a f u n d a m e n t a l

it to a l l

the

~(z)

as a s p a c e

(see R o y d e n ' s

~(z)

extend G,

the

may be

M(S)

the g r o u p

w~ =

in c o n f o r m a l

is i r r e l e v a n t

signature.

We may action

interested

ll~(z) ll~ < k < 1

set

for the

as a B e l t r a m i to

0

on

L,

coef-

the

to the B e l t r a m i

w(-i)

= -i

and

w' (-i)

= i,

81

conjugates

G

into a q u a s i - F u c h s i a n

group

G

.

The set of

U such

G

is a n o t h e r m o d u l i

space of

S;

it is the T e i c h m O l l e r

U space

T(S).

gi v e n

the

For

following

T(G)

where cell

fixed choice

{w,z}

=

of the cover g r o u p

representation

T(S):

[{w,z} I z 6 L, w- = ~w z z

is the S c h w a r z i a n

in the

of

as above}

derivative

of

3g - 3 + n

dimensional

G-invariant

bounded holomorphic

with

support

L.

deformations

S - S'

of the

of the g r o u p

G,

as i s o m o r p h i s m s

B2(G,L) for

of

G

M~bius

group.

i.e.

There

of the e m b e d d i n g S' = S"

G'

being

S'

equal

of the

of to

S

S

as d e f o r m a t i o n s

of the g r o u p

exactly

G

possible

equivalent

to

markings.

Another moduli

space of a compact

is the Torelli {S,

sidered here

space

<~i ..... 8n>~

first h o m o l o g y

extensively

as a q u o t i e n t T(S). where

g r o u p of

studied b y a l g e b r a i c (see Kra's

The u l t i m a t e

surface T(G)

S

8's

that a S"

without

the con-

without

space of

are a fixed b a s i s

The Torelli

geometers

that

by a discontinuous

It is the m o d u l i the

S.

of

into the

in the case

It is h o w e v e r

G ~ G'

in the d e f i n i t i o n

does not p r e s e r v e

obtainable

differentials

the q u a s i c o n f o r m a l

- S'

f: S"

G".

quadratic

normalization

is c o n f o r m a l l y

is a

space

occurs when

ramification,

the pair

surface

surfaces.

T(G)

complex vector

represents

G' = G"

w.

This p h e n o m e n o n

formal m a p

group,

T(G)

is an e x p l i c i t

so that

as m a r k e d

deformation

in

G, Bers has

space has b e e n

and w i l l not be con-

lecture).

objects

of our study are c o n f o r m a l

equi-

82 v a l e n c e classes of Riemann surfaces.

If we r e s t r i c t our atten-

tion to Riemann surfaces of fixed signature, is r e p r e s e n t e d

(infinitely often)

of one surface

SO

equivalence

T(S0)

in

Mp, n

in the T e i c h m O l l e r space

of that signature.

Mp, n.

the group of changes of m a r k i n g s on

T(S 0)

The relation of c o n f o r m a l

The subgroup

is the group of proper a u t o m o r p h i s m s of

fication orders.

In some cases,

is not effective.

mentioned

SO

which

Mp,n(S 0)

G O , i.e.

respect

the a c t i o n of

Mp,n(S0)

The T e i c h m O l l e r m o d u l a r group,

in Chapter 4, is defined to be exactly

d e f i n i t i o n differs

T(S 0)

is induced b y a r e p r e s e n t a t i o n of a sub-

group of the m a p p i n g class group of

every such surface

in that

Mod G O = Mod S O

its subgroup of i n e f f e c t i v e elements.

is

ramion already

Mp,n(S0). Mp,n(S 0)

Our modulo

The a r g u m e n t s of Chapter 4

may then be r e p e a t e d v e r b a t i m to show that the Riemann space of

SO,

R(G0) = R(S 0) = T ( G ) / M o d G O ,

is a n o r m a l complex space.

We have thus c o n s t r u c t e d a h o l o m o r p h i c

p a r a m e t r i z a t i o n of the c o n f o r m a l e q u i v a l e n c e classes of Riemann surfaces of a given signature. The theory of m o d u l i of n o n - s i n g u l a r a l g e b r a i c curves of genus one,

i.e. compact surfaces of genus one w i t h o u t

d i s t i n g u i s h e d points is quite classical. that the field of m e r o m o r p h i c g e n e r a t e d by the w e i e r s t r a s s where

T 6U

It is k n o w n for example,

functions on such a surface is function

~(z;l,T)

and

d~(z:l,~)/dz

= T e i c h m ~ l l e r space of Riemann surfaces of type

(I,i).

83 For surfaces of h i g h e r genus the complex n u m b e r b y a point in

Theorem i:

T(S0).

I_~f S O

T

is r e p l a c e d

A m o n g other results we have the following:

is a h y p e r b o l i c Riemann surface of finite

type w i t h signature and

S

is a surface c o r r e s p o n d i n g to a T

point

T

i__nn T(S0),

d i f f e r e n t i a l s on

S

- -

then the periods of n o r m a l i z e d A b e l i a n are h o l o m o r p h i c

functions of

T.

T

More striking results may be o b t a i n e d using fiber spaces over the T e i c h m ~ l l e r space

(see Bers

[5]).

E x c e p t in the m o s t trivial of cases, R(S)

is not compact.

possible

"natural"

(see B a i l y

This leads i m m e d i a t e l y to questions a b o u t

c o m p a c t i f i c a t i o n s of m o d u l i spaces of

their i n t e r p r e t a t i o n s m u c h too large.

in terms of the surfaces

S.

M(S)

S

and

is

The Satake c o m p a c t i f i c a t i o n of the Torelli space

[3])

adds a "boundary" of too large a codimension.

The T e i c h m O l l e r space

T(S)

as a b o u n d e d domain in a complex

vector space has a n a t u r a l c o m p a c t i f i c a t i o n . points c o r r e s p o n d to d e g e n e r a t e groups groups

the Riemann space

"represent"

But m o s t b o u n d a r y

(see Bers'

lecture).

the d i s a p p e a r a n c e of the d e f o r m e d surface

Such S.

Hence this c o m p a c t i f i c a t i o n of itself yields v i r t u a l l y no inform a t i o n a b o u t limits of d e f o r m a t i o n s of surfaces. [Before proceeding,

we note that "degeneracy"

in the

theory of K l e i n a i n groups and in a l g e b r a i c g e o m e t r y has quite d i f f e r e n t meanings.

In the following d i s c u s s i o n of c o m p a c t i f i c a t i o n

of the space of moduli,

the K l e i n i a n groups u s e d to r e p r e s e n t

d e g e n e r a t e a l g e b r a i c curves are regular b - g r o u p s and c o n s t r u c t i b l e

84

groups.

These groups We may

surface groups

S

consider

the space of m o d u l i

to be the space of c o n j u g a c y

of a fixed signature.

of a t h e o r e m

Theorem

2

of M u m f o r d

(Bers

group of the classes

[6]):

of F u c h s i a n

y [~

Let

groups

X(G)

2 + ¢ >

G

and

classes

let

of F u c h s i a n

recent

the c o m p a c t

be a finitely X(G)

isomorphic

consisting 2

of a h y p e r b o l i c

The f o l l o w i n g

characterizes

first kind,

the s u b s e t of [ trace

are not at all degenerate.]

be too

G.

of

R(S).

Fuchsian

space o_~f c o n j u g a c y Let

G'

for all h y p e r b o l i c

subsets

generated

the

o_ff groups

generalization

X

(G)

be

(G)

is

with

y 6 G.

X

e

compact.

Geodesic or slits

connecting

called admissible to 2 for some admissible

surfaces

S

removal

n

pinched.

points

of order

The c o n d i t i o n

in

R(S)

to zero.

to zero;

we

curves

lengths

of

obtained

set of a d m i s s i b l e the

be

R(S)

~(S) from

curves.

a diverg-

C

of whose

n of

R(S)

- R(S) S

b y the

In terms of the

of these a d m i s s i b l e

say that these

close

of a short

a sequence

The c o m p a c t i f i c a t i o n

is one in w h i c h the points

structures,

y

In particular,

geodesic

loops

2, w i l l be

that trace

represents

surfaces

simple

to the e x i s t e n c e

U/G'.

admissible

(topologically)

set equal

w h i c h are either

is e q u i v a l e n t

containing

of a finite

conformal been

curves.

of p o i n t s

seek

represent

U/G'

ramification

y 6 G'

converge

w h i c h we

on

curve on the surface

ent s e q u e n c e

lengths

curves

curves have

curves have b e e n

85

R(S) If

is called the a u q u m e n t e d m o d u l i space. S

is a compact surface w i t h o u t d i s t i n g u i s h e d

points, M u m f o r d and M a y e r have proved, b y a l g e b r a i c methods,

Theorem 3:

R(S)

is a compact normal complex space.

There is no c o m p l e t e p r i n t e d proof in the literature (cf., D e l i g n e and M u m f o r d

[8] w h e r e the case of p o s i t i v e char-

a c t e r i s t i c is treated). The r e m a i n d e r of this talk is a d i s c u s s i o n of w o r k in p r o g r e s s on two a n a l y t i c a p p r o a c h e s R(S)

to the p r o b l e m of d e f i n i n g

and d e t e r m i n i n g its properties.

approaches,

There are other a n a l y t i c

e.g. u s i n g the space of F u c h s i a n groups

[i0]) and u s i n g techniques

(see Harvey

from 3 - d i m e n s i o n a l t o p o l o g y

(see

M a r d e n ' s talk).

§l. P R O P E R PARTITIONS OF SURFACES A N D S U R F A C E S W I T H NODES

i.i

Let

let

{~i ..... ~ } 3

such that

S

be a h y p e r b o l i c Riemann surface w i t h s i g n a t u r e and

S\~ i

be a set of d i s j o i n t a d m i s s i b l e curves on is a u n i o n of h y p e r b o l i c surfaces

each of w h i c h is of finite type P =

IS 1 ..... Sk}

e l e m e n t of partition m

(p,n)

with

is called a proper part of

P).

It is w e l l k n o w n that if

d i s t i n g u i s h e d points,

then

S S

S 1 ..... S k

3p - 3 + n ~ 0.

is called a proper p a r t i t i o n of

P

S,

S

and each

(in the proper has genus

p

and

86

n~

If e q u a l i t y holds, is maximal,

3p - 3 - m

the p a r t i t i o n

then each

S. 3

P

is called maximal.

has type

If

P

(0,3); there are only

finitely m a n y h o m e o m o r p h i s m classes of m a x i m a l partitions. There is an obvious partial o r d e r i n g on p a r t i t i o n s and each p a r t i t i o n m a y be refined to a m a x i m a l partition.

It

follows that there are finitely m a n y h o m e o m o r p h i s m classes of proper partitions. The following theorem c o n j e c t u r e d by M u m f o r d was p r o v e d b y Bers

[7].

Theorem 4:

E v e r y surface

S

w i t h signature

admits a m a x i m a l proper p a r t i t i o n curves w h o s e

P

c =

(p,n;~l, .... Vn )

d e f i n e d bv a d m i s s i b l e

lenqths are b o u n d e d b y a constant

L

depending

only o__nn o.

If we consider a sequence same signature,

and p a r t i t i o n s

(p) n

(Sn)

of surfaces w i t h the

induced by h o m e o m o r p h i s m s

fn: S1 " Sn' then it is p o s s i b l e that the length of i.e. the sequence of d e f o r m a t i o n s a d m i s s i b l e curve

~.. l

(Sn)

of

S1

fn(~i)

- 0,

pinches the

R e p r e s e n t a t i o n s of such d e f o r m a t i o n s b y

limits of q u a s i c o n f o r m a l d e f o r m a t i o n s of K l e i n i a n groups m a y be o b t a i n e d in several ways.

A sequence of q u a s i - F u c h s i a n groups

may converge to a regular b - g r o u p in a "canonical"

fashion

Abikoff

fixed points

[ ]).

The t r a n s f o r m a t i o n p a i r i n g elliptic

in the example given in Bers'

talk

(see

(p. 9) may be d e f o r m e d to the

87

identity. Harvey

1.2

There are other m e t h o d s

such

either

to the u n i t

a g a i n to S

surface

discs,

P).

of

S

the centers case

P

but with

from the nodes

produced by point

surface w h i c h

removing

of order

f i n i t e l y m a n y parts,

hyperbolic follows,

if each part

and use on A proper

S

curves,

of

S

S

points.

we agree

are

that a

is c a l l e d of finite type.

to b e length

is d e f i n e d included 0.

S

defined

type

~

which

in w h a t

on the parts.

the a d m i s s i b l e

4 remains

is n o w the set of s i g n a t u r e s

are j o i n e d

A continuous

valid.

of the parts

in a node.)

surjection

f: S'

~ S"

if

is c a l l e d

S, t o g e t h e r w i t h a list of the pairs of r a m i f i c a t i o n

of o r d e r

a

just as before,

among

Theorem

a

(i.e. w i t h -

We a s s u m e b o t h c o n d i t i o n s

S

a

E v e r y part

is to be c o n s i d e r e d

each of finite

of

S

1 or ~), we o b t a i n

the P o i n c a r 6 m e t r i c

a n d are a s s i g n e d

(The s i g n a t u r e

~.

is.

partition

that the nodes

a node

A part

and a s s i g n

is n o n - s i n g u l a r

points;

or

(and c o r r e s p o n d i n g

If w e choose on

(integers >

ramification

isomorphic

is c a l l e d a node.

distinct

numbers

complex

to the center)

identified

S\{all nodes~.

of points,

is n o w a R i e m a n n

ramification

of

corresponding

surface w i t h nodes a n d r a m i f i c a t i o n

puncture

except

of

P

is a c o n n e c t e d

has a n e i g h b o r h o o d

with

In the second

sequence

out nodes)

it has

P 6 S

(with

to them r a m i f i c a t i o n Riemann

w i t h nodes

disc

is a c o m p o n e n t

discrete

S

that every p o i n t

to two u n i t

of

[ll] and

[i0].

A Riemann

space

due to M a s k i t

is c a l l e d a

points

88

deformation avoiding order point f

-i

if

f(node)

nodes

= node,

and r a m i f i c a t i o n

~) = r a m i f i c a t i o n of order

~ <

(ramification or a J o r d a n

and avoiding restricted

(node)

points,

~) = r a m i f i c a t i o n

point of order

arc

-i

joining

all o t h e r

= node or J o r d a n

f(ramification

p o i n t of o r d e r

~, f

point

points

of inverse points,

~,

p o i n t of o r d e r

points

of o r d e r

and nodes,

images

of

(ramification

~) = r a m i f i c a t i o n

ramification

into r a m i f i c a t i o n

-i

curve

point

of o r d e r

two r a m i f i c a t i o n

to the c o m p l e m e n t

arcs m a p p e d

f

2

a n d if

of nodes

f,

and J o r d a n

is an o r i e n t a t i o n

preserv-

ing h o m e o m o r p h i s m . If

S

is non-singular,

theoretically, surfaces without ~(S)

S'

M

can be defined,

as the set of i s o m o r p h i s m into w h i c h

ramification

by

~(S)

S

points,

classes

m a y b e deformed. and of genus

of all R i e m a n n

If

p >

set-

S

is compact,

i, we denote

. P

§2.

THE

COMPACTIFICATION

M

P

N o w we d e s c r i b e

an a n a l y t i c

construction

of

M

.

(See

P [7] and Bets'

forthcoming

paper

for d e t a i l s

and

for more

general

constructions.) We begin by choosing Fuchsian

groups

on d i s j o i n t apart, 2p - 2

G 1 ..... G 2 p _ 2

discs

that the surfaces

an integer

of s i g n a t u r e

A 1 ..... A2p_2. G. 3

generate

of type

~ ~

The discs

a Kleinian

(0,3)

and one

3

and

2p - 2

(0,3;~,~,v) should be

group

acting

so far

representing

surface

~

of type

89 (0,6p - 6). 6-gon,

Each

G. ]

has in

3 of whose vertices

number these

6p - 6

U A2p_2/G2p_2,

identified,

becomes

order

~

in Let

G. 3 t =

~'2j-i and if

GI,...,G2p_2

and

set of those

t

j.

S1

in different ways.

We

fixed.

6 C 3p-3

Define

Let

G

~

~j,

to be the identity

for w h i c h

Gt

~2j-i

containing

and let

~2j

SI, t

t. ]

and

~2j-i + t ,3

~ c C 3p-3

be the

is Kleinian and represents

and

surface

~2j

be the group g e n e r a t e d b y

t

gt,l .... 'gt,3p-3

components

be such that

gt, ~3

into

of

T

are identified w h e n e v e r

w i t h as many nodes as there t, and of Poincar~ area

The origin is an interior point of interior of

~2j-i

t r a n s f o r m a t i o n w h i c h takes

t. ~ 0. 3

a Riemann

are v a n i s h i n g

~. 3

~2j-i + t.3

(after images of

possible)

of course,

(t I, .... t3p_3) for all

respectively,

and

and

(and thus a Riemann surface

which keeps

t. = 0, and the M~bius 3

~2j-l'

~2j

the other fixed point of an elliptic element of

~2~-13 + t. 6 A. ] 3 if

We

in such a way that

with the images of

connected

(non-Euclidean)

fixed points.

~i ..... ~6p-6'

this can be done, ~j!

denote b y

a fundamental

are elliptic

points

AI/G l U ...

w i t h nodes);

A. 3

H

and the component

2~(2p-2). of the

the origin will be denoted by

X(S 1 ) . The domain

X(SI)

turns out to be a cell.

represent Riemann surfaces, d e f o r m e d into

S I.

represent

isomorphic

relation

p

on

w i t h or w i t h o u t nodes, which can be

Of course, surfaces.

X(SI).

Its points

distinct points

of

X(S I)

can

Thus we have an equivalence

This relation

is not described b y a

90

discontinuous is a n o r m a l

of a u t o m o r p h i s m s .

complex

The priate

group

of

ways,

homeomorphic

Riemann

and

parts

connected normal

spaces

can

their

form

of type

compatible

M

We also s

in

classes

it turns

U ...

out

U X(S

s

structures

becomes

s

of R i e m a n n

of n o n nodes

simply

C 3p-3

Since

it is

3p - 3

obtain

)

do

s = s(p)

, each with

X(S I) ..... X(S

and

the c o m p l e x and

s

an a p p r o -

If we

number

X ( S l ) / p ..... X ( S s ) / p .

X(SI)/p

Furthermore,

SI,...,S

isomorphism

union,

a choice:

vertices.

a certain

(0,3).

domains

are

involved

elliptic

surface

spaces

of t h e s e

S1

we obtain

bounded

complex

of

3p - 3

all possible

2p - 2

X(SI)/p

space.

construction

ordering

Nevertheless,

and

s

the p o i n t s surfaces,

we

that

)/p = M

p

of the

s

a normal

complex

can

that

space

are

space.

P

compact M

Using

Theorem

part

K., 3

such

4 one that

M

show

C Kl/P

each

U ...

X(Sj)

U Ks/P.

has

a

Hence

P

is compact. P

§3.

~(S)

AS A Q U O T I E N T TEICHMULLER

The a u g m e n t e d surface

S

with

regular

the

theorem

or

suggests

group

SPACE

Teichm~ller G

b-groups that

OF THE A U G M E N T E D

some

space

is the u s u a l

~(S)

Teichm~ller

on its b o u n d a r y . fundamental

or

set

~(G)

of a

space

together

The M u m f o r d - M a y e r for

Mod

S

or

Mod G

91 may be compactified or

~T(G).

or

~(G).

by some of the regular b-groups

We may thereby obtain a compactification Details

will appear

in the forthcoming

in

ST(S)

of

~(S)

paper of

Abikoff. We first define a topology usual

topology

on

T.

the mirror

a proper partition by a neighborhood and a positive there

is a

S' (G I) to

minus

image

S

S.

number

which

GO

of if

is a regular b-group,

A neighborhood

¢.

f

A group

admits

A T

curve.

the

a neighborhood

of a surface

G1

S

and

NK, e(G0)

lies in

it

(topologically) is determined

f

of

NK, ¢(G0)

if

S' (Go)-K

into

a locally quasieonformal S' (G I)

or is onto

In this topology,

is continuous.

filter

S' (G O ) = ((~(G0)/G 0) - S)

mapping

is either onto

group on

neighborhood

which extends

of the paired nodes on

an admissible

(i)

If

of

such that:

modular

GO .

(l+c)-quasiconformal

S' (G O )

~

We must only define

for a regular b-group represents

on

extension S' (G I)

the action

NK, ¢(G0)

of the

is a horocyclic

G O , i.e. g E Mod(G)

g(NK, e(G0))

and

g(G 0) = G O

= NK, c(G0),

(2)

if

g(G 0) = G 1

(3)

if

K, e

then

then

and

g(NK, ( G o ) )

are sufficiently

small,

is a neighborhood

of G 1

then

g(NK, ( G 0)) n NK, ¢(G 0) = ~NK, ¢(G 0)

for

g

in the modular It follows

topological

that

group. ~(-)

= ~(-)/Mod(-)

space and is, in fact,

Hausdorff.

has a well defined To show that it is

92

compact,

we must

s h o w that if

subsequence

Gn. 1 ) } _co T(G)

{gi(Gn

GO .

For

or

any

Mumford conjecture partition

P

in

S' (GI)

c u r v e s of u n i f o r m l y

C ~(G) gi

then t h e r e e x i s t s

E Mod G

gi(Gn.) c NK, c(G0) l H 6 ~(G0), a slight

noted

of

n

and a sequence

1

b-group

G

§2 y i e l d s

defined by

bounded

length.

the

so t h a t

for some r e g u l a r

extension

of

existence

3g - 3 + m

of

the a proper

admissible

For a s e q u e n c e

G n E ~(G 0)

o n l y f i n i t e l y m a n y of the c o r r e s p o n d i n g

partitions

inequivalent

to a s u b s e q u e n c e

find

gi

under

E Mod G O

Mod G O .

By passing

a

P

may be

n

we may

Gn. so t h a t gi(Gn ) are convergent 1 1 in B2(G0) and the partitions of S' (gi(Gn.)) are consistent, 1 i.e. t h a p a r t i t i o n s P are d e f i n e d b y the same c u r v e s n. l relative to a consistent marking of the S' (Gn.). A slight i generalization of Theorem 2 of Abikoff [1] s a y s t h a t t h e l e n g t h s o f the a d m i s s i b l e

and groups

curves

relative

defining

a proper partition,

gi(Gn

converge

)

to a f i x e d m a r k i n g

cannot

to a d e g e n e r a t e

stay uniformly

b-group.

and

bounded

It f o l l o w s

if the

that

l

gi(Gn the

) converge to a regular b-group or quasi-Fuchsian group 1 relative topology of ~(G 0) in B2(G0). A more refined

argument

s h o w s t h a t the c o n v e r g e n c e

~-topology

actually

occurs

in the

d e f i n e d above.

REFERENCES [ i]

W. A b i k o f f , groups,

[ 2]

Two t h e o r e m s

to a p p e a r . , to a p p e a r .

on t o t a l l y

degenerate

Kleinian

in

93

[ 3]

[ 4]

W.

L. Baily,

Jr., On moduli of Jacobian varieties,

of Math.,

7_!(1960),

L. Bets,

Holomorphic

Bull. Amer. Math. [ 5]

[ 6]

303-314. differentials

Soc.,

,

[ 7]

6_~7(1961), 206-210.

A remark on Mumford's 12(1972),

, Spaces of degenerate Groups and Riemann Surfaces, P. Deligne and D. Mumford, curves of a given genus,

[ 9]

C. Earle and A. Marden,

[i0]

W. Harvey, Surfaces,

spaces,

compactness

Acta Math.,

Ann.

B. Maskit,

[12]

O. TeichmOller,

of Math.

theorem,

400-407. Riemann surfaces, Ann. of Math.

Irreducibility

I.H.E.S.,

Discontinuous

Studies,

79(1974).

of the space of

3_~6(1969), 75-109.

to appear.

to appear in Discontinuous

[ii]

Studies,

Groups and Riemann

7-9(1974).

to appear.

quadratische Nat.

of moduli,

89-126.

Israel J. of Math.,

[ 8]

as functions

, Fiber spaces over TeichmOller 130(1973),

Ann.

Extremale

quasikonforme

differentiale,

K_!l., 22(1939).

Abh.

Preuss.

Abbildungen Akad. Wiss.

und Math.

7.

GOOD AND

BAD K L E I N I A N

GROUPS

Bernard M a s k i t SUNY,

There class

are several

of f i n i t e l y

Stony

well k n o w n

generated

kind.

More

generally,

groups

with

an invariant

classified groups

groups

characterizations

or perhaps

several

not n e c e s s a r i l y known

about

groups

which

have

sense

several

of the first

can be c o m p l e t e l y

that

these F u c h s i a n

This

different

can be used

class

Even

to d i s t i n g u i s h

component.

less

of good

characterizations.

- of good K l e i n i a n

an invariant

of the

of good K l e i n i a n

which

classified.

such groups.

is known

a class

groups

The f o l l o w i n g

characterizations

whicm

Very little

p l a n e U,

do is

about K l e i n i a n

FUCHSIAN

theorem, of r class

GROUPS

which

gives

of F u c h s i a n

several

of the m a n y

groups,

is well

known.

Theorem.

-

are not good.

§l. 1.1

component

classes

groups

is a class

the same

also has

characterizations

Fuchsian

there

can be c o m p l e t e l y

Kleinian These

in exactly

Brook

For a Fuchsian the f o l l o w i n g

group r acting

statements

on the upper half

are equivalent.

95

i) li)

r is finitely generated

and of the first kind.

U/T is a finite Riemann surface;

i.e. U/T is a

compact Riemann surface from which finitely many P0ints

have been removed and the covering U ~ U/F

is branched

iii) iv)

over finitely many points.

U/F has finite n o n - E u c l i d e a n F has a finite-slded

area.

fundamental

polygon,

and is

of the first kind. 1.2

One can (and sometimes

as being F u c h s i a n groups

- operating

disc - of the second kind. Fuchsian groups,which i.e.,

should)

There

think of K l e i n i a n groups on the 3-dimensional

is also a class of "good"

are not n e c e s s a r i l y

the finitely generated

ones.

of the first kind;

The following

theorem is

also well known. Theorem.

For a F u c h s i a n

plane U, the following i) ii)

statements

are equivalent.

F is finitely generated. U/F is homeomorphic orientable ~ranched

iii)

group F acting on the upper half-

to the interior

2-manifold

of a compact

and the covering U ~ U/F is

over finitely m a n y points.

K/F has finite n o n - E u c l i d e a n Nielsen convex region;

area, where K is the

i.e. K is the n o n - e u c l i d e a n

convex hull of A (= limit set of F). iv)

F has a finite-sided

fundamental

polygon.

96 ~2. CLASSIFICATION 2.1

OF FUCHSIAN GROUPS

F i n i t e l y generated F u c h s i a n groups

have been classically

classified;

of the first kind

we describe

one f o r m of

this classification. There is a countable ated F u c h s i a n groups

collection

[Fi]

of finitely gener-

of the first kind with the following

properties. i)

If F is a finitely generated F u c h s i a n group of the

first kind,

then for some i, F is a quasiconformal

deformation

of F i . ii) of Fj,

If i ~ J, F i is not a quasiconformal in fact the coverings

topologically iii)

U ~ U/F i and U ~ U/Fj are

distinct.

Each point in the Teichm~ller

space of quasiconformal invarlant

deformations

isomorphism

space of F i (the

of F i which keep U

and which are appropriately

to a unique

deformation

normalized)

corresponds

of F i onto some Fuchslan group;

the

m a p p i n g from the Teicnm~ller

space onto such Isomorphisms

one-to-one

(if one carefully

subclass

and real analytic

of allowable

isomorphisms,

chooses

then the mapping

is

a

is also

onto). iv)

One can topologically

as the branched

universal

describe

covering

the covering U ~ U/F i

of U/F i.

97 ~8. 3.i

CLASSIFICATION

A function

group

with an invariant function (C),

(O),

of the proof

appears

in §12.

As for Fucnsian

function cies,

groups.

satisfies

that these

~roups,

conditions

there

redundant

list of good function

invariant

component

groups

[Gi],

Ai, with the following

deformation

serve parabolic iii)

T(Gi)

space of quasiconformal a manifold

whose

space

compon-

of some G i, where

Ai onto A. deformation

f : Ai ~ Aj which

(or space

can be described

of two factors

some redundan-

would not pre-

in both directions).

The deformation

deformations)

list of

properties.

but this conjugation

elements

an

where G i has

deformation

w maps

(A),

below;

group with invariant

of Gj (there might be a homeomorphism G i onto Gj,

is a

so as to get a non-

If i @ j, G i is not a quasiconformal

conjugates

group

are all equivalent

is a countable

further

ent A, then G is a quasiconformal

product

group

in [7] contains

If G is a good function

ii)

Kleinian

in the parag,aphs

The list given

the quasiconformal

generated

anY one of the conditions

but one easily normalizes

i)

GROUPS

A good function

(S) or (F) gSven

outline

3.2

is a finitely

component.

group whica

(C'),

OF GOOD FUNCTION

T'(Gi)

universal

as follows.

and T"(Gi).

deformations covering

of quaslconformal

T'(Gi)

with support space

T(Gi)

is a

is the in Ai; it is

is the Teichm~ller

98

space of the branched T"(Gi)

universal

is the space of deformations

side A i, it is a product spaces

(see Kra's

iv)

support lies out-

of lower dimensional

deformation

Roughly,

simple disjoint

homotopically

~l ..... a n be positive covering

in terms of the covering

this description

S be a finite Riemann surface;

the elements

distinct

integers

loops on S; and let

regular

covering

The

of

the loops wj j (aj < ~) all lift to loops;

and in a natural

The statements

to those wj w~ere aj = sense,

account for all parabolic

t~e p r e c e d i n g

elements

state-

of G i.

in 3.2 do not appear as such in print,

but are easy consequences

of known results.

statement

iv) the groups G i are constructed

Statement

i) follows from Maskit

ment li) is simply normalization. iii) follows

Let

(we also allow ~i = ~)"

of G i corresponding

are parabolic;

is as follows.

let Wl,...,w n b@ a set of

A i ~ Ai/G i is the highest

Ai/G i for which

from Bers ~ technique

using quasiconformal to Kra

Teichm~ller

Each of the groups G i can be uniquely described

A i ~ Ai/G i.

3.3

whose

of AI/G i.

lecture).

up to quasiconformal

ments

covering group

[5] and Maskit

mappings; [Ii].

Starting with in [8] and

[lO] and M a r d e n

[6].

[9]. State-

The main part of statement of v a r i a t i o n of parameters

the other statements

are due

99

94. A L G E B R A I C CONDITIONS 4.1

W h e n we try to separate good f r o m bad groups,

obvious

algebraic c o n d i t i o n is to require that G be f i n i t e l y

generated. related,

the most

One m i g h t expect to ask that G be f i n i t e l y

but there is a theorem of Scott

that if G is f i n i t e l y generated, (see M a r d e n ' s

[131 which implies

then it is f i n i t e l y related

lecture).

One could v i e w the statement of G is parabolic

that a p a r t i c u l a r element

as an algebraic statement,

this added information,

but even with

we cannot separate good from bad

groups with p u r e l y algebraic

information.

For example,

tmere are degenerate groups - clearly bad - w h i c h are p u r e l y loxodromic

and which are isomorphic

to F u c h s i a n groups.

F r o m here on we will deal only with groups w h i c h are f i n i t e l y generated,

for we expect that all our good groups

will be.

§5. T O P O L O G I C A L CONDITIONS 5.1

The most obvious topological c o n d i t i o n comes from looking

at the action of a K l e i n i a n group G on the hyperbolic ball B 3 (see Marden's

lecture);

we could require that B3/G be homeo-

m o r p h i c to the i n t e r i o r of a compact 3-manifo!d and that B 3 be branched over f i n i t e l y m a n y circles One easily sees that if G satisfies f i n i t e l y generated;

and points

of B3/G.

this c o n d i t i o n then G is

n o t h i n g is known about the converse,

nor

100

is anything known about any relationship tion and questions 5.2

concerning

good and bad groups.

The next step is to look at the action of G on ~(G)

to require surfaces

where ~ is branched Theorem

over finitely many points.

[i] (also see Gardiner's

that if G is finitely generated tion.

and

that fl(G)/G be a finite union of finite Riemann

Finiteness

The converse

degenerate

groups

is false;

satisfy

§6. METRIC 6.1

between this condi-

lecture)

then G satisfies

furthermore

Ahlfors

asserts

this condi-

both Fuchsian

and

this condition. CONDITIONS

We again regard G as acting on B 3, and obtain our first

real condition. (A)

G has a finite-sided

A group

satisfying

fundamental

condition

polyhedron.

(A) is called g e o m e t r i c a l l y

finite. This condition

can apparently

that every fundamental it was shown by Beardon

polyhedron and Maskit

be strengthened

by requiring

nave finitely many sides; [2] that if G satisfies

then indeed every convex fundamental

polyhedron

6.2

Nielsen convex region,

One can also form a generalized

is finite-sided.

K, and one could ask that K/G have finite volume. dition p r e s u m a b l y far nothing

is equivalent

to condition

has been done along these lines.

(A),

This con-

(A), but thus

101

§~ 7.1

STABILITY

A K l e i n i a n group G is called stable

p h i s m from G into PSL(2;¢) and which is s u f f i c i e n t l y an i s o m o r p h i s m support

if every homomor-

which preserves

parabolic

close to the identity,

induced by a quasiconformal

elements,

is in fact

deformation

with

in ~.

We remark that it makes

sense to ask if G is stable only

if G is finitely generated. (B)

G is stable.

7.2

Theorem

fies

(A), then G satisfies

(B)

One expects

to hold even if G is not torsion-

free.

(Marden

the above

The converse

that degenerate introduced Teichm~ller

[6]).

If G is torsion free and G satis-

is not known,

groups

but it is known

are not stable;

spaces

and degenerate

see Gardiner

and Kra

§8. E X T E N D A B I L I T Y 8.1

One possible

of

groups. to certain cohomological

[4].

OF MAPS

criterion for niceness

should be determined

[3])

stability was

by Bers as part of his study of boundaries

One can also relate stability conditions;

in fact,

(Bers

is that a group

by its action on its set of discontinuity;

i.e. for any K l e i n i a n group G* other than G, if ~(G*)/G* ~(G)/G "look alike" ation of G.

then G* should be a quasiconformal

and

deform-

102

In general, groups

an isomorphism ~ : G ~ G* b e t w e e n K l e i n i a n

is called

parabolic

type-preserving

elements,

of every elliptic 8.2

if both ~ and ~-i preserve

and @ preserves

element.

A group G is q u a s i c o n f o r m a l l y

preserving conformal

the square of the trace

extendable

if every type-

isomorphism ~ : G ~ G* whicm is induced by a quasihomeomorphism

by a quasiconformal One expects pmism of ~(G)

of ~(G)

onto ~(G*)

is in fact induced

deformation.

every isomorpmism

onto ~(G*)

induced

by a homeomor-

to be type preserving;

this is known

only for good function groups. Similarly, type preserving

a group G is conformally isomorphism

is in fact induced

if every

between G and some other group G*,

which is induced by a conformal ~(G*),

extendable

homeomorphism

by a fractional

of ~(G)

onto

linear transforma-

tion. (C)

G is conformally

extendable.

(C') G is quasiconformally Using the existence one easily sees

extendable.

of solutions

that if G satisfies

the converse

is not known.

8.3

Theorem

(Marden

fies

(A),

[6]).

then G satisfies

of the Beltrami

(C), then G satisfies

If G is torsion-free (C).

equation, (C');

and G satis-

103

The converse to this t h e o r e m is not known, relationships

b e t w e e n conditions

It was shown by Bers satisfy

(C); one expects

nor are any

(B) and (C) known.

[3] that degenerate groups

do not

that they also do not s a t i s f y (C'),

but this is not known.

§9. THE L I M I T SET 9.1

One of the first possible

Anlfors

conditions was that g i v e n by

in [1] where he asks if the limit set of G n e c e s s a r i l y

has zero 2 - d i m e n s i o n a l measure.

There are n o n - f i n i t e l y gener-

ated groups whose limit sets have positive measure,

but it is

not k n o w n w h e t h e r there are f i n i t e l y generated groups w i t h p o s i t i v e area limit sets. If there were a group w h i c h satisfied

(C') but not

then that group would have a limit set of positive

9.2

For our next condition,

(C),

area.

we need some definitions.

If H is a subgroup of G, a set A c ~ is called p r e c i s e l y invariant under H if A is invariant under H, and g(A)

~ A = ~,

for all g 6 G-H. A fixed point z of a p a r a b o l i c element of G is called cussed if G z has rank 2, or if there is a set A which is the disjoint union of two open c i r c u l a r discs

(or half-planes)

where A is p r e c i s e l y invarlant under G z. A limit point z of G is a point of a p p r o x i m a t i o n if there is a point x and there is a sequence

[gn } of distinct

104

elements

of G so that the s p h e r i c a l distance

does not converge (D)

[gn(z),gn(X)]

to O.

E v e r y limit point of G is either a cusped p a r a b o l i c fixed point or is a point of approximation.

9.3

Theorem

(Beardon and M a s k i t

only if G satisfies

[2]).

G satisfies

(A) if and

(D).

§i0. M A X I M A L I T Y i0.i Or~of the most obvious groups

characteristics

of degenerate

is that ~(G)/G is in an obvious sense s m a l l e r than it

should be.

One p o s s i b l e way of d e s c r i b i n g this is in terms

of dimensions

of d e f o r m a t i o n spaces.

A K l e i n i a n group G is m a x i m a l if for every type-preserving i s o m o r p h i s m , m a p p i n g G onto some other K l e i n i a n group G*, the d i m e n s i o n of the d e f o r m a t i o n space of G is not less than that of G*. (E)

G is maximal.

Unfortunately,

all that is k n o w n about this c o n d i t i o n is

that the usual bad groups

don't s a t i s f y this c o n d i t i o n either.

There are several other m a x i m a l i t y conditions w h i c h are e q u a l l y well understood.

§Ii.

CONSTRUCT~BIL!TY

II.i T~ere are two simple c o n s t r u c t i o n s w h i c h one can use to build more complicated groups from simpler ones;

we outline

105

these

here,

details

appear

If H is a F u c h s i a n B is an open

[12].

or q u a s i _ F u c h s i a n

topological

under H in G,

in

disc w h i c h

then we call

subgroup

is p r e c i s e l y

B a regular

of G,

and

invariant

disc for H if

n(H) n ~B c n(G). Combination iant

under

group

H.

I.

Let ~ be a s i m p l e

the f i n i t e l y Let

B 1 and B 2.

B 1 is a regular

disc for H in G 2.

by G 1 and G 2 is K l e i n l a n and G 2 via C o m b i n a t i o n

Combination curves

II.

bounding

B i is a smooth finitely there

topological

disc for H i in GI, Fuchsian

the group

discs

i = 1,2;

f w h i c h maps

generated

simple

from G 1

B 1 and B2,

where

here H i is a group.

B 1 onto

Then G,

closed

Suppose

the complement

the group g e n e r a t e d

and we say that G is formed

from G 1

II.

G is c o n s t r u c t i b l e

groups

by y be

for H in G 1 and B 2

or q u a s i - F u c h s i a n

H 1 into H 2.

by G 1 and f is Kleinian, and f via C o m b i n a t i o n

is invar-

I.

disjoint

of B 2 and conjugates

tions

Then G,

bounded

and we say that G is formed

is a t r a n s f o r m a t i o n

elementary

disc

which

or q u a s i - F u c h s i a n

discs

Let Y1 and Y2 be disjoint

generated

A group

curve

Fuchsian

the two open t o p o l o g i c a l

Suppose

is a regular

generated

closed

and F u c h s i a n

I and II a finite

number

if it can be built triangle of times.

groups (This

up from

using

Combina-

definition

is

106

not quite corect, complicated technical modifications are needed for groups with parabolic elements.) (F)

G is constructible.

One easily proves that if G satisfies fies (D), and hence (A). satisfies

(F), then G satis-

One can also easily show that if G

(F) then it satisfies (B).

There are tme usual

examples of groups that do not satisfy (F). §12. FUNCTION GROUPS For function groups it was shown in [I0] that (E) and (F) are equivalent.

Using the Combination

one easily proves that (F) implies

(D).

Theorem [12],

We have already

remarked that (D) is equivalent to (A) [2] which implies (B), (C) and (C')

[6].

One shows that (C) ~ r (C'))implies

(F) by using [I0] and the non-uniqueness of degenerate groups due to Bers [3].

There is little doubt that (B)

similarly implies (F), but there is no proof in the literature.

107

REFERENCES l°

L . V . Ahlfors, Finitely generated Kleinian groups, J. Matm. 86 (1964~ 413-429.

Amer.

2.

A. Beardon and B. Maskit, Limit points of Kleinian groups and finite-slded fundamental polyhedra, Acta Math. to appear.

3.

L. Bers, On boundaries of Teichm~ller spaces and on Klelnian groups: I, Ann. of Math. 91 (1970), 570-600.

4.

F. Gardiner and I. Kra, Stability of Kleinian ~roups, Indiana Math. J. 21 (1972), 1037-1059.

5.

I. Kra, On spaces of Kleinian groups, Helv. 47 (1972), 53-69.

6.

A. Marden, The Geometry of finitely generated groups, Ann. of Math., to appear.

7.

B. Maskit, Uniformizations of Riemann surfaces, to appear in Contributions to Analysis, Academic Press, New York, 197q.

8.

, Construction of Kleinian groups, Proceedings of the Conference on Complex Analysis, Minneapolis, 1904, Sprlnger-Verlag, Berlin, 1905, pp. 2~I-29b.

9.

, On Boundaries of Teichm~!!er spaces and on Kleinian groups: II, Annals of Math. 91 (1970~ 607-639.

i0.

, Decomposition of certain Kleinian groups, Acta Math. 180 (1973), 248-263

ii.

, Self-maps 98 (1971), 840-856.

12.

, On Klein's Combination T~eorem, llI, Advances in the Theory of Riemann Surfaces, Annals of Math. Studies 66 (1971~ 297-316.

18.

G. P. Scott, Finitely generated presented, J. London Math. Soc.

Comment. Math.

on Kleinian groups,

Kleinian

Amer. J. Math.

$-manifolds are finitely (2) 6 (1973), 487-440.

8.

K L E I N I K N GROUPS AND 3-D-IMENSIONAL T O P O L O G Y A SURVEY Albert Marden 1 U n i v e r s i t y of M i n n e s o t a U n i v e r s i t y of M a r y l a n d

§i.

INTRODUCTION

In an 1883 paper in Acta Mathematica, the general theory of Kleinian groups.

Poincar@ formulated

He based his approach

on that which he had s u c c e s s f u l l y applied to Fuchsian groups. The starting point was the r e c o g n i t i o n that a M@bius transformation acting in the plane can be regarded via stereographic p r o j e c t i o n as acting on the 2-sphere e x t e n d e d to the

3-ball

MSbius t r a n s f o r m a t i o n s

@

as well.

~

and then

Given a group

acting d i s c o n t i n u o u s l y on

showed there were fundamental p o l y h e d r a in

~

G

~,

for

of Poincar@

G

exactly

analogous to the fundamental polygons for F u c h s i a n groups.

At

this point h o w e v e r his general analysis ended. In r e t r o s p e c t this is not surprising.

For Fuchsian groups

only involve surface t o p o l o g y and m a t h e m a t i c i a n s have been dealing with this for a long time.

In contrast,

Kleinian groups

involve 3 - d i m e n s i o n a l t o p o l o g y which is incredibly more complicated.

A c t u a l l y it is only r e l a t i v e l y r e c e n t l y that a

sufficient amount of i n f o r m a t i o n has been accumulated about

iThis w o r k was Foundation.

supported

in part

by the N a t i o n a l

Science

109

3-manifolds for this theory to be an e x t r e m e l y useful tool in the study of K l e i n i a n groups.

The purpose of this report is

to suggest why this is the case.

It is based m a i n l y on [7] to

which the reader is r e f e r r e d for details and complete references.

§2, E X T E N S I O N TO

Each MSbius transformation,

6

acting on the sphere

is the product of an even n u m b e r of r e f l e c t i o n s on

SB.

circle

Let X

o

be the sphere orthogonal to

and set

o 0 = o n 6.

be e x t e n d e d to a r e f l e c t i o n of

in circles

SB

A r e f l e c t i o n in 6

group of all MSbius t r a n s f o r m a t i o n s

in

o 0.

~,

along the ~

can

In this way the

can be extended to act on

With respect to the h y p e r b o l i c metric

ds2/(l-lxl2) 2

in

~.

Z,

one gets the group of all o r i e n t a t i o n p r e s e r v i n g isometrics.

§3. D I S C R E T E N E S S AND D I S C O N T I N U I T Y

A group

G

of MSbius transformations is discrete if it

contains no sequence tending to the identity transformation. G

is discrete if and only if it acts d i s c o n t i n u o u s l y in

(has no limit point in

6).

but the converse is false:

6

Kleinian groups are always discrete The limit set may be all of

~

(sometimes these are called Kleinian groups of the first kind). One advantage of our a p p r o a c h is that it applies equally well to all discrete groups. A t h e o r e m for 3-manifolds due to G.P. dently proved by P. Shalen) g e n e r a t e d d i s c r e t e group

G

Scott

[i0]

(indepen-

implies that every finitely is a c t u a l l y finitely presented.

110

So far, generated

there

groups

is a r e a s o n a b l e

and we will

addition

we will always

(contain

no e l l i p t i c

essential A.

Selberg

index. the

However

sort

free

that

subgroup

theoretic

assume

because

there

Bers'

our groups Usually

free

involving

are torsion

free

from a result subgroup

of

of finite

for example

passing

of

to a torsion

the very d i f f i c u l t which

In

is not an

counting,

area theorem,

involved

for f i n i t e l y

case here.

this

it follows

does not resolve

questions

only

to this

is a torsion

in matters

involving

stick

elements).

restriction

theory

in general

group-

have not been

solved.

§4. THE

If

G

is d i s c r e t e

3-MANIFOLD

we can form the

3-manifold

9~I(G) = 8 U ~(G)/G w h i c h has

a natural

from that

of

3-dimensional

6 U ~(G).

is a u n i o n of R i e m a n n

surfaces

addition

(If

would would of

still be true that ~i(~) ~ G.)

~(G)

component

is a sphere

G

inherited

= a(G)/G but

were

~(G) G

has the disk

Examples. fuchsian

If

structure

The b o u n d a r y ~ICG)

~i(~) ~ G.

conformal

is empty allowed

if

~(G)

to have

is a 3-manifold

is not elementary,

as u n i v e r s a l

cover.

= ~.

In

torsion

it

but no longer

each c o m p o n e n t In p a r t i c u l a r

no

or a torus.

(1)

If

G

is a f i n i t e l y

generated

group with

~0

one of its invariant

quasi-

components

then

111

~G)

~ (~0/G)×l

(here

I : [0,i]).

For Fuchsian groups this

can be seen directly. (2)

If

G

is a Schottky group of genus

is a h a n d l e b o d y of genus

g.

That is

~I(G)

g,

then

is h o m e o m o r p h i c

to the compact region bounded by a surface of genus in

9n(G)

g

embedded

~. (3)

~I(G) ~

If

G

is a d e g e n e r a t e group it is unknown w h e t h e r

(~(G)/G)

× [0,I).

This f u n d a m e n t a l p r o b l e m will be

d i s c u s s e d later in §12.

§5.

Given a point with center at

FUNDAMENTAL P O L Y H E D R A

0 ( ~

0

the Poincar6 f u n d a m e n t a l p o l y h e d r o n

is defined as

@ = {x ( ~: d(x,O) Here

d(.,')

< d(x,T(O)),

is the h y p e r b o l i c distance.

analogous case for Fuchsian groups, in pairs and the orbit of o v e r l a p p i n g interiors.

~

under

T ~ G}.

Exactly as the

the faces of G

covers

With its opposite

provides a model for

all

P ~

are a r r a n g e d without

faces identified,

~I(G).

§6. THE ROLE OF PA_RA~OLIC T R A N S F O R M A T I O N S

Suppose

p ( ~8

is a parabolic

maximal p a r a b o l i c subgroup at

p

fixed point of

M

The

is defined as

Mp = {T ~ G: T(p) It is known that

G.

= p}.

consists of parabolic t r a n s f o r m a t i o n s P

with a common fixed point and is either free abelian of rank two

or infinite cyclic.

112

The role played in

~I(G)

by the rank two

M's

can be

P

d e s c r i b e d precisely. lifting to Mp's

9~(G) °

U

(via

b e t w e e n c o n j u g a c y classes of these rank two

and solid cusp tori in

submanifold in

~)

There is a one-one c o r r e s p o n d e n c e

~(G).

in the interior

A solid cusp torus is a

~(G) °

whose r e l a t i v e b o u n d a r y

is a torus, w i t h ~

{z

o <

( ¢:

]z[

< l}×S

~.

The tort c o r r e s p o n d i n g to distinct conjugacy classes can be taken to be m u t u a l l y disjoint. The role played in

9rt(G)

by the cyclic

M's P

c o m p l i c a t e d and is not completely understood. a s s o c i a t e d w i t h punctures on

~gL(G).

is more

They are closely

A puncture is an ideal

b o u n d a r y component of a c o m p o n e n t of

~Dt(G)

which has a

n e i g h b o r h o o d c o n f o r m a l l y equivalent to the once p u n c t u r e d disk. C o r r e s p o n d i n g to each puncture c o n j u g a c y class of a cyclic

(p,q)

if there is a s u b m a n i f o l d

~Z)

is the

M . But there may be more Than P to a given class.

one p u n c t u r e c o r r e s p o n d i n g Two punctures

(via lifiing to

on

39A(G)

~

of

are said to be paired

9rt(G), which is called a

solid cusp cylinder, with the properties that ~ {z ( ¢: 0 < Izl < i} × [0,1] and of

~ n ~9]L(G) q

is a union of a n e i g h b o r h o o d of

p

and one

each of which is c o n f o r m a l l y e q u i v a l e n t to the once

p u n c t u r e d disk. a cylinder.

The relative b o u n d a r y of

F u r t h e r m o r e if

p

and

no third puncture paired w i t h either

q

~

in

97t(G)°

is

are paired there is p

or

q.

The solid

113

cusp cylinders corresponding to distinct pairs can be taken mutually disjoint. A pair of punctures corresponds to the conjugacy class of a cyclic

M .

In general, however,

this correspondence goes

P

only in one direction. §7. GROUPS OF COMPACT TYPE This class is defined to consist of those non-elementary discrete groups with a finite sided Poincare fundamental polyhedron. The group

It is important because of the following fact [7]. G

is of compact type if and only if

the following structure.

~(G)

has

There are a finite number of

mutually disjoint solid cusp cylinders and solid cusp tori so that their complement in if

G

~(G)

is compact.

has no parabolic transformations

simply that

9~(G)

In particular

the condition is

be compact.

One of the critical junctures in the theory of Kleinian groups was the discovery by Leon Greenberg

[4] that degenerate

groups are not of compact type. On the other hand it follows from a result of Selberg and Garland-Raghunathan

that if

of finite hyperbolic volume PSL(2,~)

G

has a fundamental polyhedron

(i.e. the coset space of

has finite volume) then

Recently Wielenberg

G

G

is of compact type.

[14] found an elementary proof of this

in the context of M~bius groups and in fact proved a much stronger,

in

local form of the result.

114

§8.

THE C L A S S I F I C A T I O N PROBLEM;

Suppose that words that

G

is compact and

each component of

is injective,

~(G)

in other

[13],

~(G)

along each of

that is, that

is simply connected. has a hierarchy:

introduce n o n d i v i d i n g surfaces in T~G)

~ #,

For technical reasons assume too that the

~l(~h)+~l(~)

to W a l d h a u s e n

~

is a purely loxodromic K l e i n i a n group of

compact type. inclusion

'Dr(G)

THE COMPACT CASE

~I(G)

these surfaces

finite n u m b e r of steps

~I(G)

According One can s u c c e s s i v e l y

so that, cutting

in succession,

is reduced to a ball.

after a Once

this is a c c o m p l i s h e d the steps can be r e v e r s e d thereby reforming

~I(G)

from a ball.

With L. G r e e n b e r g and P.

we observed that this r e b u i l d i n g process

Scott

can actually be

started with a h a n d l e b o d y of genus two (a Schottky group of genus two) r a t h e r than just a ball.

The r e c o n s t r u c t i o n can

be d e s c r i b e d in terms of the K lein-Maskit combination theorems

[8, 9]. In studying the a p p l i c a t i o n of

the h i e r a r c h y to K l e i n i a n

groups one finds that a basic p r o b l e m is to u n d e r s t a n d which of the t o p o l o g i c a l operations

one can use to form a new 3-

m a n i f o l d from some

~(G)

can be carried out in the context

of K l e i n i a n groups

(by using c o m b i n a t i o n theory).

example c o n s i d e r a solid torus on

~

~

For

w h i c h arises from the action

by a cyclic group of loxodromic transformations.

two disjoint disks in 3-manifold.

~

Take

and identify t h e m to form a new

This can be carried out in the context of

115

K l e i n i a n groups and one obtains a Schottky group of genus two. On the other h a n d suppose we take two disjoint, p a r a l l e l annuli in

~

r e p r e s e n t i n g a n o n - t r i v i a l element in

~ (U). l

One can i d e n t i f y these to obtain a new 3-manifold but this prooess cannot be carried out in the context of K l e i n i a n groups.

§9. D E F O R M A T I O N AND D E G E N E R A T I O N T H E O R Y

Given a group

G

of compact type what happens w h e n one

varies the entries in the matrices for

G

group?

and uses these deformed m a t r i c e s to generate a new More precisely,

homomorphisms

Hom G

p a r a b o l i c elements space

of a set of generators

one wants to consider the space of all

of

G

into

which preserve

(actually a small m o d i f i c a t i o n of the W e l l

R(G,PSL(2,~))).

as a q u a s i - p r o j e c t i v e

Geometrically

Hom G

a l g e b r a i c variety

i n t e r e s t e d in the subset

T(G)

of

groups g e o m e t r i c a l l y similar to is a connected open subset of a n a l y t i c manifold.

PSL(2,~)

V(G).

V(G)

G.

One is e s p e c i a l l y

consisting of discrete

It turns out that

V(G)

T(G)

and in fact a complex

The proof [7] depends on proving the

equivalence of the two topologies derived from conformal deformations of T(G)

can be i n t e r p r e t e d

G,

and

b)

a)

the quasi-

the t o p o l o g y of

is the d e f o r m a t i o n or p a r a m e t e r space of

G.

V(G).

From a

different point of view the theory concerns conformal deformations of the conformal structure of structure r e m a i n i n g the same. [7] that the d i m e n s i o n of ~(3gi+bi-3)

T(G)

~t(G) °,

the t o p o l o g i c a l

This t h e o r y yields the result (which after n o r m a l i z a t i o n is

in the n o t a t i o n of §ii) depends only on

~9~t(G),

116

not on

the internal structure

~9~c(G) = ~,

then

tion of Mostow's

T(G)

of

9~[(G).

is a point

If in particular

(this fact is an elabora-

rigidity theorem).

As an open subset of the quasi-projective T(G)

has a relative boundary

proved that points to

G

on

3T(G)

3T(G)

in it,

although not n e c e s s a r i l y Kleinian.

discrete

groups

on

and proofs

[3]

isomorphic

are in fact discrete

Recently T. J ¢ r g e n s e n

of these results

from the theorems ~T(G)

which

of the classical modular group description

I n the 3-manifold

3~I,

its general form,

analogies with the cusps

a cusp has to do

simple

two annuli in

of compact type

Approaching

in [6]).

along one or more mutually

where

0A(G),

disjoint

a submanifold

{e < Izl < i} x~I

becomes

The

see JCrgensen's

~L(G)

{e < Izl < i} × I ,

[5].

(for instance

with pinching loops.

to groups

of certain

of

also follow

of his elegant paper

correspond

are called cusps because

geometric

V(G),

a very general method of treating convergence

immediately points

Chuckrow

correspond to groups

and we noticed that these groups

discovered

variety

is the union of

a solid cusp cylinder

this degeneration

(§6).

In

theory is s%ill in its

infancy. I n the case of a Fuchsian shown that

T(G)

Teichm~ller

space.

T(G) U ~T(G)

is esentially

surface

group

G,

T ×T

where

T

The compactification

gives the Bets b o u n d a r y

3T

Bers has is the ordinary

of a slice of

T.

T

in

Using methods

suggested by this general point of view one can extend the action of the T e i c h m ~ l l e r

modular group from

T

to the cusps

117

on

~T.

This p r o g r a m is currently b e i n g carried out in joint

w o r k w i t h Clifford Earle.

It yields a c o m p a c t i f i c a t i o n of the

moduli space.

§10. THE C L A S S I F I C A T I O N

PROBLEM;

THE N O N - C O M P A C T CASE

Now we c o n s i d e r the case of an a r b i t r a r y finitely generated,

torsion free K l e i n i a n group

is to find out how

9b/G)

G.

The p r o b l e m here

is related to a compact manifold.

We w i l l start by listing two questions. (I)

Does the i n t e r i o r

a submanifold (i)

M

isomorphism, if

N

of every

~,(M) ÷ n , ( ' ~ ( G ) )

in

is a component of

"~(G) °

inclusion

Z

is an

~%(G)°-M ° ~N

is connected,

~l(~N) + ~I(N)

isomorphism, in

contain

and

then the relative b o u n d a r y N

~(G)

with the properties

the i n c l u s i o n

(ii)

91t(g)°

the

is an

and each lift of

divides

~

of

3N

into two com-

ponents at least one of w h i c h is a ball? (2)

If

G

is a d e g e n e r a t e group and

S = ~(G)/G,

is

~t(G) ~ Sx [0,i)? We w i l l

discuss

affirmative w h e n G.P. M

G

(2) in

§12.

The answer t o

(1)

is

is of compact type or is a function group.

Scott [I0] showed that the answer is a f f i r m a t i v e and that

can be taken to be compact for all groups

G

such that

G

cannot be w r i t t e n as a free p r o d u c t of n o n - t r i v i a l subgroups.

118 He also showed that in the general case one can find a compact M

satisfying

(i).

The significance is a group for which can completely degenerate

of these questions (i) holds.

understand how

groups and groups

complicated

to describe

understanding

of

G

and

is structurally

of compact type.

made up of

This is rather

and we refer to [7] for details. ~bl~G)

The

is such that if (2) were

it would follow that

9~(G) °

is

F I N I T E N E S S THEOREM AND BERS' I N E Q U A L I T Y

Consider a finitely elliptic elements. by removing

of the inclusion H,(~L')

G

to the interior of a compact manifold.

§ii. AHLFORS'

0OI(G)

Suppose

Then in a certain sense one G

known to be true in addition, homeomorphic

is this.

Let

generated K l e i n i a n

G

without

Trt, denote the manifold obtained

the solid cusp tort.

An elementary

of first integral homology

÷ H,(OOL')

group

from

study

groups

shows that

I gl ÷ c ~ N where c

gi

is the genus of the

i th

is the number of solid cusp tort

number of conjugacy

classes

the number of generators compared with Bets'

of

component of

897L(G),

(or equivalently

of rank two G.

inequality

the

M 's), and N P This formula should be

is

[2]

[ (gl + bl/2 - I) ! N - 1 where

bi ~ 0

ponent of

is the number of punctures

~0~L(G).

on the

i th

com-

[7]

119 In order to investigate necessary to assume that Then it is possible

Bers'

~L(G)

situation: genus

N

ponents

(Is of

G

inequality

only in the following

91L is compact with

~b

with

~TL

connected of

then a Schottky group?) or all the com-

~G)

have punctures

from

of which

It shows for example that

and these are arranged

in pairs so that when the corresponding are removed

(I) of §I0.

to deduce a master inequality

in Bets'

Either

further it is

has the property

the two above are special cases. there is equality

inequality

~I(G),

a compact

there remains

solid cusp cylinders a compact m a n i f o l d

surface of genus

9%

N.

§ 12. THE FINAL EXAM We believe both K l e i n i a n determining aspects

that the most important

group and T e i c h m ~ l l e r

the structure

theory

of degenerate

of this are finding the measure

finding the t o p o l o g i c a l of view of K l e i n i a n

of

is the problem of

groups

G.

The two

of the limit set and

7h(G).

groups this information

their classification of T e i c h m u l l e r

structure

unsolved problem in

From the point is important

in

(see §i0) and from the point of view

theory

it is necessary

for a fuller under-

standing of the boundary. Lest the reader be left with soaring e n t h u s i a s m in the power of 3-dimensional

topology to solve problems

groups, we close with the following G.P. S

Scott based on wDrk of Tucker

be a closed surface

important

in K l e i n i a n

example of

[12] (see also [ii]).

of genus > I.

According to Scott

Let

120

(personal

communication)

there exists

a 3-manifold

M

with the

following properties. (i)

(ii) (iii)

3M

:

S

The inclusion The universal

upper half space (iv)

If

T

~,(S) + ~,(M) covering

space of

is any cover transformation T

is the closed

and



is

then

H/ ~ (S I × ~ ) ×

[0,i).

M ~ S × [0,I).

Degenerate But what about

groups (v)?

of course have properties

Consider the case that

The corresponding

thing for Mobius

generated by say

z~-+z+l,

know that

M

H.

the group generated by

(v)

is an isomorphism.

~I(G) ~

groups

z~-+z+i.

S

(i) - (iv). is a torus.

is the group

G

But here at least we

S × [0,I).

References i.

L.V. Ahlfors, Finitely generated K l e i n i a n groups, Amer. J. Math. 86 (1964), 413-429 and 87 (1965), 759.

2.

L. Bets, I n e q u a l i t i e s groups, Jour. d'Anal.

3.

V. Chuckrow, On Schottky groups with application K l e i n i a n groups, Ann. of Math. 88 (1968), 47-61.

to

4.

L. Greenberg, Ann. of Math.

Fundamental polyhedra 84 (1966), 433-441.

groups,

5.

T. J~rgensen, to appear.

On discrete

6.

T. JCrgensen,

On reopening

7.

A. Marden, The geometry groups, Ann. of Math.,

for finitely generated K l e i n i a n Math. 18 (1967), 23-41.

groups

for K l e i n i a n

of Moebius transformation,

of cusps, to appear.

of finitely generated K l e i n i a n 99 (1974).

121

8.

B. Maskit, On Klein's combination theorem, Trans. A.M.S. 120 (1965), 499-509 and 131 (1958), 32-39.

9.

B. Maskit, On K lein's combination theorem III, in Advances in the Theory of Riemann Surfaces, Annals of Math.'-~udl~-~s 66, Prz"-nceto----~U--niversity Press, Princeton, N.J.

i0.

G.P. Scott, Compact submanifolds of 3-manifolds, J. London Math. Soc. 7 (1973), 246-250.

Ii.

G.P. Scott, An introduction of Maryland Lecture Notes.

12.

T.W. Tucker, Some non-compact 3-manifold examples giving wild translations of ~3, to appear.

13.

F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968),

to 3-manifolds,

University

56-88.

14.

N. Wielenberg, On the fundamental po!yhedra of discrete Moebius groups, Thesis, University of Minnesota, 1974.

THE C U R V A T U R E

OF T E I C H M U L L E R

SPACE

H o w a r d Masur U n i v e r s i t y of M i n n e s o t a (Abstract)

It is k n o w n (punctured) plane;

torus,

in this

metric,

that

case

surface

metric

of genus

negative

there

is a unique g e o d e s i c

is s t r a i g h ~ w h i c h means

direction.

A geodesic

Busemann

be of

curvature.

negative

the m i d p o i n t Then

the inequality

for all choices a Riemannian coincides

of points

metric,

with

It was It is p o s s i b l e

P, Q, R.

this d e f i n i t i o n

any

showed

two points

in either line.

in a s t r a i g h t

points

P-Q

d(Q,R)

Kravetz

image of the real

curvature

of the segment

of a closed

g

However

is an isometric

defined

T

to infinity

G i v e n any three n o n - c o l l i n e a r

PR.

space

g ~ 2, the

that b e t w e e n

extending

in the c o m p l e x

is the Polncare For

is not Riemannian.

T

g

metric

in the T e i c h m d l l e r

g

space of a

to the unit disc

the T e i c h m ~ l l e r

that

space.

the T e i c h m ~ l l e r

is isomorphic

and thus has

Teichm011er

TI,

P, Q, and Ro

and

R

let

the m i d p o i n t

< ~ d(Q,R)

is to h o l d

If a straight of n e g a t i v e

space has

curvature

the usual one. asserted

that

Tg,

to prove h o w e v e r

g ~ 2, has

that

T

g

negative

curvature.

does not h a v e

negative

curvature. The

idea

at the same point

is to find ~

which

two geodesic satisfy

rays

d(x,s)

r,

s

~ M < ~

originating for

x 6 r,

123

which sees

is to say that

r

and

s

do not diverge.

that not all t r i a n g l e s w i t h sides on

vertex

P

can h a v e

r

T h e n one

and

s

and

the n e g a t i v e c u r v a t u r e p r o p e r t y .

By the c l a s s i c a l

t h e o r e m of T e i c h m O l l e r ,

on a ray t h r o u g h the p o i n t

the p o i n t s

P, w i t h u n d e r l y i n g R i e m a n n s u r f a c e

S, are d e t e r m i n e d b y the T e i c h m ~ l l e r e x t r e m a ! maps on with dilatation

k~/1

~

I, w h e r e

0 ~ k < 1

holomorphic quadratic differential The c r u c i a l e l e m e n t quadratic differentials have

closed horizontal

tensively by S

Strebel.

on

on

trajectories Strebel's

parameter.

is d i v i d e d

is to find the

"right"

and h a v e b e e n

structure

into annuli

s t u d i e d ex-

t h e o r e m says

that

of such a q u a d r a t i c

each e q u i p p e d w i t h a n a t u r a l

The T e i c h m ~ l l e r m a p then can be d e s c r i b e d .

s i m p l y an a f f i n e

s t r e t c h of each annulus,

d i s t i n g u i s h e d parameters, An e x i s t e n c e differentials are p a i r w i s e determine

is a

The q u a d r a t i c d i f f e r e n t i a l s

cut a l o n g the c r i t i c a l t r a j e c t o r i e s

differential

~

S.

in the p r o o f S.

and

S

t h e o r e m of S t r e b e l ' s

freely h o m o t o p i c .

the r e q u i r e d rays

are p a i r w i s e not diverge.

to these

o n t o the image surface.

exist w h i c h d i v i d e

the T e i c h m 0 1 1 e r m a p

with respect

It is

r

the s u r f a c e

that d i s t i n c t

into annuli

that

These distinct differentials and

s.

is e a s i l y d e s c r i b e d

freely h o m o t o p i c ,

says

one shows

Details will appear

Using

the fact

and that that

elsewhere.

r

that

the annuli and

s

do

SOME U N S O L V E D

PROBLEMS

Compiled by William Abikoff

The p r o b l e m s lecturers problems

shortly are,

b e e n made

or

after

to indicate

with

I)

The c l a s s i f i c a t i o n

of p r ob l e m s

References

to the

the The

No attempt

has

difficulty

list of open prob-

given below literature

is more

are g i v e n

each problem.

Ahlfors' If

Zero Measure G

[$],

[9] and

II)

Teichm011er i)

group

G,

Abikoff's

Problem:

is a finitely

two-dimensional

measure

~],

of

Abikoff

is

generated A(G)

Kleinian

equal

[i] and Kra's

group,

to zero?

is the

(See A h l f o r s

lecture.)

Spaces:

For a finitely T(G)

or i n f i n i t e l y

starlike

in the Bets

generated

Fuchsian

embedding?

(See

lecture.)

2) Teichmdller same

unsolved.

an e x h a u s t i v e

by

meeting.

our sense of the r e l a t i v e

or to give

less arbitrary.

submitted

the San F r a n c i s c o

as far as we know,

of the p r o b l e m s lems.

s t a t e d b e l o w were

Let

G

space.

be a F u c h s i a n

group

Is the C a r a t h e o d o r y

as the Kobayashi

metric?

and

T(G)

metric

(See Royden's

on

its T(G)

lecture

the

and Earle

[17] .) 3) first kind,

If

G

is a f i n i t e l y

is there a natural

generated

notion

Fuchsian

of K o b a y a s h i

group metric

of the on

125

(G)

or on

lecture,

~(G) ?

(~(G)

and

~(G)

are d e f i n e d

in A b i k o f f ' s

the K o b a y a s h i metric and other r e l e v a n t notions may

be found in the lectures of Royden and Masur.) 4)

If

G

is a fixed point free finitely g e n e r a t e d

F u c h s i a n g r o u p of the first kind, is isomorphic to elements,

T(G')

then the Bers

for some

G'

If

G

has elliptic

then in g e n e r a l no such i s o m o r p h i s m exists.

finite number of cases the q u e s t i o n is open. lecture;

fiber space

in particular,

reference

17 of his

For a

(See Kra's lecture lists

the open cases.) 5)

Do all d e g e n e r a t e b - g r o u p s

lie on the b o u n d a r y of

a finite d i m e n s i o n a l T e i c h m ~ l l e r space? and Bers

If

f(z)

is a schlicht

function in the lower h a l f -

does the S c h w a r z i a n d e r i v a t i v e

closure of [13] and

lecture

[15].) 6)

plane,

(See Bets'

T(1),

{f,z}

lie in the

the u n i v e r s a l T e i c h m ~ l l e r space?

(See Bets

[15] .) 7)

Let

G

the first kind and

b e a finitely g e n e r a t e d F u c h s i a n g r o u p of T(G)

be given the Bets embedding.

groups w i t h a c c i d e n t a l p a r a b o l i c t r a n s f o r m a t i o n s dense b o u n d a r y of 8)

T(G)? Let

G

(See Bers'

lecture and Bers

and

be as in Problem II-6.

T(G)

Are in the

[15].) Mod G

is a normal family of h o l o m o r p h i c a u t o m o r p h i s m s of the b o u n d e d domain

T(G).

Let

gn 6 Mod G

with

d e g e n e r a t e h o l o m o r p h i c m a p p i n g of

g = lim gn"

T(G)

into

g

ST(G).

is then a Can

g

126

be n o n c o n s t a n t w i t h o u t b e i n g onto a b o u n d a r y T e i c h m ~ l l e r space? (Boundary Teichmi~ller spaces are d i s c u s s e d also A b i k o f f ' s 9)

Let

G

and

T(G)

is a s e q u e n c e of elements of holomorphic mapping

be as in Problem II-6.

g

F

Mod G

Does

so that if

gn

c o n v e r g i n g to a d e g e n e r a t e

then either:

g(T(G))

(ii)

the diameters of

(See the references

for

Mod G

(i)

i0)

[6] ; see

lecture and his forthcoming paper.)

there exist a fundamental set

or

in A b i k o f f

is a b o u n d a r y T e i c h m ~ l l e r space gn(F)

c o n v e r g e to zero?

for p r o b l e m II-9)

Is the image of the c a n o n i c a l

i n j e c t i o n of

into the affine algebraic v a r i e t y of h o m o m o r p h i s m s finitely g e n e r a t e d K l e i n i a n group

G

into

M6b

T(G)

for the

a submanifold?

(See I
III)

(See Chu

[16] and A b i k o f f ' s

T(G)

in the Bers

lecture.)

A u t o m o r p h i c Forms and Eichler C o h o m o l o q y i)

for

Find or estimate the inradius of

g > 2

Is the c a n o n i c a l map and

r

B

q

(Q,F)

infinitely g e n e r a t e d ?

~ HI(F,~2q_2)

injective

(See G a r d i n e r ' s

lecture for notation.) 2)

For finitely g e n e r a t e d K l e i n i a n groups,

c o h o m o l o g y groups

HI(F,~2q_2 )

can be d e c o m p o s e d

the Eichler

into direct

sums of cusp forms and q u a s i - b o u n d e d Eichler integrals. such a d e c o m p o s i t i o n for infinitely g e n e r a t e d groups? Gardiner's

lecture for d e s c r i p t i o n and references.)

Is there (See

127

3)

W h a t does the p r e s e n c e of trivial Eichler integrals

tell us about the s t r u c t u r e of the K l e i n i a n group? Gardiner's 4)

lecture and the references given there.) Let

G

b e a K l e i n i a n group.

structive characterizations operator?

IV)

(See Bets

Are there any con-

. J of the k e r n e l of the Polncare theta

[12] and Ahlfors

[8].)

G e o m e t r y add T o p o l o q y of K l e i n i a n Groups i)

are known, Maskit

Few p r o p e r t i e s of totally d e g e n e r a t e K l e i n i a n groups find more.

[19] and 2)

(See A b i k o f f

[5], Bers

[15], M a r d e n

[18],

[21].)

If one allows q u a s i c o n f o r m a l d e f o r m a t i o n s

s u p p o r t e d on the limit set, Bers

(Again see

are d e g e n e r a t e groups

to b e

stable?

(See

[14].) 3)

Abikoff

C l a s s i f y the finitely g e n e r a t e d web groups.

[2] and 4)

[3] and A b i k o f f and Maskit

Which 3-manifolds

(See

[7].)

are u n i f o r m i z a b l e b y K l e i n i a n groups,

i.e. admit metrics of constant n e g a t i v e c u r v a t u r e ?

(See Marden's

lecture.) 5)

For a general

are conditions

(i.e. non-function)

(A), ( B ) , (C),

(E) and

conditions are stated in Maskit's 6)

For a K l e i n i a n g r o u p

K l e i n i a n group,

(F) e q u i v a l e n t ?

(The

lecture.) G,

let

K

b e the reglon in

B3

d e f i n e d as the i n t e r s e c t i o n of all h y p e r b o l i c h a l f - s p a c e s w h o s e boundaries

lie in

~(G).

finite h y p e r b o l i c v o l u m e ?

For w h i c h groups does (See Marden's

(B3-K)/G

lecture and M a r d e n

have [18].)

128

7)

Exactly as in the d e f i n i t i o n of c o n f o r m a l l y extendable,

one can define

the n o t i o n of t o p o l o g i c a l l y extendable.

there exist a K l e i n i a n group which

(not n e c e s s a r i l y

Does

finitely generated)

is t o p o l o g i c a l l y e x t e n d a b l e but not q u a s i c o n f o r m a l l y

extendable?

(See Maskit's

lecture and Maskit

[20] for the

r e l e v a n t notions.) 8)

Do Maskit's c o m b i n a t i o n theorems p r e s e r v e Bers

s t a b i l i t y w h e n the a m a l g a m a t i n g subgroups or c o n j u g a t e d subgroups are of the second kind? 9)

Let

G

(See A b i k o f f

be a finitely g e n e r a t e d K l e i n i a n group of

the first kind w h o s e q u o t i e n t has Does

it n e c e s s a r i l y

[4].)

follow that

infinite h y p e r b o l i c volume. G

has a finitely g e n e r a t e d

d e g e n e r a t e or n o n - c o n s t r u c t i b l e web subgroup?

(Kleinian groups

of the first kind are d e f i n e d in Ahlfors

for the other

notions see Problem IV-6 Abikoff

[3] .)

[I0],

and the r e f e r e n c e s given there and

129

REFERENCES [ i]

W. Abikoff,

Some remarks on Kleinian groups,

the Theory of Riemann Surfaces,

Ann.

Advances

of Math.

in

Studie___~s,

6_9_6(1970), pp. i-7. [ 2]

W. A]Dikoff, Math.,

[ 3]

Residual

130(1973),

W. Abikoff,

[ 4]

W. Abikoff, groups, Math.

[ 5]

Contributions

Discontinuous

W. Abikoff,

Groups

Two theorems

Academic

Press,

and Bers stability of Kleinian and Riemann Surfaces,

on totally degenerate

III,

Ann.

of

Kleinian

W. Abikoff and B. Maskit,

[ 8]

L. Ahlfors,

Finitely

of T e i c h m ~ l l e r

spaces and on

to appear.

[ 7]

to appear.

generated

Kleinian groups,

Amer. ~. o__ff

8__66(1964), pp.413-429.

L. Ahlfors, Conference Groups,

[I0]

t_ooAnalysis,

of

to appear.

Kleinian groups,

[ 9]

and d e f o r m a t i o n

7__99(1974), pp.3-12.

W. Al)ikoff, On b o u n d a r i e s

Math.,

Acta

pp.l-10.

Constructibility

Studies,

groups, [ 6]

1974,

groups,

pp.127-144.

On the d e c o m p o s i t i o n

Kleinian groups, New York,

limit sets of Kleinian

Some remarks

on ~ u a s i c o n f o r m a l

(1965),

L. Ahlfors, Festband

on Kleinian groups, MaPg,

Proc.

Tulane

Moduli and Discontinuous

pp.7-13.

Kleinische

70. G e b u r t s t a g

Gruppen

in der Ebene und im Raum,

R. Nevanlinna,

p.7-15,

Springer,

1966. [ii]

[12]

L. Ahlfors,

Remarks

on the limit point set of a finitely

generated

Kleinian group,

Surfaces,

Ann.

L. Bers,

of Math.

Automorphic

infinitely generated 8_J_7(1965), pp. 196-214.

Advances

Studies,

in the Theory o_~f R i e m a n n

66(1971),

pp.19-26.

forms and Poincare series Fuchsian groups,

Amer.

for

J. of Math.,

130

[13]

L. Bers,

Universal

i__nnMathematical

Teichm~ller

Physics,

space,

Analytic

Gordon and Breach,

Methods

(1970),

pp.65-83. [14]

L. Bers,

Spaces of Kleinian groups,

155(1970), [15]

L. Bers,

Springer,

On b o u n d a r i e s

Kleinian groups, [16]

On the outradius

spaces,

Discontinuous

Studies,

C. Earle,

Studies,

of Math.,

spaces

and on

9_!1(1970), pp.570-600.

of finite-dimensional

Groups

and Riemann

Teichm~ller

Surfaces,

Ann.

of

7__99(1970), pp.75-80.

On the C a r a t h e o d o r y

Discontinuous

[18]

I, Ann.

in Math.,

pp.9-34.

of TeichmOller

T. Chu,

Math. [17]

Berlin,

Lecture Notes

Groups

metric

in Teichm~ller

and Riemann Surfaces,

Ann.

spaces,

of Math.

7__99(1974), pp.99-i04.

A. Marden,

Geometry of finitely

generated

Kleinian groups,

to appear. [19]

B. Maskit,

[20]

B. Maskit,

On boundaries

Kleinian groups,

Math., [21]

93(1971),

B. Maskit,

If, Ann.

Self-maps

of Teichm~ller of Math.,

to appear.

and on

9_!1(1970), pp.607-639.

on Kleinian groups,

pp.840-856.

spaces

Amer.

J. of