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
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)- --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)

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

~

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

~
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