Discontinuous Groups and Riemann Surfaces (AM-79), Volume 79: Proceedings of the 1973 Conference at the University of Maryland. (AM-79) 9781400881642

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Annals of Mathematics Studies Number 79

DISCONTINUOUS GROUPS AND RIEMANN SURFACES Proceedings o f the 1973 Conference at the University o f Maryland

EDITED BY LEON GREENBERG

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1974

Copyright © 1974 by Princeton University Press A L L RIGHTS RESERVED

L.C. Card: 73-16783 ISBN: 0-691-08138-7

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

Printed in the United States of America

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

PR EFACE

This volume contains almost all of the papers that were read at the Conference on Discontinuous Groups and Riemann Surfaces, which was held at the University of Maryland during May 21-25, 1973.

The conference

was the third of a sequence of conferences on this subject in recent years. The other two were held at Tulane University (1965) and SUNY at Stony Brook (1969).

The Proceedings of the Stony Brook Conference appeared

as Advances in the Theory of Riemann Surfaces (Annals Study 66). Thanks are due the National Science Foundation for supporting the con­ ference; the University of Maryland for making its facilities available; the Princeton University P re ss for assuring rapid publication; and of course the participants of the conference for creating the mathematics contained herein.

L. GREENBERG

v

C O N TE N TS

P r e f a c e ..........................................................................................................

v

Constructability and Bers Stability of Kleinian Groups by William A bikoff ................................................................................

3

Algebraic Curves and H alf-canonical Linear Series by Robert D. M. A c c o la ........................................................

13

Sufficient Conditions for Quasi-conformal Extension by Lars V. Ahlfors ...............................................................................

23

Fundamental Domains for Kleinian Groups by A. F. Beardon ..................................................................................

31

Spaces of Degenerating Riemann Surfaces by Lipman B ers ....................................................................................

43

Mapping C la s s Groups of Surfaces: A Survey by Joan S. Birm an..................................................................................

57

A Note on L 2(T G) by Su-shing Chen ..................................................................................

73

On the Outradius of Finite-dim ensional Teichmiiller Spaces by T. C h u ................................................................................................

75

Some Direct Lim its o f Prim itive Homotopy Words and of Markoff G eodesics by Harvey C ohn......................................................................................

81

On the Caratheodory Metric in Teichmiiller Spaces by Clifford J. E arle ..............................................................................

99

Some Refinements o f the Poincare' Period Relation by L . Ehrenpreis and H. M. Farkas .................................................... 105 Remarks on Automorphisms of Compact Riemann Surfaces by Hershel M. F a rk a s ............................................................................ 121

viii

CONTENTS

The Structure of P S L ~ (R ); R, the Ring of Integers in a Euclidean Quadratic Imaginary Number Field by Benjamin F i n e .................................................................................. 145 Quasi-conformal Mappings and L eb e sg u e Density by F. W. Gehring and J. C. K e lly ........................................................ 171 On the Moduli of Compact Riemann Surfaces with a Finite Number of Punctures by Jane G ilm a n ...................................................................................... 181 Maximal Groups and Signatures by Leon G re e n b e rg ................................................................................ 207 Commensurable Groups of Moebius Transformations by Leon G re e n b e rg ................................................................................ 227 Chabauty Spaces of Discrete Groups by William H arvey .................................................................................. 239 Monodromy Groups and Linearly Polymorphic Functions by Dennis A. H e jh a l.............................................................................. 247 C ollars on Riemann Surfaces by Linda K een ........................................................................................ 263 Deformations of Certain Complex Manifolds by Garo K. Kiremidjian ........................................................................ 269 On the A (T ) C B (T ) Conjecture for Infinitely Generated Groups by Joseph L e h n e r .................................................................................. 283 The Fundamental Groups of Certain Subgroup Spaces by A. M. Macbeath ................................................................................ 289 Modular Groups and F iber Spaces over Teichmiiller Spaces by C. Maclachlan .................................................................................. 297 Universal Properties of Fuchsian Groups in the Poincare Metric by Albert Marden.................................................................................... 315 Remarks on Complex Multiplication and Singular Riemann Matrices by Henrik H. Martens ............................................................................ 341 Intersections of Component Subgroups o f Kleinian Groups by Bernard Maskit ................................................................................ 349 Polynomial Approximation in the B ers Spaces by Thomas Metzger and Mark Sheingorn ........................................... 369

CONTENTS

ix

Simple Illustrations o f the U s e s of E xplicit Computation of Theta Constants by Harry E. Rauch ................................................................................ 379 On the Relation between L o c a l and Global Properties of Boundary V alu es for Extremal Quasi-conformal Mappings by Edgar R e ic h ........................................................................................ 391 Symmetric Embeddings of Riemann Surfaces by Reto A. R u e d y .................................................................................. 409 On the Trajectory Structure of Quadratic Differentials by Kurt Strebel ...................................................................................... 419 The Maximal Inscribed B a ll of a Fuchsian Group by Jacob Sturm and Meir Shinnar ....................

439

DISCONTINUOUS GROUPS AND RIEMANN SURFACES

C O N S T R U C T A B IL IT Y A N D BERS

S T A B IL IT Y OF

K L E IN IA N G R O U PS William A bik off*

The study of small deformations of Kleinian groups was initiated by Poincare, who claimed without proof that a small deformation of a Fuchsian group results in a discontinuous group.

The deformations of surfaces of

finite Poincare area w as studied by Teichmiiller and he gave this deforma­ tion space a real analytic structure.

Later, Ahlfors and Bers used the

results of Teichmiiller, to parametrize the homotopy equivalence c la sse s of finite Riemann surfaces. It was, however, the beautiful result of Teichmiiller that there is an intimate tie between the deformations of a surface and the bounded quadratic differentials that live on that surface.

Ahlfors

finiteness theorem yields the fact that the quotient of the set of discon­ tinuity Q (G )

of a finitely generated Kleinian group by the group

finite union of conformally finite Riemann surfaces.

G

is a

It therefore becomes

relevant to consider whether each small deformation of a finitely generated Kleinian group may be induced by deformations of the quotient fl(G )/ G , or, more specifically, infinitesim ally by the bounded quadratic differentials on that quotient. of G

T h ese deformations are the quasi-conformal deformations

whose dilatation is supported on the ordinary set of G. Bers [4 ], Kra

[6 ], and Maskit [10] have studied these deformations and have woven an elegant theory of those quasi-conformal deformations of Kleinian groups. It is however not clear when all small type-preserving deformations of a finitely generated Kleinian group G

are induced by quasi-conformal defor­

mations whose dilatation is supported on its ordinary set.

Such groups are

called Bers stable. *

R esearch partially supported by the N a tio n a l S cience Foundation.

3

4

WILLIAM ABIKOFF

Bers [3] first raised questions of stability in the study of B-groups, and proved that a finitely generated quasi-Fuchsian group is Bers stable. Gardiner and Kra [5] have derived a cohomological condition for Bers stability.

Marden [8] showed that a finitely generated Kleinian group with­

out torsion is Bers stable if it is geometrically finite, i.e., has a finite sided Dirichlet region in the upper half-space. relationship between Bers

We w ill examine here the

stability and the construction algorithms

developed by Maskit and used by him to discover some of the deepest re­ sults in Kleinian group theory. been announced in [2],

Some of the results presented here have

The combination theorems as stated there have

not appeared in print but w ill appear shortly in M askit’s book. We w ill c a ll a group Maskit-constructible if it is finitely generated and may be constructed from cyclic groups using the Maskit algorithms.

The

principal results are:

THEOREM 1. A Maskit constructible finitely generated Kleinian group is Bers stable.

COROLLARY.

A totally degenerate group is not Maskit constructible

from cyclic groups.

Since constructibility is a topological property Theorem 1 has a strong­ er form, conjectured by Bers [4], namely

THEOREM 2.

Any quasi-conformation deformation of a Maskit constructi­

ble group is Bers stable, i.e., is uniformly stable.

T hese theorems are direct consequences of Theorem 3 and 4.

In Sec­

tion 3, we show

THEOREM 3.

If G

IS a finitely generated Kleinian group, formed by

application of Maskit Combination I to its subgroups both G j

and G2 are Bers stable, then G

Gj

and G 2, and

is Bers stable.

CONSTRUCTABILITY AND BERS STABILITY

5

Section 4 is devoted to the proof of

THEOREM 4.

If G

is a finitely generated Kleinian group which is

generated by a subgroup Combination II and G j

and an element y is Bers stable, then G

by application of Maskit is Bers stable.

We note that the statements of the combination theorems given in Sec­ tion 2 are not the most general given by Maskit. or H j

and

in Combination I

H2 in Combination II are non-elementary quasi-Fuchsian of

the second kind, the theorems remain valid. that Bers

If H

While it may be conjectured

stability is preserved in the above cases, the techniques given

here do not yield a proof.

No Kleinian groups known to this author are

constructible using the most general combination theorems and not con­ structible using the combination theorems as stated here. §1.

Kleinian groups and stability A Kleinian group is a discrete subgroup of the Mobius group = Jfi = / z t-> — —k | a, b, c, d, e C, ad-bc = 1, z e C L ( cz+d

which acts discontinuously at some point of C

and which is neither

trivial nor a finite extension of a cyclic group.

If 11(G) is the subset of

C

on which the Kleinian group

the set of discontinuity of G. G

is elementary if A (G )

G

acts discontinuously,

A (G ),

the limit set of G,

has le ss than three points.

H (G ) is

is called

C — H (G ).

We define the pro-

jection map 77 = 77q

11(G) -> i2^G V G •

A ll groups considered here have finite quotient, i.e.,

Vq

is a

finite union of conformally finite Riemann surfaces. A Kleinian group domain B

G

is called quasi-Fuchsian if there is a Jordan

so that y (B ) = B

for a ll y e G.

A (G ) = (9B and of the second kind otherwise.

It is of the first kind if We say that Q j

is a

6

WILLIAM ABIKOFF

component of G

if f lj

is a component of £2(G).

stability subgroup of A, y (A ) = A.

If

C —

denoted

then the

is the set of those y e G

is a component of G,

then J = dQ>1 H

tor for G

G^

If A C C,

and O j j

so that

is a component of

1 is a quasi-circle and is called a separa­

(se e Abikoff [1 ]).

A subset F

of fl(G )

is called a fundamental set for G GF =

U

if,

y (F ) 3 Q (G )

ye G

and F fl y F = 0 If f : C -» C for G
for y e G — H.

If G

for

has finite quotient then

7

CONSTRUCTABILITY AND BERS STABILITY

we claim

H is trivial, cyclic or a finitely generated quasi-Fuchsian group

of the first or second kind.

Since

is either a subset of A (G )

or J fl 12(G) is contained in a single com­

ponent of 12(G).

In the first case

B f) A (G ) =

J is a separator and in the latter case,

the image of J fl 12(G) in (B )

, it follow s that J = d B

0

is a finite set of disjoint simple loops.

is a topologically finite (p ossibly ramified) surface.

77

now coalesce.

The two cases

If J is a separator then H is finitely generated as is

shown in [1],

Otherwise

H

is the cover group of a planar cover of a

finite surface, hence is finitely generated.

It must be either elementary or

quasi-Fuchsian of the second kind by Marden [7] and Maskit [9]. §2.

The Maskit combination theorems The combination theorems are techniques for building a Kleinian group

from simpler groups. group

Combination I gives a geometric condition that a

G be describable as a free product of two groups

with amalgamated common subgroup

H.

Hj

and G 2

Combination II gives such a con­

dition for adjoining a transformation y a subgroup

G^^

to a group

of GQ into a subgroup

G Q which conjugates

H 2. We proceed to state the

theorems.

2.1 C o m b i n a t i o n T h e o r e m I. with a common subgroup H,

G x and G 2

Let

and B 1

and B 2

be Kleinian groups

precisely invariant discs

under H with J = d B j = 2.

(T p

could just as w e ll stand for T orelli space or the Jacobian locus of the Siegel upper half sp ace.) Hp be the locus in Tp the dimension of Hp is 2).

The (com plex) dimension of Tp

is

3p — 3. L et

corresponding to hyperelliptic surfaces.

is

2p — 1;

Then

that is, the codimension of Hp

in Tp

p — 2. Riemann surface admitting a g *(2 ) We first consider a Riemann surface,

F p _ 1 where the

Wp,

admitting a plane model

g (2 ,p —1) cut out by the lines of

By formula (1 ) d = (p —1) (p —6)/2

since

n = p — 1.

P 2 is half-canonical. It follow s that the

g (2 ,p —1) is half-canonical if and only if there is an adjoint of degree p —6, f p _ 6 [4]. divisor.

Thus, if L

is a line then f p _ 6L 2 cuts out a canonical

We give three examples.

(i)

p= 6; g*(2 , 5).

F 5 is a non-singular plane quintic.

(ii)

p= 7, g * (2 ,6 ).

F 6 has three collinear nodes.

9

9

F 4 F j — F^ (iii)

has nodes where

p= 9, g*(2 , 8). example,

Fg

F 1 and

For example,

F 3 intersect.

has twelve nodes lying on a cubic.

F 2 Fg — F 2. But for this example the given

For g * (2 ,8 )

is not complete. L et g *(r).

G * (r )

be the locus in Tp

of surfaces admitting a complete

We w ill now try to compute the dimensions of the various compo­

nents of G *(2 ).

Let

simple

A s a guess we give

g * (2 )’s.

component of

G * (2 )

G *(2 )

be the components ofG *(2 )

G *(2 )

is the locus in

df) ... = ^ - ( e ; B ) = 0 for some half period

%

an argument thatindicates that each

may have codimension three in Tp.

may seem surprising since

containing the

This result

Tp where

- (e; B ) =

^ e where 6 has odd order. Thus

the codimension, three, is considerably le ss than p, tions defining the locus in terms of the theta function.

the number of equa­

ROBERT D. M. ACCOLA

16

If we first consider

F

^s

with

( p - l ) ( p - 6 ) / 2 (=d) nodes, but with

no restriction on the position of the nodes, then Severi’s result gives the dimension of this family of curves to be the nodes to lie on a

(p —1) (p+2)/2 — d.

Since we wish

f p _ 6 , the number of degrees of freedom for the

nodes is reduced by d — (p —6) (p —3)/2;

for the first

(p —6 )(p —3)/2

nodes

determine f p _ 6 and the remaining nodes have but one degree of freedom since they must lie on the f p _ 6 so determined. F p _ i ’s

Thus the family of

with d nodes on a fp_ 6 has dimension

(p—l)(p + 2 )/ 2 — d —

(d — (p—6) (p—3)/2) = 3p + 2.

Since every Riemann surface has at most

a finite number of

the dimension of G *(2 ) w ill be

g * (2 )’s,

3p + 2

minus the dimension, eight, of the groups of collincations of P 2. Thus we guess that G *(2 ) has dimension

3p — 6 ( = dim Tp — 3).

The method of arriving at the guess furnishes a proof only for p = 6, 7 and 8.

For

p = 9 the twelve nodes may not be chosen arbitrarily on a

f3; only eleven can be chosen arbitrarily.

The guess has been verified

for p = 6, 7, 8, 9 and 10. The guess does not really deserve the status of a conjecture since there is some counter evidence. G *(3 )

of different dimensions.

For

p = 10 there are two components of

A lso for p = 6 there is a question of the

dimension of a ll components of G *(2 ). Wp e G *(2 )

and

(ii) every

For a

W6 is hyperelliptic if and

only if

(i)

g *(2 )

G *(2 ).

But it is not clear, at least to the author, whether or not each com­

ponent of H6 is a component of G *(2 )

is composite.

Thus

H6 C

or whether H6 is in G *(2 ).

If

the former case held, then those components of G *(2 ) — G *(2 ) would have codimension 4,

the known codimension of Hg

Of course, for p = 5, codimension three in Tg. 3).

in Tg.

G *(2 ) = H 5, so all components of G *(2 ) A lso every

g *(2 )

have

on a W5 is composite.

Syzygetic families of half-canonical linear series

D E F IN IT IO N .

Let

order 2 ^ (0 < V < p).

G be a subgroup of the half-periods of J(W ) A coset,

H,

of G

of

in the half-periods of J(Wp)

w ill be called a syzygetic coset if the orders of vanishing of 0(u; B )

at

the points of H all have the same parity (odd or even).

2^

If there are

ALGEBRAIC CURVES AND LINEAR SERIES

complete

g *(r^ )’s

on Wp, i = 1,2,

17

> 0; r- = rj (mod 2),

sponding to the points of H then the

g * (r p js

corre­

w ill be called a syzygetic

family. For example, any complete family of order one and

(V = 0).

w ill be considered a syzygetic

g*^ )

and

g *(r2)

are distinct, complete,

rj = r2 (mod 2) then they form a syzygetic family of order two ( V = l ) .

g *(r-), i = 1, 2, 3, 4, (i)

If

g *(r)

w ill form a syzygetic family of order four (V = 2) if

they are distinct and complete,

(iii)

if

(ii)

is an arbitrary divisor from

Di + D2 + D3 + D ,

= r2 = r3 = r4 (mod 2), g * (r j)

and

we have that the divisor

is bicanonical.

Riemann surfaces admitting automorphisms of period two (involutions) usually admit extraordinary syzygetic fam ilies.

Our purpose in this s e c ­

tion and the next is to exhibit Riemann surfaces as plane curves which admit syzygetic fam ilies but which do not admit obvious involutions. 3.0. V = 0; g *(p / 3 ),s : A theorem due essen tially to Castelnuovo [3] asserts that

g *(r, p—1) is composite for

r> 2

if p < 3r:

If 3 1p a

problem arises of exhibiting Riemann surfaces admitting a simple For

p = 6 and

g *(4 )

p = 9 the examples given in Section 2) suffice.

on a W12 can be exhibited by considering a

ordinary triple point.

g*(p/3). A simple

F 7 with a single

Conics through the triple points cut out a

g *(4 )

since sp ecial adjoints are quartics with double points at the triple point of Fy.

Similarly, arbitrary conics cut out a

F ? (p = 15). genera

g *(5 )

on a non-singular

In a sim ilar manner examples can be constructed for higher

p where

3|p.

Now we ask whether the existence of a simple p lies the existence of such a plane model. simple

g * (2 )

model.

For

For

g *(p /3 )

on Wp im­

p = 6 the existence of a

im plies, by definition, a non-singular plane quintic for a p = 9 the existence of a simple

whose equation is shows that the

g * (3 )

implies a plane model

F^ F 2 — F 4. The method of Castelnuovo’s theorem

g *(3 )

gives a space model for W9 which is the complete

ROBERT D. M. ACCOLA

18

intersection of a quadric and quartic surface.

Then eliminating one vari­

able (projecting from a point not on the quadric) gives an equation of a plane curve of the indicated form. 3.1.

V = 1; two g * ((p —2 )/3 )’s : By an extension of the theorem of

Castelnuovo cited above in subsection 3.0 a Riemann surface of genus cannot admit two simple

g * (r )’s

amples of plane curves of genus g * ((p —2 )/ 3 )’s If A

A , B, C,

ear.

L et

P

point where For

Ex­

p = 2 + 3r admitting two simple

are now given.

and B

Let

for p — 2 < 3r (p = 4 is excluded).

p

are points of

and D

and

L BD

with ordinary triple points at P lines through Q cuts out a

half-canonical.

be the line connecting them.

Lq

L AB

and

L CD

meet and let

Q be the

meet.

p = 5 consider a curve

special adjoint if

L AB

be four points of P 2 no three of which are collin-

be the point where L AC

P 2 let

F 7 with nodes at A, B, C, and Q

g (l,4 )

(formula (1 )).

on F ?. Since

La

P

D

and

The pencil of ^C D ^Q

b

is a line through Q we see that this

Similarly, the lines through

and

cut out a

*s a

g (l,4 )

g * (l).

is

It can be

shown that every Riemann surface of genus five without involutions admit­ ting two For

g * ( l ) ’s

p = 8 consider

points at B conics,

admits such a plane model [2].

and

C

P , Q,

adjoint and so the F 2 ’s

3.2.

and

with nodes at A

and D,

and ordinary four points at P

F 2 , through

through P , Q

Fg

and B.

cut out a

Then

g *(2 )

and

ordinary triple Q.

Consider

L AC L CD F 2 *s a s Pe c ^a ^

on F 7. Similarly for conics

C.

V = 2; four g * ((p —3 )/ 3 )’s : By an extension of the theorem of

Castelnuovo cited above in subsection 3.0, a surface of genus

p cannot

admit a syzygetic family of four simple

Examples

g * (r )’s

for 3r > p — 3.

are now given to show this bound is sharp. U sin g the notation of 3.1 let

R be the point where

meet, that is, the third diagonal point for the quadrangle

L AD

and

L BC

A, B, C, D.

ALGEBRAIC CURVES AND LINEAR SERIES

19

Consider

F g with ordinary triple points at A, B, C ,

and D

at P , Q,

and

is a line passing

through A of such C

R.

p = 6 (formula (1)).

fj

then L BC L CD L BD L A

L a ’s

and D

Then

cut out a

g * (l)

If L A

and nodes

is a special adjoint and so the pencil

on F g . U sing analogous notation for B,

we see that

l a l b l c l d l ab

l ac

l ad

l bc

l bd

l cd

is a double adjoint and so the sum of the four linear series is bicanonical. It can be shown that a W6 without involutions admitting a syzygetic family of four

g * ( l ) ’s

admits such a plane model.

Consider a F 10 with ordinary four points at A, B, C nodes at and D. out a

P, Q, Then

g *(2 )

and R.

Then

L A B L A C L AD

p = 9.

Let F A

is a sp ecial

D

be aconic through adjoint so the

and B, C

F A ’s cut

on F 1Q. Since

f a f b f c f d l ab

l ac

l ad

l bc

is a double adjoint, the sum of the four g * (2 )’s 4).

and

l bd

l cd

is bicanonical.

Construction of more general syzygetic families We now give a more general procedure for producing plane curves

which have fairly obvious syzygetic fam ilies. for V = 1 and Let

k and

A ll the previous examples

2 are sp ecial cases of this procedure. t be positive integers with k > t.

ducible plane curve with

t tacnodes at points

which are collinear.

L ^ .^ L j.

Let

Let

F ^+4

be an irre­

P 1, P 2 , ...,P^.

no three of

be the t tacnodal tangents and sup­

pose they are distinct and all meet in a single point Q. has an ordinary k-fold singularity at Q.

Suppose

Since each tacnode contributes

two to d of formulas (1 ) and (2) we have (3) (4)

F ^ +4

d = 2t + k(k—1)/2 and

so

2p — 2 = (k + 4 ) ( k + l ) - 4t - k ( k - l ) = 6 k - 4 t + 4

ROBERT D. M. ACCOLA

20

Special adjoints have degree

k+1

and must be tangent to L j

addition to the usual condition at Q. must consider separately the cases

at P j

in

In the following computations we

k even and odd although the final

results are the same for both cases. Suppose the

(2 s )

l x, l

k is odd.

Pick an integer

possible ways, choose

2, ...,

l

2 s . Then

( k + 1 — 2s)/2.

2s

s

so that 0 < 2s < t.

of the lines

f ( k + i _ 2s)/2

g (R ,N )

k+1

if m =

Pa sse s through P 2 s + l , , , , , P t and

has a (k — 1 — 2s)/2-fold singularity at Q. a sp ecial adjoint.

say

r\

L 1 L 2 . . . L 2 s fm w ill have degree

Suppose

In one of

Then

The “ free points’ ’ of such

L ^ L ^ .-L g f^

fm’s

w ill be

cut out on

a

where

N = (k + 4 )(k + 1 — 2s)/2 - 2 ( t - 2 s ) -

k (k -l-2 s )/ 2

= p -

1

and R > (k + 1 — 2 s )(k + 7 — 2s)/8 - ( t - 2 s ) -

( k - 1 - 2 s )(k + 1 - 2s)/8

or R > k - t+ 1 . Thus such

f (k + i -2 s )/ 2 s

choices of 2s

cut out

(2 s ) g * (R )’s.

of the lines we get 2 *~ 1 f = ^

be shown to form a syzygetic family. R = k — 1+1 For

for a ll the

k even and

lines, say adjoint if

Considering all

( 2 s ) ) g * (R )’s

which can

It is to be expected that in general

g * (R )’s.

s

chosen so that 0 < 2s

L,1, L 2, ..., L 2s+1. Then m = (k — 2s)/2,

a

r\

2s + 1

b 1 L 2 ... L 2s+1 fm w ill be a special

fm passes through P 2s+2> •••> P t anc* ^m ^as

a ( k — 2s — 2)/2-fold singularity at Q. again cut out on F ^ +4

+ 1 < t,choose

The “ free points” of such

g *(R ), R > k —1+1, and

syzygetic family of order 2*-1

^

(2s + 1)^

fm’s

so we again obtain

a

.

There remains, of course, the problem of the actual existence of irre­ ducible algebraic curves scribed singularities. if for any t+ 1

points

F ^ +4

of genus

p = 3k — 2t + 3 having the pre­

C a ll a pair of integers ...,P ^ ,Q ,

(t, k), k > t > 0, admissible

no three of which are collinear,

ALGEBRAIC CURVES AND LINEAR SERIES

21

there exists an irreducible algebraic plane curve

F ^ +4

of genus

p =

3k — 2t + 3 having the prescribed singularities.

By using Bertini’s

theorem (concerning irreducible curves in a linear family of curves) and other standard methods we can obtain the follow ing results about adm issi­ ble pairs: (1 )

(t, t + w )

(2 ) If (t, k) Now set

is adm issible if

t < 13 + 5w (w > 0)

is adm issible so is

(t, k + 1).

V = t — 1 for the follow ing theorem.

THEOREM 1.

For integers

V, k so that (V + l , k )

exists a Riemann surface of genus family of 2 ^ g * (R )’s

is admissible, there

p = 3k — 2V + 1 admitting a syzygetic

where each R > k — V.

The c ases considered in subsection 3.1 occur when V = 1 and or 3.

k= 2

One transforms the curves derived in Theorem 1 by a standard quad­

ratic transformation with fundamental points at the two tacnodes and at any third non-singular point of F j ^ . 3.2 occur when

V = 2 and

The cases considered in subsection

k = 3 or 4.

One transforms

by a

standard quadratic transformation with the fundamental points at the three tacnodes. The methods used for the above constructions can be considerably ex­ tended by considering singularities of F on each line

L

where several tacnodes occur

and/or more complicated tacnodes occur on each

restriction on adm issible pairs

(t, k)

L.

The

given in (1 ) immediately preceding

the statement of Theorem 1 seems inherent in the method of construction but hardly seems inherent in the nature of things.

N evertheless, the

above constructions give a sharpness for the follow ing generalization of Castelnuovo’s theorem which subsumes a ll the sp ecial instances cited in Section 3).

We omit the proof [1].

22

ROBERT D. M. ACCOLA

THEOREM 2.

For

V = 1, 2,

or 3 and any k > V ,

a surface of genus

p < 3k — 2V -i- 1 admitting a syzygetic family of g *(k —V ) ’s must admit an involution and either no

g *(k —V )

of order 2^

of the syzygetic family

is simple or Wp is hyper elliptic.

Castelnuovo’s theorem and its extension can also be used to show that all

R ’s

of Theorem 1 are equal to k — V

namely

V = 1, 2, 3,

and

for low values of V,

4.

UNIVERSITY OF MANCHESTER AND BROWN UNIVERSITY

REFERENCES

[1].

A ccola, Robert D. M., Vanishing properties of theta functions for abelian covers of closed Riemann surfaces.

[2].

Part III (to appear).

A ccola, Robert D. M., Some loci of Teichmiiller space for genus five defined by vanishing theta-nulls (to appear).

[3].

Castelnuovo, G., Sur multipli du una serie lineare di gruppi di punti. Rendiconti del Circolo Matematico di Palermo Vol. VIII (1893), pp. 89-110.

[4].

Kraus, Note iiber ausgewohnlicke Specialgruppen auf algebraischen Curven.

[5].

Mathematischen Annalen Vol. XVI (1880), p. 310.

Severi-Loffler, Vorlesunger iiber algebraische Geometrie, L e ip z ig 1921 (C h elsea).

S U F F IC IE N T C O N D IT IO N S F O R Q U A S I-C O N F O R M A L E X T E N S IO N Lars V. A h lfors*

1.

A locally homeomorphic mapping of a sphere into itse lf is automatically

a homeomorphism.

This simple topological fact can be used to prove that

a function on a subregion is schlicht.

Indeed, it w ill be schlicht if it can

be extended to a locally homeomorphic mapping of the whole sphere.

The

easiest way to show that a mapping is locally homeomorphic is to show that the Jacobian is

^ 0.

This w ill be so if the mapping f

satisfies

fz ^ 0 and

(D

I%l < H f z l

for some k < 1.

Hence the importance of quasi-conformal extensions.

To be specific, suppose that f satisfies (1) together with fz ^ 0 in A == \\z\ < 1}.

If k < k '< 1,

extended to a k '—q.c.

under what additional conditions can

mapping of the Riemann sphere?

f be

The question

may be regarded as consisting of two parts: 1) When does there exist a of A * = { \z\ > 1! such that f to

k -q .c. and

f*

locally homeomorphic mapping f* have equal continuous extensions

\z\ = 1. 2) If f and f *

have this property, when do they together form a

locally homeomorphic mapping of the sphere, and hence a

k -q .c .

homeo­

morphism of the sphere on itself. There are some trivial answers to the second question. if f and

f*

disks

and A * ,

*

A

For instance,

extend to locally homeomorphic mappings of the closed then they determine a global homeomorphism.

R esearch partially supported under N S F Grant GP-38886.

23

More

LARS V. AHLFORS

24

refined results would be interesting, but seem to be difficult to come by. At present the best method is to approximate f n with good boundary behavior that allow

f by a sequence of functions

kn-q.c.

extensions

ffl*.

If

kn -» k' it is p ossible to extract a subsequence which converges to a k '—q.c. 2.

extension of the original f.

Not much generality is lost by assuming that k = 0 or, in other words,

that f is conformal.

Indeed, by the general existence theorem for quasi-

conformal mapping there exists a q.c.

homeomorphism h of the sphere

on itself that preserves symmetry with respect to the unit circle and sa tis­ fies

h^-/hz = fz /fz

in A .

This means that f ° h_1

and if f has a quasi-conformal extension so does

is conformal in A , f o h- 1 , and vice

versa. Of course, since less tractable.

h is not explicitly known, the general case remains

It is therefore le ss by choice than of necessity that we

limit ourselves to the discussion of holomorphic f. has a 3.

k-q.c.

If a holomorphic f

extension to the sphere we shall say that f is

Two separate sufficient conditions are known.

W eill [1] proved that f (2)

k-schlicht.

In 1962 Ahlfors and

is k-schlicht if its Schwarzian derivative sa tisfies |{f,z}| ( l - | z | 2) 2 < 2k .

This result has been important for the theory of Teichmiiller spaces and Kleinian groups.

In 1972 Becker [2] proved the sufficiency of the condition

(3 )

|zf7f'| ( l - | z | 2) < k .

This w as an improvement on a result by Duren, Shapiro, and Shields [3]. Actually, Becker proves much more.

Under the condition (2 ) or (3 ) he

shows that f is the initial link in a chain of subordinate schlicht func­ tions.

This leads to a Lowner-type differential equation and to more

intrinsic proofs than by the original method.

In this paper we do not pur­

sue this interesting aspect, but we prove the following generalizations of conditions (2) and (3).

25

QUASI-CONFORMAL EXTENSION

THEOREM.

In order that f

(A )

be k-schlicht it is sufficient that either

z p ( l - | z | 2) + c|z|2| < k

for some complex constant c (B )

| j f f , z i ( 1 - |z|2) 2 -

c ( l - c ) z 2| < k |c|

|c —11 < k.

with

Note that (2) is (B ) with 4.

with |c| < k, or that

c = 1,

and (3 ) is (A ) with

c = 0.

We proceed directly to the proof, but under the preliminary condition

that f is holomorphic on the closed disk A .

It is convenient to define

the extension in the form (f(z ) = ;

f (z )

(g (l/ z ) where If we

g is sense reversing

and

for

|z| < 1

for

|z| > 1

k-q.c.

in

A withg = f

set g = f + uthe boundarv condition

is

on {|z| = l }.

u = 0,and thequasi-

conformality is expressed by (3 )

|f'+ uz | < k |Ug-1 .

In addition we require that

0 in A

and that u is sufficiently regu­

lar, for instance real-analytic on

|z| = 1.

evident that F

homeomorphic extension of f.

w ill be a k-q.c.

In these circumstances it is

Nothing prevents us from making u dependent on f.

If we choose

u = i '/ o condition (3) becomes (4)

and

| x l > x2 >

The closed region

•

D n bounded by the Euclidean polygon with vertices

x n’ xn+l ’ x n+l + i/xn +lplaced by its image

xn " °

1 + i/xn+l>

Hn(D n).

1 + i/xn- x n + i/xn’ xn may be re*

If this is done for a ll

mental domain for G which has

n we obtain a funda­

z = 1 as the only limit point on its

boundary and this is not a parabolic fix-point.

40

A. F. BEARDON

F inally we remark that, as we have seen, locally finite.

D

may be convex but not

Example 1 shows that this is so and in this case

has genus one.

A */ G

The author can prove that in the case of the Modular

group (and certain other groups) any convex fundamental domain is neces­ sarily locally finite.

It remains an open question, however, as to whether

or not there exists a group A */ G §6.

G

has genus zero and (ii)

and a fundamental domain D D

such that (i)

is convex but not locally finite.

Kleinian groups Let

G be a Kleinian group with set of discontinuity fl.

a maximal inequivalent set of components of fl, subgroup of A j

and Dj

A parabolic fix-point ft/G:

let

Gj

Let A j

be

be the stability

a connected fundamental domain for Gj

in A j.

p is a cusp if it corresponds to a puncture on

then there are horocyclic neighborhoods of p lying in some image

of A j. It is evident that the preceding results are true in much greater gener­ ality than stated and that the proofs can easily be modified where neces­ sary.

We add (as u sual) the cusps to the A j

true for Dj

and the three theorems are

in A j.

The same arguments apply to finitely generated Kleinian groups. suppose that °o is a cusp whose stabilizer contains that H = {y > t!

n).

P : z -> z + 1 and

lies in A j : we suppose also that D- fl H can only

meet a finite number of sets of the form H fl P n V (D j) varying

We

With this supposition, if G

locally finite, then Dj

(for a fixed

is finitely generated and

is “ finite sid e d .’ ’

V and Dj

is

FUNDAMENTAL DOMAINS FOR KLEINIAN GROUPS

41

REFERENCES

[1].

Beardon, A. F ., and Maskit, B ., “ Limit points of Kleinian groups and finite sided fundamental polyhedra,” (unpublished).

[2].

Greenberg, L ., “ Fundamental polygons for Fuchsian g ro u p s/ ’ /. cVAnal. Math., 18 (1967), pp. 99-105.

[3].

H eins, M., “ Fundamental polygons of Fuchsian and Fuchsoid groups/ ’ Ann. Acad. Sci. Fenn., 1964, p. 30.

[4].

Marden, A ., “ On finitely generated Fuchsian gro u p s/ ’ Comment. Math. H elv., 42 (1967), pp. 81-85.

SPACES OF D E G E N E R A TIN G RIEMANN SU R FA C ES* Lipman Bers T h is is a prelim inary report on work in progress, prepared for the Maryland co n fe re n c e .**

P ro o fs w ill appear elsew h ere.

Since the manuscript is not com­

pleted, I prefer to la b e l a ll statements of results conjectures.

The aim is to con­

struct an alogu es of Teichm iiller sp a c e s for a lg e b ra ic curves which have or acquire ordinary double points.

§1.

RIEM ANN S U R F A C E S WITH N O D ES A Riemann surface with nodes,

such that every either to the set

S,

is a connected complex space

P e S has arbitrarily small neighborhoods isomorphic |z| < 1 in C

C 2. In the second case,

P

or to the set

|z| < 1,

is called a node.

complement of the nodes is called a part of S, singular) Riemann surface. sphere with

0,

|w| < 1,

zw = 0 in

Every component of the it is an ordinary (non­

S is called exceptional if a part of S is a

1 or 2 punctures, or a torus with

0 or 1 punctures.

If S is non-exceptional, every part carries a canonical Poincare metric. If S is also compact, the Poincare"area of every part is a multiple of 2n, and the Poincare" area

A

of S is a multiple of 477. The genus

compact non-exceptional S is defined by the relation number k of nodes sa tisfie s the inequality 3p — 3 nodes, it is called terminal. finitely many terminal S,

A = 4z7(p—1);

0 < k < 3p — 3.

If S has

p.

with or without subscripts, denotes a com­

pact non-exceptional Riemann surface with nodes, of a fixed genus denote by Mp the space of isomorphism classes of a ll such *

**

the

There are, up to isomorphisms, only

for a given

From now on the letter S,

p of a

Work partially supported by the N a tio n a l S cience Foundation.

p. We

S.

_

International Conferen ce on Riemann Surfaces and D iscontinuous Groups, U n iv e rsity of M aryland, C o lle g e Park, Maryland, May 21-25, 1973.

43

44

LIPMAN BERS

Let

q > 0 be an integer.

phic form of type

A regular q-differential on S is a holomor­

(q, 0) on each part of S,

which has poles of order at

most q at the punctures, the “ residu es” at any two punctures joined in a node being equal (if

q is even) or opposite (if q is odd).

of a holomorphic form w

of type

(q, 0) at a point P

Riemann surface is the residue, at P , where z

[T h e residue

of a non-singular

of the Abelian differential w z^

is a local parameter with z = 0

at P .]

The number of linearly independent regular q-differentials is q = l

and

(2q—1) (p—1) for q > 1.

p for

Every b asis of regular q-differentials

defines a holomorphic (q -canonical) embedding of S into and exhibits

1dz,

P (2 q -l)(p -l)-l

S as a “ sta b le ” (in the sense of Mayer and Mumford) alge­

braic curve with nodes, of arithmetic genus A continuous surjection

p.

f : Sj -> S w ill be called a deformation if the

image of a node is a node, the inverse image of a node is a node or a Jordan curve avoiding a ll nodes, and the restriction of

f

to the comple­

ment of the inverse images o f the nodes is a sense preserving homeomor­ phism. We choose once and for all an integer throughout this paper.

Tw o deformations

v > 3;

it w ill remain fixed

fj : Sj -> S and f2 : S2 ->

be called equivalent if there exist a homeomorphism such that ^1 = ^2 °

g of Sj

onto

Sw ill S2

anc* & *s h ° motopic to a product of v-th powers of

Dehn twists about Jordan curves mapped by

f^ into nodes, follow ed by

an isomorphism. Note that a deformation which is not a homeomorphism increases the number of nodes.

A terminal

S admits no deformations except for iso ­

morphisms. §2. R IEM ANN S U R F A C E S WITH N O D E S A N D B O U N D A R IE S O F T E IC H M U L L E R S P A C E S Before describing our present approach, we report briefly on the con­ nection between Riemann surfaces with nodes and boundaries of T eich-

45

SPACES OF DEGENERATING RIEMANN SURFACES

miiller spaces. tinuity,

In this section, which may be omitted without lo ss of con­

S denotes a non-singular Riemann surface,

and 77p its fundamental group.

S

its mirror image,

By a sense reversing deformation we mean

a deformation followed or preceded by the canonical mapping of a Riemann surface onto its mirror image. By the theorem on simultaneous uniformization, equivalence c la s s e s of sense reversing homeomorphic deformations

S -» S1 can be represented

by conjugacy c la sse s (in the Mobius group) of quasi-Fuchsian groups iso ­ morphic to 77p, and by points of the Teichmiiller space that T (S )

T (S ).

It is known

is a complex manifold which admits a standard embedding as a

bounded domain in C 3P- 3 , viewed as the space of regular 2-forms on S , and that every boundary point if/ e 0 non-invariant non­

conjugate components A j , . . . , A f with stability subgroups such that A q/G

is a non-singular Riemann surface

A r/G r are the parts of a Riemann surface

Sj

with

S

G j ,..., Gf ,

and A 1/ G 1,.. . ,

k > 0 nodes, two

punctures joined into a node corresponding to conjugate parabolic sub­ groups of G.

The group G

sing deformations

S ->

defines an equivalence c la ss of sense rever­

.

The construction of regular b-groups representing a given sense re­ versing deformation is due to Maskit.

Maskit also showed that d T (S )

contains regular b-groups representing deformations of S

onto terminal

surfaces, and that such groups yield new, non-standard, embeddings of T (S )

into C 3P- 3 . Important recent results are due to Marden (who uses

three-dimensional topology), E arle and Marden, Abikoff (who relies on twodimensional and function-theoretical methods) and Harvey (who operates mainly within the framework of Fuchsian groups).

In view of their investi­

46

LIPMAN BERS

gations, which are to appear, one knows that every regular b-group is in­ deed a boundary group, that is, occurs on the boundary of a Teichmiiller space, and that such a group is determined, up to a conjugacy in the Mobius group, by the equivalence c la ss of the corresponding deformation. One can extend the action of the modular group from T (S )

to the part of

(2T(S) consisting of regular b-groups. §3.

S P A C E S O F R IEM ANN S U R F A C E S WITH NODES We proceed to construct a complex domain parametrizing a ll Riemann

surfaces with nodes which can be deformed into a given surface

S,

sur­

faces with nodes occurring not on the boundary but in the interior of this domain.

The construction depends on various choices, which w ill later

turn out to have been irrelevant, as w ill the complex structure of S, though not its topological structure. Let

S have

have genus

pj

r parts and

5 ^ ,...,^

and k nodes

nj punctures.

P ^ ...,? ^

L et

2j

Then

2k = n1 + ... + nf , p = p1 + ... + pf + k + 1 — r .

We choose

r Fuchsian groups

disjoint closures, such that Gj

G ^ .-.jG has

nj

acting on discs

non-conjugate maximal elliptic

subgroups, each of the same fixed order v > 3, A j/ G j,

A 1, . . . , A r with

the Riemann surface

with the images of a ll elliptic vertices removed, is conformally

equivalent to £ j,

and

G 1, . . . , G f generate a Kleinian group G which is

their free product and has an invariant component A Q such that A q/G a compact Riemann surface A q/G

2

of genus

being ramified over 2k points.

p1 + ... + p , the mapping A Q -> (Our conditions can be satisfied,

in view of the limit circle theorem of Klein and Poincare"and of K lein ’s combination theorem.) Now we assign to each node Pelliptic subgroups T '-, T"-

r\ C Gj

and T ".

C

Gg.

of G,

of S two non-conjugate maximal so that if

P-

joins

Xj

to 2g,

then

Two elliptic vertices, not in A Q, w ill be called

related if they are fixed under elliptic subgroups conjugate to either r ' j

is

47

SPACES OF DEGENERATING RIEMANN SURFACES

or to r ^ .

The T -

are chosen so that the union of the A j/ G j,

with the

images of any two related elliptic vertices identified, is isomorphic to S. If s i e C,

\s{ \ > 0 and sm all, then there exists a unique loxodromic

Mobius transformation group F ] , (j

g: c 1,

which conjugates the group T - into the 1

has multiplier s-,

and £ being as before).

and has fixed points in A j

Set s = ( s 1, . . . , s^).

small, let G Q g denote the group generated by with

Sj ^ 0.

If G

and Aj?

\s\ = max |s-|

is

and a ll elements

By M askit’s second combination theorem,

GQ g

g^ g ,

is a

Kleinian group. Let C

s

be as before, and let W be a quasi-conformal automorphism of

such that W leaves

and W|Aq is conformal.

0 ,1 , oo fixed, Then

WGQ SW_1

W|Aj, j = l , . . . , r ,

is a Kleinian group, defines an element rj

of the Teichmiiller space T (G -), which we represent as a bounded domain 3 p--3 +n : J in C J J. If 0, set tj = a* — a^ where a^ is the repelling fixed point of W °g : °W - 1 i,s* If s- = 0,

and

a-

1

the fixed point of

Wr'- W” 1 in A*. 1 j

set t- = 0. The point (7 ,t) = (r x,

tx , . . . , t k) r C 3p_3

determines the group WGQ SW—1 completely; we denote this group by Gf

andwe denote the set of all

fined by

Xa (S).

Here a

(r, t)

for which a group

represents the choices made:

Gf t is de­

the groups

Gj

and the subgroups F '-, F "-. I.

Xa (S )

is a hounded domain in C 3P ~ 3 .

We denote by Qa (r ,t )

the part of the region of discontinuity of Gf ^

which does not map onto 2 , of the set of elliptic vertices. conjugation

G _ _ W G W- 1 O^o OjO

elliptic vertices in

and by &a( T, t) the complement in The images of the pair T '-,

under the

w ill be denoted by T 'T r, t), r " . (r , t). Two 1 1

t) are called related if they are fixed under

elliptic subgroups conjugate to either F ^ (r, t) or to T ^ (r t t). tient &a( T, t)/G(r, t),

t)

The quo­

with images of related elliptic vertices identified,

is a Riemann surface with nodes

Sf

The genus of Sr t is

p,

and

48

LIPMAN BERS

Sr j. has as many nodes as there are zeros in (t1? . . . , t f). each

Sr ^ is equipped with a deformation

ff ^ : Sr ^

S,

Furthermore, determined up to

equivalence. II.

Xa (S )

is in a natural one-to-one correspondence with the set of

equivalence classes of deformations The fiber space ( M ) f Xa (S ), III. Y a (S )

Y a (S) over

S'

Xa (S )

S. is the set of points

(r ,t ,z )

with

z f QaV t ) .

is a domain in C 3P ~ 2.

L et A (r,t,z )| d z |

denote the Poincare metric in

t).

It projects

onto the Poincare" metric on Sf

Set d = d/dz,

IV. A ll functions dm(9n A,

n > 0, are continuous in Y a (S).

m > 0,

d = d/dz.

The proof depends on standard potential theoretic estimates. If S is non-singular, space

T (S ),

and

Xa (S )

may be identified with the Teichmiiller

Y a (S ) with the standard fiber space of Jordan domains

over T(S ). §4.

A U TO M O R PH IC FORMS One would like to attach to each point (r ,t )

of Xfl(S )

a b a sis of

regular q-forms on ST t, which depends holomorphically on (r, t). present we can do this only after removing from Xa (S) (except when S V.

For every

q,

At

a “ sm all” set

is non-singular or q = 1). 1 < q < v , and every

set o C XQ(S) — j(r °, t°)S

(r °,

t ° ) e XQ(S), there is a closed

which is either empty or of real codimension

2,

and d = (2q—l ) ( p —1) holomorphic functions j(r,t;z) defined for (r, t) e Xa (S ) — a, on Sf

(If t° =

z € &a( T, t), 0, o

which form a

basis of regular q-differentials

is an analytic subvariety. If S

is non-singular,

o may be chosen as empty.) We shall sketch the construction of these functions for k > 0. k = 0,

the result is known from Teichmiiller space theory.

If

49

SPACES OF DEGENERATING RIEMANN SURFACES

If t° = 0,

the first d—k functions j are Poincare" series

< £ j(M ;z ) = 2 $ j ( g ( z ) ) g ( z ) q , g e Gr t ,

where the O j tions 0 j

are appropriately chosen polynomials.

are defined as follow s.

of the loxodromic transformation

Let

a-(r, t),

The next k func­

b^(r, t) be the fixed points

gj r t in the group Gr ^ which conju­

gates one of the two groups r ^ (r , t),

T 'f e , t) onto the other.

These are

holomorphic functions which, for t- -> 0, become the two fixed points of r j(r ,t ),

lying in fia (r, t).

Set

I'jC’’, t; z ) = [z - a j( t , t )]—q [z - bjfr, t )]- q and oo

^ i+ d -k =

2

v i

-

£=o where y 1, y 2 , - - -

runs through all

Gr t if

t^ = 0,

and through a complete

list of right coset representatives of Gr ^ modulo the subgroup generated bv

g; r f if t* ^ 0. 1 ,/,L 1

In the case just considered, the functions 4>i are de1

fined in the whole domain

Y a (S),

and a is the divisor of the Wronskian

of the functions cf>l f ..., 0^. We say that a point (r, t) e Xa (S ) t- = 0 implies that t? = 0, If t° ^ 0,

lies not below

(r °, t ° ) e Xa (S)

if

i = l,...,r .

let kQ be the number of zeros among t^\ ...,t£ .

The first

d — kQ functions are Poincare series formed with d — kQ functions N

^jO", t; z ) = ^

a| [z — A (r , t )]—1 [ z - B ( r , t ) ] _1 [z - C q (j, t )]_1

£=i where a| are appropriately chosen constants and A(r, t),

B(r, t),

CjCr, t), ..., C N (r, t) appropriately chosen holomorphic functions defined for (r ,t )

not below

fixed points of Gr defined above, with

(r °,t °),

and representing N 4 - 2

The remaining kQ functions cj>j t? = 0.

distinct loxodromic are the

k

LIPMAN BERS

50

We need now a modification of the Petersson scalar product. Let 8 be a geodesic Jordan curve or a node of S. of area

2a,

A collar about 8,

in the union of two doubly connected regions on S,

Poincare"area

a,

each of

each bounded by 8 and by a Jordan curve 8 ' freely

homotopic to 8 and orthogonal to the pencil of geodesic arcs entering 8 if 8 is a point, orthogonal to 8 if 8 is a curve. We choose, once and for all, sufficiently small positive numbers and £ q , and a non-increasing function a(t), and

a(£ Q) = 0.

Let

denote the Riemann surface

removed every collar of area 2a(£)

0< t< +

with

aQ

a (0 ) = aQ

S from which we

2aQ about a node and every collar of area

about a geodesic Jordan curve of length

£ < £Q. (Cf. the paper by

L. Keen in this volume.) Let 0 (t)d t^ A(t)|dt|

and i/r(t)dt^ be two regular q-forms on S,

be the Poincare"metric on S;

set

here

and let

t is a local parameter.

We

n /> = i

I

I

0 (t ) 0Xt) A(t)2 -2 q d t A d f .

A basis of regular q-forms on S is called orthonormal if it is so with respect to this scalar product. formal structure of U sing

Such a b a sis is determined by the con­

S, except for a unitary transformation.

IV and V and the compactness of the unitary group one estab­

lishes VI.

If $T' i s

not isomorphic to Sf " ^ ,

then the points

(r', t ') and

N ' and N " such that no S f with (r,t) € N ' • >*■ is isomorphic to a SG g with (cr, s ) e N".

( t", t " )

have neighborhoods

U sin g VII.

V

one can also prove

There are p holomorphic functions 0 ( r , t ; z )

defined in Y (S)

J

which form a basis of regular

1-differentials on Sr

The proof involves normalizing regular 1-differentials on Sr t by the condition:

51

SPACES OF DEGENERATING RIEMANN SURFACES

f

0

k(t)d t =

sjk

where )S1,. . . , /3p are simple closed curves which either define non­ vanishing homology c la s s e s or are homotopic to nodes. One may use this result to study the behavior of a period matrix under degenerations. §5.

F E N C H E L -N IE L S E N C O O R D IN A T E S A partition of S is a set of 3p — 3 — k disjoint geodesic Jordan

curves on S,

where

k is the number of nodes.

The Jordan curves and

the nodes are called the boundary elements of the partition. ary element has two banks.

Every bound­

A partition is ordered by ordering the 6p — 6

banks, with the provision that every bank of a boundary element which does not separate which does.

S has a lower number than any bank of an element

An ordering of the partition induces an ordering of the bound­

ary elements, and induces a direction on every boundary element which is not a node.

F ollow in g the celebrated manuscript by Fenchel and N ielsen,

we can describe the conformal structure of a Riemann surface, endowed with an ordered partition, by such that we have

3p — 3 complex numbers

|£*|

is the length of the j-th boundary element, and for ^ 0 i O' = l£jl e J where the twist parameter 6j is determined by the

position of endpoints of certain geodesic arcs joining and orthogonal to distinct boundary elements. L et

S q be a terminal Riemann surface; it is partitioned by its nodes.

If we order this partition, then a deformation partition on S,

obtained by deforming every Jordan curve on S,

into a node of SQ, into a geodesic. mapping g : (r, t)

(£ x ,

N ielsen coordinates of by

g °fr f

g : S -> SQ induces an ordered

We call

£ 3p_ 3)

For where

(r ,t ) e Xa (S),

mapped

one obtains a

(£ j, . ..,£ 3p_ 3)

are the Fenchel-

t with respect to the ordered partition induced

g a Fen ch el-N ielsen mapping.

It is never holomorphic.

LIPMAN BERS

52

VIII.

Let

g : Xq (S ) -> C 3P

k < 3 p — 3 nodes,

3

g(Xa (S ))

be a F en ch el-N ielsen mapping.

If S has

is the complement of 3p — 3 — k coordinate

hyperplanes, and g is a covering ramified over k coordinate hyper­ planes, a universal covering, if k = 0,

a i/3 P" ~ 3 -sh eeted one if k = 3p—3.

§6. R E L A T IO N S B E T W E E N T H E SP A C E S

Xfl(S)

Statement VIII has several important consequences. IX. A homeomorphic deformation f .S^ -» S2 induces an isomorphism (o f complex manifolds) f^ : XQ(S 1) -> X^g(S2). Thus we may write

X (S)

instead of

Xa (S).

distinguished analytic hyper surfaces, the sets X.

Note that X(S) has t^ = 0 in XQ(S).

A deformation f : Sj -> S2, where S2 has I

more nodes than S j,

induces a holomorphic mapping f^ : X (S1) -> X(S2) the complement of I

k

such that f 5f;(X (S 1))

distinguished hypersurfaces and

is

is a universal

covering. Another consequence of VIII is XI. X (S)

is a cell.

Question:

is

X(S)

a domain of holomorphy?

If Sx has no nodes, so that

X (Sx)

there alw ays is a deformation S^ -> S, X (S1) -> X °(S ),

where

hypersurfaces of X(S). Mod (S 1),

X °(S )

can be identified with

T ^ ),

and therefore a universal covering

is the complement of a ll distinguished

The subgroup of the Teichmiiller modular group

which normalizes the covering group of this covering, induces

a group of holomorphic self-m appings of

X °(S );

this extends to a group

r(S)

of holomorphic self-m appings of

XII.

F (S )

X(S)

induced by topological automorphisms of S.

X(S).

is discrete, and is the group of holomorphic automorphisms of

53

SPACES OF DEGENERATING RIEMANN SURFACES

§7. M O D ULI S P A C E S Let II = n s (r ,t ) e Xa (S)

into the isomorphism c la ss of Sf ^. It turns out that R o y = 11

for y e T (S ), by T (S )

denote the canonical mapping X (S) -> Mp which takes

but that there are points in X (S ) which are not equivalent

and have the same image under Ilg.

(finite) isotropy group, in T (S ), point represented by XIII.

L et

r Q(S ) denote the

of the origin in X (S),

that is, of the

id : S -> S.

The origin in X (S)

such that if X j, X2 e N,

has a neighborhood N, then I K x j) = II(x 2)

stable under r Q(S),

if and only if y (x x) = x2

for some y e r Q(S). U sing statements IX - XIII one obtains XIV.

Mp has a Hausdorff topology and a structure of a normal complex

space (and a V -manifold) which makes all mappings II

holomorphic.

Note that isomorphism c la s s e s of Riemann surfaces with k > 0 nodes form an analytic subvariety of Mp, XV.

There is a number L ,

of codimension

depending only on p,

k. such that every S has

a partition in which no boundary element has length exceeding

L.

This is a geometric lemma which generalizes results by Mumford and by Keen. There are finitely many non-isomorphic terminal Riemann surfaces, S i,

of genus

p,

Ilg (X (S p ),

i = 1, ...,m .

and Mp is contained in the union of all By XV, each

X (S p

has a compact subset

such that M^ is contained in the union of a ll P

EL (X ;), o• i

X^

i = 1, ...,m . Thus

we obtain XVI.

Mp

is compact.

That the space of moduli of non-singular algebraic curves of genus can be compactified by adding moduli of stable curves with nodes was announced by Mayer and Mumford 10 years ago.

The desire to establish

p

54

LIPMAN BERS

this result without using higher dimensional algebraic geometry was the original stimulus for this investigation.

A P P E N D IX .

SC H O T T K Y G R O U PS

B esid es the spaces constructed in Section 2, there are many other spaces of Kleinian groups representing Riemann surfaces with nodes.

A

particularly simple example is that of Schottky groups. Let

0,1, oo, £'2, £3 , £'3, ..., £p,

tended complex plane, and let

be

2p distinct points in the ex­

t1? ...,tp

be complex numbers,

such that the loxodromic Mobius transformations anc* multipliers course,

tj

with fixed points

are free generators of a Schottky group (of

£ 1 = 0, £\ = 00 anc* C2 =

(* )

gj

0 < |tj| < 1,

Such (3p—3)-tuples

( C 2X 3, - , C p, t 1, . . . , t p) e C 3P - 3

form the Schottky space.

We form the extended Schottky space Sp by

adding points with some or all of the tj with tj ^ 0 generate a Schottky group.

being

0,

provided that the

g^

A point in Sp represents, in an

obvious way, a Riemann surface of genus

p,

with one part, and with as

many nodes as there are zeros among t^, ...,tp . Many of the results stated above have valid analogues for the spaces Sp.

We shall describe only the construction of regular q-differentials,

for q > 1. Let

G be the group determined by the point (* ); it is understood that

if t- = 0,

there is no i-th generator.

Now set

$ 1(z ) = z ~ q , O j(z ) = ( z - £ ir q( z - C ir q ,

if t = (t 1, . ..,t p) = 0,

and OO

^ (z ) =

^ e=i

< W z ))/ e (z )q

i = 2 ,..., p ;

SPACES OF DEGENERATING RIEMANN SURFACES

if t ^ 0,

where y 1, y 2, ...

a lis t ° f right coset representatives of G

modulo the subgroup generated by G,

55

g-,

if

t- ^ 0,

a list of a ll elements of

if t{ = 0.

Also set F j(z) = zj+ 1 -q [ ( z - l ) ( z - £ '2) ( z - £ 3)... (z -£ p )]1-cl, and V j (2 )= 2

F j(g (z ))gm

j

= l,...,(2 q -l)(p -l)-p .

gf G Then the functions

0 1(z ), .

tions of all 3p—2 variables

0 (2 q _ i)(p _ i)(z)

are holomorphic func­

£'9, ..., £ p , £ 'p , tii , ..., tp , z, z

q-differentials on the Riemann surface represented by (lC'2 » •••»tp) independent.

avoiding an analytic subvariety The subvariety

COLUMBIA UNIVERSITY

o

,

is presented by < a 1,

then M

i= l

a , b j , ...,b g i

is the group of c la s s e s

(mod Inn ^ S

S

automorphisms of

r1S , that is, automorphisms induced by free substitug which map

JJ

itse lf (but not a conjugate of its inverse).

1 -1

a -jb j

The group MQ is, of course, trivial. Since

77

^

) of proper S

7

tions on the generators

easy.

The homeo-

of Sg is the group of all isotopy c la s s e s of s e lf-

homeomorphisms of Sg.

in Hg,

g.

If

[a ^ b ^ ]

onto a conjugate of

g = 1 the situation is still

is free abelian of rank 2, the group Mj

57

is simply

JOAN S. BIRMAN

58

Aut7r1S1. If a ,b

are a canonical ba sis for

of 7r1S1 is specified by the images respectively. the 2 x 2

then an endomorphism

a

This mapping determines an element of

matrix

group of 2 x 2

||n-||

has determinant +1,

hence

is precisely the

integral matrices with determinant -fl.

been studied extensively in the c la ssic a l literature. esting case is

if and only if

This group has Thus the first inter­

g = 2.

Generators and defining relations for the group M2 were determined by the author and H. Hilden in [6].

Their proof w as based upon a demon­

stration that the group M2 is closely related to the mapping c la ss group of S2 with 6 punctures, and also to the mapping c la ss group of the 6-punctured sphere.

We w ill d iscuss that result in Section 4 below, but

first we review the related questions of (i) the determination of generators for Mg and (ii) the relationship between the mapping c la ss groups of closed and punctured surfaces. §2.

Generators for Mg Let

c

be a simple closed curve on a surface

Sg.

L et

neighborhood of c which is homeomorphic to a cylinder. N c has cylindrical coordinates tc

about c

(y ,0 ),

—1 < y < + 1,

is defined by the identity map outside

(y ,0 + 7r(y + l)).

N c be a

Assum e that

0 < 0 < 2n. A twist

N c , with

tc(y, 0 ) =

(See Figure 1.) Observe that isotopic deformations of the

F ig . l C y lin d rica l neighborhood of the curve c.

N( c

Image of N c

under the tw ist

MAPPING CLASS GROUPS OF SURFACES

C2g+2

F ig . 2.

C3g—1

Generators for

curve c w ill not alter the isotopy c la s s sense of

59

M^.

rQ of a twist

tc

about c.

The

depends on the choice of a positive direction for 0 (but not

for y )

in the parametrization of N , and may be fixed for every c f S u & by assigning an orientation to and using a rule such as “ positive twists move points on a directed line segment which is approaching c to the right.’ ’ Such a rule w ill be valid for points on either side of c. Note that the sense of rQ does not depend on the orientation of c. It was proved by Dehn [12] that

is generated by twists.

Dehn’s

result was later reproved and sim plified by Lickorish [18,19], who showed that the particular set r

, ...,r

illustrated in Figure 2 w ill do. It 3g — 1 may be shown that the mapping c la s s groups of punctured surface (where C 1

the punctures are either kept fixed as a set, or kept fixed individually) are also generated by twists [3].

This is still true if we remove discs

instead of points, requiring either that the boundaries of the discs be kept fixed pointwise or that they be kept fixed setw ise [30], and again it is true (with a single exception) for non-orientable surfaces [9].

It is not

known whether any sim ilar result holds for higher dimensional manifolds; one suspects not.

A lgebraically, there is a c lo se relationship between

the automorphisms induced by twist maps and elementary N ielsen trans­

60

JOAN S. BIRMAN

formations [24].

Intuitively, one may explain the universality of these

particular generators for mapping c la ss groups of surfaces by observing that non-abelian surface groups are either free or are one relator groups, and that in the latter case every automorphism is induced by a free substi­ tution; in the abelian case, the only surface group which is not free is free abelian of rank 2,

and every automorphism of a free abelian group of rank

2 is induced by a free substitution [24]. We illustrate with an example why twist maps are so pleasant to work with.

A s noted above,

M0 is generated by r

be p o ssib le to express the isotopy c la ss curve

,...,r

. Hence it must

C1

z

5

r£ of the twist tf about the

f in Figure 3 as a product of powers of these generators.

now show how to find this power product explicitly. c and d are simple closed curves on Sg, of Sg which maps

c onto d,

We w ill

F irst observe that if

and if h is a homeomorphism

then

a)

td = h t c h - 1 .

Second, we observe that if c

and d are simple closed curves on Sg

which meet in precisely one point, with algebraic intersection 1, effect of the twist tc where it meets

c,

on the curve

d is obtained by cutting the curve

and inserting a copy of c.

examine the effect of the twist product t_

t_ 5

(se e Figures 3 (ii), 3 (iii) and 3 (iv )). S0 the curve 1

t_

t_ 5

t_ 4

t_ 3

(c * ) 2

then the

With this rule in mind, we t_

4

d

t_ 3

on the curve

c.,,

2

From Figure 3 (iv) one sees that on is isotopic to f.

This fact, together

1

with equation (1), implies the ‘ ‘relation’ ’

rc Tc Tc rc 1 rc 1 Tc 1 rc 1 rf1 = Tc 5 Tc c 4 u 3 c 2 C1 2 u3 4 c5 in M2 . With a little practice it is p ossible to discover very complex rela­ tions in Mg by an escalation of this simple technique.

In particular, we

note two general types of “ lo c a l” relations which occur in a ll of the groups Mg,

which are easily verified by the method just described.

They are:

MAPPING CLASS GROUPS OF SURFACES

F ig . 3.

E x p re s sin g

T^ as a power product of T\>T2 , T 3>T4>TS'

61

62

JOAN S. BIRMAN

(3)

^

if

rc rd = rd rc

c

n

d = 0

if c an(i ^ ^ave algebraic intersection

Tc Td rc = Td Tc Td

1 .

We observe that Artin’s braid group [5,24] may be defined by a system of relations which are all of type (3) or (4).

This is the first hint one has

of a connection between braids and twist maps, and indeed this connection has proved to be a very important one. A non-trivial twist map is, of course, alw ays of infinite order.

It is of

interest to note that Maclachlin has found a set of elements, each of which has finite order, which also generate

[20], and this result has been

extended by Patterson [28] to surfaces with punctures, for a ll except cer­ tain exceptional cases.

This fact was used by Maclachlin [20] and again

by Patterson [28] to prove that the space of moduli is in most cases simplyconnected.

These two papers are interesting examples of how knowledge

about the mapping c la ss group has been applied to Riemann surface theory. §3. The relationship between mapping class groups of closed surfaces and of surfaces with punctures Let

Q = Sq1? . . . , q n S denote an arbitrary but henceforth fixed set of

points on the surface cla ss group of S^,

S . The group M is defined to be the mapping & &>n with adm issible maps and isotopies restricted to those

which keep the set

Q

fixed.

Pa the “ pure” mapping &>n class group, is defined by the condition that adm issible maps and iso ­ topies keep each point

q-

A second group,

fixed.

It is immediate that M

from P

by extending by the full symmetric group &>n tion first on P _ The relationship between the groups

P

and P

is obtained

2 . We fix our attenn

__,

m ^ n,

was

studied by the author in [3]:

THEOREM (Birman).

Let

be the natural homomorphism from P

to Per n 1 ( defined by “ filling in ” the n puncture). Then &>n—1 is canonically isomorphic to 771(S^—q 1, ••.,q n_ 1) / center.

ker O

&>n

&>n

63

MAPPING CLASS GROUPS OF SURFACES

T h is result has an easy intuitive explanation.

Suppose that a

is a

non-trivial element of ker O g n> Then a may be represented by a s e lfhomeomorphism of (S^—q1, . . . , q n) (S g—q ^ . - . , q n),

which is not isotopic to the identity on

but is isotopic to the identity on (Sg—q ^ .• .,q n__1).

is the isotopy, then the orbit « t(q n) ^iCSg—q i ,

qn_ i )

If

of qn defines an element of

which we may associate with a.

Looking at this from another point of view, we w ill locate in Mg n a set of maps which generate a subgroup isomorphic to ^ ( S For sim plicity, first take

n = 1,

simple closed curves on S

q = 1.

L et

a 1, . . . , a g ,

—q 1? . . . , q n_ 1).

b^-.^bg

which form a canonical basis for n.S . Let b

o

a-j^ j , ..., ag j , h1 1, ..., bg ^ be a second such system, with each chosen to be homotopic to a j ?b^ on Sg, the puncture q j

be

on Sg—

. Then

a ^ ,b^

but separated from a^, b^ by

ker O g j

is generated by the

2g

twist products V o r 7 1 , ru 1 J I a i a i,i b i b i,i

1 < i < g r • -

It is not difficult to convince oneself that these

2g surface mappings

satisfy the same relations in P g 1 as do { a ^ - ; 1 < i < g! similar construction holds for arbitrary

n;

one chooses

{ a i l , b ii, a ij ; i = 1, ..., g, j = 2, ..., ni with

a^

ai> bj

but homotopic to a-

by the punctures

closed surface.

\q1 U q2 U ... U q^i,

Then, if

n > 1, g > 1,

ker $

and b ^

in z^Sg.

A

2g+n—1 curves separated from on the

is generated freely by

the 2g+n—1 pairs

V a ra 1 ’ Tb rb 1 >ra 'a ’* ’ I i i,l i i,l a l , j - l a l , j Knowledge of the group

ker O

1 < i < g, 2 < j < n > . )

may be used to find a system of deb> n

fining relations for P

b>n

, and hence for M

b>n

, by an iterative procedure,

if one knows defining relations for P g Q = Mg Q. (The action of P g n ^ on ker O

b>“

may be determined by the method described in Section 2 of

this paper.) T h is procedure easily goes over to a procedure for finding defining relations for the groups

M

since the latter may be obtained & >n

64

JOAN S. BIRMAN

from P g n by extension by

£ n. For explicit results for

trary

n,

see [3]; for g = 0 and arbitrary

(using a different method) [21].

g = 1 and arbi­

n see [29], or [30], and also

These results are not available for

because the group P g 0 = Mg Q is not well-understood.

g > 2

U sin g recent re­

sults in [6] it should, however, be possible to determine presentations for M2

for arbitrary

n.

In addition to indicating a technique for finding defining relations for the groups

P

or M our theorem also yields insight into the struc&>n & * 11 ture of the group P a n, for it tells us that P may be built up from &>n &>n P 0 by repeatedly extending ^ ( S g - q j , q j { _ 1) by P g k _ i for each k = 1 ,2 ,..., n—1.

This yields, for example, a unique normal form in P „ &>n (modulo a normal form in P g 0). The n-string braid group of a surface is defined to be the fundamental group of the space of unordered n-tuples of distinct points of S.

(If S

is taken to be the Euclidean plane, this yields the c la s s ic a l braid group of Artin [24]; in the more general case, it yields a related group.) Let 'Pg n denote the natural homomorphism from Mg n to Mg Q defined by fillin g in the punctures.

Then our theorem may be restated “ the kernel of

'Pg n is isomorphic to the n-string braid group of Sg modulo its center.” The significance of this connection between braid groups of surfaces and mapping c la ss groups lies mainly in the fact that braid groups have been studied extensively, both from a topological and from an algebraic point of view, and the possibility exists for applying some of this knowledge to Riemann surface theory.

For a general introduction to the theory of braid

groups, see [5] and [32].

For an explicit development o f the connections

between

ker 'Pcy

and M

, see [2, 3, 29].

For a discussion of the re-

lationship between braid groups of surfaces and the n-fold product group ^ jS g X ... x

Sg see [13].

For generalizations to higher dimensions, see

[11].

§4. Lifting and projecting homeomorphisms In this section we discu ss a method for attacking the problem of find­ ing defining relations for the groups

Mg Q. The method w as used by the

65

MAPPING CLASS GROUPS OF SURFACES

author and Hilden in [ 6 ] to determine a presentation for the group but if

M2 0,

g > 3 the method yields, at best, a presentation for a “ la rg e ” sub­

group of Mg 0 (probably alw ays of infinite index).

The problem of find­

ing defining relations for the full group Mg Q is open if g > 3.

The

methods described in this section do, however, indicate relationships which exist between subgroups of Mg Q and subgroups of Mg/ suitable triplets L et £

(g, g', n')«

be a hyperelliptic involution on the surface

space of Sg under the action of £ 7T exhibits the image (S q — B )

Sg B =

for

Sg.

The orbit

is a sphere, and the collapsin g map

as a 2-fold ramified covering of SQ, with branching over 77

(B )

of the

2g+2

fixed points

B

of £ .

If h : ( S Q — B )

is a homeomorphism, then h alw ays lifts to Sg,

and its lift h

w ill be “ £-sym m etric,” that is: (5)

h £ h = £

.

If h represents the c la ss of the identity in MQ 2g+2 , then between

h and the identity map on SQ — B

is °t°P y

also lifts, and its lift

w ill have the property that h^ is also £-symmetric for each

h^

0 < t < 1.

Let M ( £ ) denote the group of £-symmetric mapping c la s s e s of S , that o o is adm issible maps and adm issible isotopies are required to commute with £.

One would not expect M g (£ )

to embed as a subgroup of Mg Q, because

the requirement that two homeomorphisms

h, k be £-sym m etrically iso ­

topic is a stronger requirement than isotopy.

Nevertheless, it w as d is ­

covered in [ 6 ] that: (i)

If h : Sg -> Sg is an £-symmetric homeomorphism which is isotopic to the identity map of Sg, between

(ii)

If

g = 2,

h and

then there is an £-symmetric isotopy

id.

every element of Mg Q has an £-symmetric representa­

tive. A s a consequence of (i) and (ii) one obtains almost immediately: (i ) *

Mg(£)

maps homomorphically onto MQ 2g+ 2 ,

subgroup of order (i i )* M 2( £ )

2 generated by the c la ss of £ .

coincides with

M2 Q.

kernel the

JOAN S. BIRMAN

66

This allow s us to lift the known presentation for MQ 6

to M2 Q and so

obtain defining relations in M2 Q. Even more, it shows the clo se rela­ tionship between the group M2 Q (and its appropriate generalizations to ^-symmetric subgroups of Mg) and the mapping class group MQ 2 g+ 2 the punctured sphere, which is in turn a close relative of Artin’s braid group (cf. Section 3 above).

Thus, for example, knowledge of the exist­

ence of elements of finite order in MQ 6

may be applied to describe the

elements of finite order in M2 Q [s e e 16]. These ideas were extended to the case where £

is replaced by any

element of finite order in Mg Q by the author and Hilden in [7, finite subgroups of and Maclachlan, in [16].

8

], and to

Q (with a somewhat different emphasis) by Harvey If G

is a finite group of homeomorphisms of Sg,

then one may define a symmetric mapping c la ss group M g(G) the requirement that adm issible maps

(6 )

h G h_ 1

of

by

h satisfy

= G .

Note that if h, k are two representatives of an element of Mg(G), the isotopy

hj. joining

h to k must satisfy ( 6 ) for each

0

< t
3,

then for every finite set of points

sphere, and for every element a there is a Riemann surface

B

in the automorphism group of

S which covers

He e s ­

on the 771

(Sq —B),

SQ and is ramified over B,

MAPPING CLASS GROUPS OF SURFACES

such that a does not lift to an automorphism of

^ S.

67

The connections

77

between these ideas and c la s s ic a l work of Hurwitz [17], and also recent work of V. I. Arnold [1] has also been explored by Magnus in [ 2 2 ], and by Magnus and P elu so in [23]. §5. D oes Since

Mg/[Mg, Mg]

have order

1 or 2,

if g < 3?

Mg is generated by tw ists, and since these twist generators

are all in a single conjugacy c la s s , it follow s immediately that Mg/[Mg,Mg] is a cyclic group. Mg/[Mg,Mg]

Known relations between these twist maps imply that

has order 2 or

1 if

g > 3 [se e Section 4], however it re­

mains an unsolved problem whether Mg/[Mg,Mg]

does, in fact, have order

To prove this it would be adequate to find any relator in Mg which is a product of an odd number of twists; alternatively, to show that Mg/[Mg,Mg] has order M

0

2

, it would be adequate to exhibit a single homomorphism from

onto a group of order 2.

The problem of finding such homomorphisms

w ill be discussed in the next section. §6 . Finite representations of Mg A group

G

is said to be residually finite if for every

a homomorphism

rj: G -> Q,

with

Q finite, such that

g f G

(g ) ^ 1.

77

there is R esidual

finiteness is a pleasant property for an infinite group to have, because it offers the possibility that many aspects of the group

G

may be investi­

gated in finite quotients of G. One collection of finite quotients of our group Mg which has been known for a long time is the c la ss of groups

\S (2g, Z

P

p

); p prime, n > 1|.

In 1965 it w as established by Mennicke [26] that this c la ss exhausted the finite quotients of Mg which factor through

Sp(2g, Z ),

however until very

recently it was not known whether Mg had any finite quotients which do not factor through

Sp(2g, Z ).

For this reason, it is particularly interesting

that E. Grossman has established that Mg is residually finite [14]. R. Gilman has suggested a concrete method for constructing finite representations of Mg,

and it is not difficult to arrange this construction

so that the representation does not factor through Sp(2g, Z ).

We w ill give

1

.

JOAN S. BIRMAN

68

a brief description of Gilm an’s construction, and of its geometric meaning. Let

Q be an arbitrary but henceforth fixed finite group, and let \r}^;\eA\

be the collection of a ll homomorphisms from

onto Q.

T hese homo-

morphisms may be listed explicitly by a straightforward procedure in which one tests every p o ssible mapping from the generators of ni^ g to see whether the single defining relation in homomorphism rj^

ni^ g

°nto

is satisfied.

Q

A

may be defined explicitly by an ordered array of

2

g

elements of which are the images of a standard set of generators of ^ S g under

^.

If

77

then 177^

: ni ^ g ^ Q

is one such representation, and if t e Aut Q,

w ill be another such representation, hence we may divide our

representations into equivalence c la s s e s

(mod Aut Q).

collection of all such equivalence c la s s e s .

L et

N

Each element of N

be the may be

described geometrically as the covering space of Sg which belongs to the subgroup

ker

^

77

of ^i^g*

Thus

N

may be regarded as the c o lle c ­

tion of all Riemann surfaces which are regular coverings of a fixed surface Sg,

with group of covering transformations isomorphic to Q. The mapping c la ss group Mg

may now be allowed to act on the set

N

as a group of permutations, by the rule that if a e Mg is represented by a * e Aut to

ttj

^ a^.

77

Sg,

then the permutation na

induced by a maps each

^

77

It is not difficult to check that na is independent of the choice

of representative a

that na preserves equivalence c la s s e s (mod Aut Q);

and finally that na^ = TTa nf i ’ a finite representation of Mg tions of the finite set

6

^g*

Therefore one obtains in this way

as a subgroup of the group of a ll permuta­

N.

While these representations were introduced because it was hoped that they would provide answers to certain explicit questions about Mg (such as the question which was posed in Section 5 above), they are also of con­ siderable interest from a purely group-theoretical point of view. group Mg is a perfect group (i.e .,

Mg/[Mg,Mg] =

If the

then all of its finite

quotients w ill also be perfect groups, hence Gilman’s construction would give an explicit method for finding infinite sequences of finite perfect groups.

Tw o such finite representations have been studied by the author

69

MAPPING CLASS GROUPS OF SURFACES

(unpublished).

The first uses the group

£ 3

of permutations on 3 letters

as the defining subgroup, to obtain a transitive representation of M3

as

a group of permutations of 2,530 letters. The second uses the the quaternion group of order

8

as a group

, to obtain a transitive representation of M3

of permutations of 5,040 letters.

REFERENCES

[1].

Arnold, V. I., “ Remarks on the branching of hyperelliptic in tegrals,” Functional A n alysis and its Applications (E n glish translation), V ol. 2 (1968).

[2 ].

Birman, Joan S., “ On Braid G roups,” Com. Pure and App. Math., 22 (1969), pp. 41-72.

[3].

Birman, Joan S., “ Mapping c la ss groups and their relationship to braid groups,” Com. Pure and Applied Math., 22 (1969), pp. 213-238.

[4].

Birman, Joan S., “ A belian quotients of the mapping c la ss group of a 2-m anifold,” Bull. AMS, 76 (1970), pp. 147-150; Errata: ibid., 77 (1971), pp. 479.

[5].

Birman, Joan S., “ Braids, links and mapping c la ss groups,” a re­ search monograph, to appear in Annals of Mathematics Studies Series, early 1974.

[ 6 ].

Birman, J., and Hilden, H., “ Mapping c la ss groups of closed sur­ faces as covering s p a c e s ,” Annals of Math. Studies # 6

6

, “ Advances

on the theory of Riemann S u rfaces,” ed. Ahlfors et a l., Princeton University P re ss, 1972. [7].

Birman, J., and Hilden, H., “ Isotopies of homeomorphisms of Riemann surfaces and a theorem about A rtin's braid group,” Bull. AMS, 78, No.

[ 8 ].

6

, Nov. 1972, pp. 1002-1004.

Birman, J., and Hilden, H., “ Isotopies of homeomorphisms of Riemann su rfa c e s,” Annals of Math., to appear.

[9].

Birman, Joan S., and Chillingworth, D. R. J., “ On the homeotopy group of a non-orientable su rfa c e ,” Proc. Camb. Phil. Soc., 71 (1972), pp. 437-448.

JOAN S. BIRMAN

70

[10]. [11].

Cohen, David, “ The Hurwitz Monodromy G roup/’ Ph.D. thesis, 1962. Dahm, David, “ A generalization of braid theory,” Princeton Univ. Ph.D. thesis, 1962.

[12].

Dehn, M., “ Die Gruppe der A bbild u n gsk lassen ,” Acta Math., 69 (1938), pp. 135-206.

[13].

Goldberg, C ., “ An exact sequence of braid groups,” to appear in Math. Scand.

[14].

Grossman, Edna K., “ On the residual finiteness of certain mapping cla ss groups,” to appear.

[15].

Harvey, W., “ C yclic groups of automorphisms of a compact Riemann su rface,” Quart. Jnl. of Math., 17, No.

[16].

6 6

(1969), pp. 86-97.

Harvey, W., and Maclachlan, C ., “ On mapping-class groups and Teichmiiller s p a c e s ,” to appear.

[17].

Hurwitz, A ., “ Uber Riemannsche Flachen mit gege benen V erzw eigungs punkten,” Math Annalen 39 (1891), pp. 1-61.

[18].

Lickorish, W. B. R ., “ A representation of orientable, combinatorial 3-m anifolds,” Annals of Math. 76 (1962), pp. 531-540.

[19].

Lickorish, W. B. R ., “ A finite set of generators for the homeotopy group of a 2-manifold,” Proc. Camb. Phil. Soc., 60 (1964), pp. 769778.

[20].

A lso , Corrigendum, 62 (1966), pp. 679-681.

Maclachlan, C ., “ Modulus space is sim ply-connected,” Proc. AMS, 29 (1971), pp. 85-86.

[21].

Magnus, W., “ Uber Automorphismen von Fundamental gruppen Berandeter F lach en ,” Math Annalen 109 (1934), pp. 617-646.

[22].

Magnus,

W., “ Braids and Riemann su rfa c e s,” Comm,

on Pure and

Applied Math., 25 (1972), pp. 151-161. [23].

Magnus,

W., and P ellu so, A ., “ On a theorem of V. I.

Arn old,” Comm.

Pure and Applied Math., 22 (1969), pp. 683-692. [24].

Magnus, K arass and Solitar, “ Combinatorial Group T heory,” Inter­ science, John Wiley and Sons, 1966.

[25].

Mangier,

“ Die klassen von topologischen Abbildungen einer

geschlossenen Flache auf s ic h ,” Math Zeitsch, 44 (1939), pp. 541554.

MAPPING CLASS GROUPS OF SURFACES

[26].

71

Mennicke, “ Zur Theorie der Siegelschen Modulgruppe,” Math Annalen 159 (1965), pp. 115-129.

[27].

N ielsen , J., “ Untersuchen zur Topologie der geschlossenen Z w eiseitigen Flachen I , ” Acta Math. 50 (1927), pp. 184-358, also III, Acta Math 58 (1932), pp. 87-167.

[28].

Patterson, D ., “ The fundamental group of the moduli s p a c e / ’ Mich. Jnl. of Math., to appear.

[29].

Scott, G. P ., “ Braid groups and the group of homeomorphisms of a su rface,” Proc. Camb. Phil. Soc.

[30].

6 8

(1970), pp. 605-617.

Sprows, David, “ Isotopy classification of homeomorphism of multiplypunctured compact 2-m anifolds,” thesis, Univ. of Penna., 1972.

[31].

Zieschang, H., “ On the homeotopy groups of su rfa c e s,” Math Annalen, to appear.

[32].

Magnus, W., “ Braid groups; a su rvey ,” to appear.

A N O T E ON

L 2 (r \ G )

Su-shing Chen

Let

G be a locally compact topological group and T

subgroup of G T

of G

such that T \ G

on L 2 ( r \ G )

is compact.

is unitary and

be a discrete

The regular representation

L 2(T \ G )

splits into the direct

sum of a countable number of invariant irreducible subspaces with finite multiplicities. Consider the collection of all subgroups r.

Then each T a \ G

If T o C

is compact.

embedding L 2 (Tp \ G ) C L 2 (F a \ G ) . L 2 (T a \ G ) ( a * A ) lows.

We set r

is denoted by q

= r,

r ; = r

L 2( r / a _ 1 \ G ) C L 2( r /a \ G ) . of the subspace

F 2 ( r /a _

1

sum of the Hilbert spaces representation of

G

H. 1

, then we have a natural

This space can be described as fol­

n . „ n r a . Then Ha

and

c r

/a _

1

and

the orthogonal complement

in L 2 (F 'a \ G ) .

L2 (r \ G )

on H.

of finite index in

The direct limit of the spaces

Denote by

\G)

F a (a e A )

Then

Ha (a e A ).

H

is the direct

There is a natural

Furthermore, the representation on H splits

into a countable direct sum of irreducible unitary representations. The above account is given on pp. 19-20 of [1 ], where the follow ing question for G = SL(2, R )

is raised:

Is the multiplicity with which an

irreducible unitary representation occurs in this decomposition finite? In this note, we sh all explain why the answer is negative. It is shown on p. 78 of [1] that when T ments, the multiplicity

N “ (F )

discrete series occurs in T

does not contain elliptic e le ­

with which the representation

is given by the formula:

77 where pi(r \ G )

denotes the volume of T \ G .

73

T“

of the

SU-SHING CHEN

74

Consider a sequence r i+ 1 ^ r .

and

1

{ F - J of subgroups of T

< [r * : r . +1] = Aj < oo, for

such that Tq = T ,

i = 0 ,1 ,2 ,...

. The existence

of such a sequence follow s from the fact that the fundamental group of a surface of genus of any order. covering,

g >

1

can be mapped homomorphically onto a cylic group

Since the projection

/u(Fi+ 1 \ G ) = X j / r C T G )

N ^ (T j+1) = A qA j^

THEOREM.

Let

...A . n J cT ),

T j+1\ G -> T j \ G and

N n ( r i+1) =

Therefore

and consequently we have:

n be a positive odd integer.

with which the representations

is a A -sh e e te d

T^

Then the multiplicities

of the discrete series occur in the

decomposition of H are infinite.

UNIVERSITY OF FLORIDA GAINESVILLE, FLORIDA 32601

REFERENCES

[1].

G e l’fand, I. M., Graev, M. E ., and Pyatetskii-Shapiro, I. I., Repre­ sentation theory and automorphic functions, Saunders, Philadelphia, London, and Toronto, 1969.

ON T H E O U T R A D IU S OF F IN IT E -D IM E N S IO N A L T E IC H M U L L E R S P A C E S T. Chu

The universal Teichmiiller space, functions

T (l),

may be defined as the set of

l i

which are Schwarzian

derivatives of mapping functions admitting quasi-conformal extension to C. T (l)

has a natural embedding in the linear space

II0 1 1 =

for (f> € B 2 ( l ) .

s u j ^ (|z |2 — l

)2

B2 (l)

with

|0 (z)|

It follow s from well-known results of Nehari, H ille and

Ahlfors and W eill that

0

(1) =

id )

o (l)

=

Sup

0fT(l)

l|0 1| =

inf

0 f B2(D—T(l)

6

||0 || =

2

.

is the radius of the sm allest b a ll about O containing T ( l )

called the outradius of T ( l ) . O contained in T ( l ) .

i(l)

and is

is the radius of the largest ball about

It is also known that the function

k (z) = — - —-

-6

(Z_1)

i_ = [k] = ------------ with norm 6 . k (z) is the (z 2- l f usual Koebe function composed on the right with the Mobius transformation has a Schwarzian derivative

z -» z - 1 .

e d T (l)

since there are holomorphic

defined on A * , [k n(z )] e T ( l )

and

( l — pf)

||[kn] — [k] || -» 0.

75

’ **1

roots

kn(z )

T. CHU

76

If G

is a Fuchsian group leaving the unit circle invariant, we define B 2 (G ) = \ e B 2 (l)| (< £ ° y )(z )y '(z ) 2 = 0 ( Z) , for z £ A *

T (G ),

the Teichmiiller space of G,

and y t Gi . is the component containing 0

T ( l ) fl B 2 (G ).

T (G )

and only if

is finitely generated and of the first kind.

G

of

is a topological c e ll which is finite dimensional if

The major result presented here is a construction lemma leading to an asymptotic estimate for o(G ),

where

o (G ) =

sup \\\\ . 0 £ T (G )

The construction theorem is a generalization, suggested by Akaza, of a technique of Klein.

The application to estimates of o (G ) w as proposed

by Abikoff.

The precise statement is

Theorem .

For each 8 > 0,

Fuchsian group G

there is a finitely generated hyperbolic

of the first kind, for which o (G ) >

—e.

6

Before proceeding, we need the following definition. curves

and J 2

are said to have Frechet distance

8

Two Jordan (J 1 ,J 2) < £ ,

there are parametrizations a^(t) of J-, i = 1, 2, 0 < t < 1 so that a 2 (t)| < e

for each

t e [0,1].

if

la ^ t ) —

We w ill state the two basic lemmas, show

that the theorem follow s easily from them, and then give their proofs.

LEMMA 1. For each positive is a 1 +
0 so that, if Jj

is a Jordan curve with 5 (J 1 ,J2)

onto Ext J 2

||[f]|| >

satisfies

Given a rectifiable Jordan curve J and e 3 > 0,

—8 .

there exists

a finitely generated purely loxodromic quasi-Fuchsian group G first kind so that S(J, A (G )) < e 3>

6

of the

77

OUTRADIUS OF TEICHMULLER SPACES

Proof of Theorem. Each

^1 + —^ -le v e l curve

Jn of k is a rectifiable

Jordan curve hence we may find finitely generated purely loxodromic quasiFuchsian groups

G n k so that

conformal map of A * f'n k(°°) >

Lim ^ (J n> A (G n k)) -> 0.

onto Ext A (G n k)

normalized by

L et

fn k be the

fn j^C00) =

a ^d

00

It follow s from the Caratheodory convergence theorem [1]

that f n k converges normally to f R, the conformal map of A * Ext Jn with

fn( oo) = oo and

yields that [ f n

f'n(oo) > o.

onto

The Cauchy Integral theorem then

converges normally to [ f n] Lim ||[ffl k]|| < ||[fn]|| > k ->oo ’

hence 6

- £ .

It is w ell known that f ^ k ^ n k^n k is a Lnitely generated purely hyper­ bolic Fuchsian group

H n k of the first kind and

[f fl k] e T (H n k)

which

proves the theorem. Proof of Lemma

1

. We may assume

conditions are satisfied by rj of, [f].

f(°o) = oo and f '(° ° ) > 0 since these

r/ a Mobius transformation and

We now assume that the Lemma is false.

for each

£ and £^ and a ll s 2 > 0,

lar, if e n

[77

of] =

It would then follow that

the conclusion is false.

In particu­

, J i n denotes the ( 1 + s n)-le v e l curve of k (z) and J 2 n ’ 1 * f is a Jordan curve with