Advances in the Theory of Riemann Surfaces. (AM-66), Volume 66 9781400822492

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Annals of Mathematics Studies Number 66

ADVANCES IN THE THEORY OF RIEMANN SURF ACES Proceedings of the 1969 Stony Brook Conference

EDITED BY

LARS V. AHLFORS LIPMAN BERS HERSHEL M. FARKAS

ROBERT C. GUNNING IRWIN KRA HARRY E. RAUCH

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY

1971

Copyright © 1971, by Princeton University Press ALL RIGHTS RESERVED

LC Card: 72-121729 ISBN: 0-691-08081-x AMS 1968: 3001

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

Printed in the United States of America

PREFACE The Stony Brook Conference (June 1969) was a gathering of mathematicians working in certain inter-related fields: quasiconformal mappings, moduli, theta-functions, Kleinian groups. All of these fields belong directly or indirectly to the theory of Riemann surfaces. The first such gathering was the conference at Tulane University in May, 1965, sponsored by the Air Force Office of Scientific Research. The Tulane proceedings were published informally and are not generally available. This may have been a mistake, and it was decided not to repeat the mistake after Stony Brook. The present volume contains all but two of the papers read at the conference, as well as a few papers and short notes submitted afterwards. We hope that it reflects faithfully the present state of research in the fields covered, and that it may provide an access to these fields for future investigations. In the name of all participants, we thank the State University of New York at Stony Brook for its hospitality, the Office of Naval Research for the financial support which made the conference possible, and Princeton University Press for assuring rapid publication of the results.

The Editors

CONTENTS Some Remarks On Kleinian Groups by William Abikoff ......................................................................... . Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces by Robert D. M. Accola ..................................................................

7

Remarks on the Limit Point Set of a Finitely Generated Kleinian Group by Lars V. Ahlfors ..... .. ....... .... .............. . .... ..... ... .. ........ .... .. .... . ........ 19 Extremal Quasiconformal Mappings by Lipman Bers ........... .... .............. ..... .............. ............. ..... ............. 27 Isomorphisms Between Teichmiiller Spaces by Lipman Bers and Leon Greenberg ..... .... ..... ................... ... ...... ... 53 On the Mapping Class Group of Closed Surfaces as Covering Spaces by Joan S. Birman and Hugh M. Hilden ......................................... 81 Schwarzian Derivatives and Mappings onto Jordan Domains by Peter L. Duren ........................................................................... 117 On the Moduli of Closed Riemann Surfaces with Symmetries by Clifford J. Earle ........................................................................ .119 An Eigenvalue Problem for Riemann Surfaces by Leon Ehrenpreis ......................................................................... 131

Relations Between Quadratic Differentials by Hershel M. Farkas ...................................................................... 141 Deformations of Embeddings of Riemann Surfaces in Projective Space by Frederick Gardiner ..................................................................... 157

vii

viii

CONTENTS

Lipschitz Mappings and the p-capacity of Rings in n-space by F. W. Gehring ............................................................................. 175 Spaces of Fuchsian Groups and Teichmilller Theory by William J. Harvey ....................................................................... 195 On Fricke Moduli by Linda Keen ................................................................................. 205 Eichler Cohomology and the Structure of Finitely Generated Kleinian Groups by Irwin Kra .................................................................................... 225 On the Degeneration of Riemann Surfaces by Aaron Lebowitz ...................................................................... 265 Singular Riemann Matrices by Joseph Lewittes .......................................................................... 287 An Inequality for Kleinian Groups by Albert Marden ............................................................................ 295

On Klein's combination Theorem III by Bernard Maskit ........................................................................... 297 On Finsler Geometry and Applications to Teichmilller Spaces by Brian O'Byrne ........................................................................... .317 Reproducing Formulas for Poincare Series of Dimension -2 and Applications by K. V. Rajeswara Rao ................................................................. .329 Period Relations on Riemann Surfaces by Harry E. Rauch ..........................................................................341 Schottky Implies Poincare by Harry E. Rauch ......................................................................... .355 Teichmilller Mappings which Keep the Boundary Pointwise Fixed by Edgar Reich and Kurt Strebel ................................................... .365 Automorphisms and Isometries of Teichmilller Space by H. L. Royden ............................................................................ .369 Deformations of Embedded Riemann Surfaces by Reto A. Rtiedy .......................................................................... .385 Fock Representations and Theta-functions by Ichiro Satake ............................................................................. .393 Uniforrnizations of Infinitely Connected Domains by R. J. Sibner ................................................................................407

ADVANCES IN THE THEORY OF RIEMANN SURF ACES

SOME REMARKS ON KLEINIAN GROUPS Willaim Abikoff In this note, we will construct four Kleinian groups. The first is finitely generated and possesses limit points which are not in the boundary of any component of the ordinary set of the group. The construction yields a counterexample to the following assertion of Lehner [ l]: If G is a discontinuous group with limit set "L and ordinary set Q, then "L

=

u Bd (Qi)

where the

Qi are the components of Q. The remaining examples are of infinitely generated groups whose limit set has positive area. Each example will show that a statement, either proved or believed to be true for finitely generated groups, is not true for infinitely generated groups. The First Group

We construct a group G 1 having a limit point which does not lie in the boundary of any component of the ordinary set of the group. Let G 1 ' be any finitely generated Fuchsian group of the first kind whose elements all possess isometric circles. The Ford fundamental region then has two components Rb and R00 C and the second a neighborhood of

-

the first inside the principle circle

oo.

Choose a Mobius transformation T

with the following properties: a) the fixed points of T are x and x ', x < Int Rb and x' < Int R00 , b) the isometric circle of T, I(T) C Int Rb , c) the isometric circle of T- 1 , I(T- 1 ) C Int R00 and

oo

is in the

exterior of I (T- 1 ). The free product G 1 of G 1 ' and the group generated by T is discontinuous, Furthermore, x is in the limit set of G 1 but is not in the boundary 1

2

WILLIAM ABIKOFF

of any component of the ordinary set of G 1 . To see this, suppose x is in the boundary of 0 1 , a component of O(G 1 ). Then there is a point z

Iz- xi

=

0 1, a > 0. But for sufficiently large n, Tn(C), which is a subset of f

l, will separate z from x. Hence the component containing z cannot have x as a boundary point.

The Second Group It is a conjecture of several years standing that the limit set of a

finitely generated Kleinian group has zero area. We first recall that an Osgood curve is a Jordan curve of positive area; a construction is described in Osgood [2]. We will construct an infinitely generated Kleinian group whose limit set and Ford fundamental region both contain two Osgood curves. The following statement of Ahlfors [3] is thus shown not be extended to infinitely generated groups: if G is a finitely generated Kleinian group, then the portion of the limit set of G lying in the exterior of all isometric circles has area zero. Let C and C' be disjoint Osgood curves with disjoint interiors, and such that one may be obtained from the other by translation. By Schoenflies' version of the Jordan curve theorem, there exist homeomorphisms of the plane f and f' which map U, the open disc, onto the interiors of C and C' respectively and map the unit circle onto C and C' respectively. Let !Uil be a filling up of the unit disc by disjoint discs such that for every point on the unit circle, there exists a subsequence (U/) of (U) which converges to that point. Let xi and x; be the images of the center of Ui under f and f' respectively. There exist discs Ci and about xi and

C[

contained in f(Ui) and f'(Ui)

x{ Further, every point on the curves C and C ' is a limit

of some subsequence of the Ci or the

c;.

Pick a Mobius transformation Ti

whose isometric circle is Ci and whose inverse has isometric circle Ci'· Let G 2 be the group generated by the Ti. G2 is discontinuous in the common exterior of C and C ', since that set is contained in the Ford fundamental region of G 2 . Furthermore, the isometric circles of the elements of G2 accumulate at every point of C and C '; hence C U C 'C l, where

SOME REMARKS ON KLEINIAN GROUPS

3

I is the limit set of G2 • We have shown that there exists a Kleinian group whose limit set has positive area. The Third Group

Our first construction showed that N(G)

=

I (G) - U Bd (0) need not

be void even for finitely generated groups. We now show that for infinitely generated groups N(G) may have positive area. Let C and C' be as in the previous example. We consider four sequences (J i), (J i'), (Ki) and (K{) of Jordan curves having the following properties: a) all curves are disjoint from each other and from C and C ', b) J i C lnt J i + 1 C Int C C lnt Ki + 1 C lnt Ki , J { C Int J i ~ 1 C Int C 'C Int K[ + 1 C Int K[ c) the sequences (J i) and (Ki) converge uniformly to C and the sequences ( J i') and (Ki') converge uniformly to C' (i.e., (J i) converges uniformly to C if for each positive c, there is an i sufficiently large so that max Clx-Jil' lz-CI) < c for each x on C and z on Ji). Using the techniques of the previous construction, we get a group G3 having each curve in the four sequences in the limit set of the group, and by the same argument as in the first construction, C U C 'C N(G 3). Since both C and C' have positive area, we have constructed a group for which N(G) has positive area. The Fourth Group

Maskit [4] has shown that every finitely generated Kleinian group G leaving a Jordan curve invariant, that is every quasi-Fuchsian group, is the quasiconformal deformation of a Fuchsian group. We now exhibit an infinitely generated quasi- Fuchsian group which is not the quasiconformal deformation of a Fuchsian group. We construct a group G4 whose limit set I is an Osgood curve C. Thus G4 leaves C invariant. If G4 were the quasiconformal deformation of a Fuchsian group, C would have zero area

WILLIAM ABIKOFF

4

(Ahlfors [ 5), p. 33) which it does not. In the previous two constructions the existence of Osgood curves was sufficient for our purposes, but we now need some specific properties of Osgood's example. Osgood constructs not a Jordan curve but a Jordan arc of positive area. With the addition of two Euclidean line segments, the arc becomes a Jordan curve J. J is then the closure of a countable number of Euclidean line segments, Ik, which are pairwise disjoint. The measure of J

is supported on

J- U Ik, which is a measurable set. The idea of the example is to construct a group whose limit set is a curve passing through each point of J- U Ik . The curve J cannot be the limit set of a Kleinian group since as a function of a parameter t, a subarc of J is differentiable. In this case J would have to be a circle (Appell-Goursat [6], p. 82). We first construct a reflection group G having the following properties: a) the centers of the generating circles lie in U Ik , b) the generating circles have disjoint interiors, c) a generating circle intersects exactly two other generating circles, each in a single point. d) except for its endpoints, each Ik is covered by the closed discs corresponding to the generating circles whose centers lie on Ik, e) as we approach an endpoint of Ik, the radii of the generating circles tend to zero, f) no circle whose center lies on Ik

intersects a circle whose center

lies on Ik, for k /-. k '. We let G4 be the subgroup of G consisting of the orientation preserving motions. It is of index two in G and therefore has the same limit set as G. The interior of each generating circle contains one (in fact many) limit points of G4 hence ~ (G 4 ) '") J - U Ik and ~ (G 4 ) has positive area. G4 is discontinuous, so we must only show that ~(G 4 ) is a Jordan curve. Let K be an arbitrary generating circle whose center lies on Ik. Then by considering reflections in K and the generating circles tangent to K, we note that K nik C ~(G 4 ). Further, no other points of ~(G 4 ) lie on K. K n Ik

SOME REMARKS ON KLEINIAN GROUPS

5

consists of exactly two points a and b. It suffices to show that the part of I (G 4 ) contained in the closed disc bounded by K is a simple arc. The argument is tedious but straightforward. One looks at the a's and b's corresponding to all other generating circles, reflects them into the inside of K and uses the images to define densely the arc between a and b. The arc carries an induced parametrization. Simplicity of the arc is guaranteed by an argument similar to the nexted interval theorem. REFERENCES

[l] J. Lehner, Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., 1964 p. 105. [2] W. F. Osgood, A Jordan curve of positive area, Trans. A mer. Soc., 4 (1903), p. 107-112. [3] L. V. Ahlfors, Some Remarks on Kleinian Groups, Proceedings of the

Conference on Quasiconformal Mappings, Moduli and Discontinuous Groups, Tulane University, 1965. [4] B. Maskit, On Boundaries of Teichmuller Spaces and on Kleinian Groups, II, Ann. of Math., (to appear). [5] L. V. Ahlfors, Lectures on Quasiconformal Mappings, Von Nostrand, 1966. [6] P. Appell and

E.

Goursat, Theorie des fonctions algebriques et de

leurs integrales, Vol. 2, Gauthiers- Villars, 1930.

Bell Telephone Laboratories and Polytechnic Institute of Brooklyn

VANISHING PROPERTIES OF THETA FUNCTIONS FOR ABELIAN COVERS OF RIEMANN SURF ACES (unramified case) Robert D. M. Accola 1 1. INTRODUCTION. The vanishing properties of hyperelliptic theta func-

tions have been known since the last century [3]. Recently, Farkas [l] discovered special vanishing properties for theta functions associated with surfaces which admit fixed- point free automorphisms of period two. The author has discovered other vanishing properties for special surfaces admitting abelian automorphism groups of low order. The purpose of this report is to give a partial exposition of a theory that will subsume most of the above cases in a general theory. Due to limitations of time and space, a full exposition must be postponed. Let W1 be a closed Riemann surface of genus p 1 , p 1 ~ 2, admitting a finite abelian group of automorphisms, G. The space of orbits of G, W/G (= W0 ), is naturally a Riemann surface so that the quotient map, Q, is analytic. In this report we shall develop the theory in the case where no element of G other than the identity has a fixed point; that is, the map

Q : W1

-+

W0 is without ramification. We shall, however, state some theorems

in a more general context especially when proofs are omitted.

1 The research for this report has been carried on during the last several years during which the author received support from several sources. 1) Research partially sponsored by the Air Force Office of Scientific Research, Office of Aero· space Research, United States Air Force, unqer AFOSR Grant No. AF-AFOSR· 1199-67. 2) National Science Foundation Grant GP-7651. 3) Institute for Advanced Study Grant-In-Aid.

7

ROBERT D. M. ACCOLA

8

II. REMARKS ON GENERAL COVERINGS. Let !!_ : W1

->

WO be an arbi-

trary n- sheeted ramified covering of closed Riemann surfaces of genera p 1 and p 0 respectively. Let M1 be the field of meromorphic functions on W1 and let M0 be the lifts, via

£.

of the field of merom orphic func-

tions on W0 • Then M0 is a sub field of M1 of index n. We now define an important abelian group, A, as follows: Definition: A , {f

f

M; I fn

f

M~ 1/M~

.

2

Now let MA be the maximal abelian extension of M0 in M1 . LEMMA 1: A is isomorphic to the '(dual of the) Galois group of MA over Mo. PROOF: (omitted). A proof in the case where M1 "" MA will follow in Section V. Now, fix a point in W1, z 1 , and let z 0 "" £(z 1 ). Fix canonical homology bases in W1 and W0 and choose bases for the analytic differentials dual to these homology bases. Thus maps u 1 and u 0 from W1 and W0 into their Jacobians, J(W 1 ) and J(W 0 ), are defined:

b

3

The maps u 1 and u0 are extended to divisors in the usual way. A map a is now defined from divisors on W0 to those of W1 as follows: for x 0 f W0 ,

is the inverse image of x 0 under £ with branch points counted according to multiplicity. Thus !!_Xo always has degree n. !!_ is extended by

!!_Xo

2

If K is a field, K* will stand for the multiplicative groups of non-zero elements

in K. 3 The symbol .2. will be used consistently to denote homomorphisms from

w0 into the corresponding abelian groups associated with W1 • The particular group will be clear from the context.

abelian groups associated with

VANISHING PROPERTIES OF THETA FUNCTIONS

9

linearity to arbitrary divisors on W0 . Now we define a map from J(W 0)-+ J(W 1), again denoted by !!. as follows: if D 0 is a divisor on W0 of degree zero, then !!_u 0(D 0) = u 1 (!!_D 0). !!. is easily seen to be a homomorphism. Let Mu A be a maximal unramified abelian extension of M0 in M1 ; thus M0 C MUA C MAC M1 . LEMMA 2: The kernel of a: J(W 0)-+ J(W 1) is isomorphic to the Galois group of MuA over M0 . PROOF: (omitted).

A proof in the case when MUA

= M1

will follow in

Section VI. With the homology bases and the dual bases of analytic differentials chosen, let (rr i E, B 0)p0x 2 Po and (rr i E, B 1)p1x 2 P1 be the corresponding period matrices where E is the appropriate identity matrix. Finally let IHxoHu; Bo) and

e[x 1](u; B1)

be the corresponding first order theta

functions with arbitrary characteristic. LEMMA 3: For any characteristic X 1 there is an exponential function E(u) so that E( u) e[x 1H!!_ u; B1 ), as a multiplicative function on J(W 0), is an nth order theta function. PROOF: (omitted). The proof is an immediate adaptation of the simplest parts of transformation theory.

III. RESUME OF THE RIEMANN VANISHING THEOREM. The proofs of the results in this report depend on Riemann's solution to the Jacobi inversion problem. We summarize here those portions of the theory that will be needed later. 4 Let W be a closed Riemann surface of genus p, p;:: l, let a canonical homology basis be chosen, let a dual basis of analytic differentials be chosen, let a base point be chosen, and let u be the map of W into J(W). 4 The material in this section is covered in Krazer [3]. For a complete and more modern treatment see Lewittes [ 4].

ROBERT D. M. ACCOLA

10

Riemann's theorem asserts the existence of a point K in J(W) so that if we choose any e

f

J(W), then there is an integral divisor D on W of

degree p so that u(D)+K If O(e)

1:

= e(modJ(W)).

0, then D is unique. If O(e) = 0, then the above equation can

be solved with an integral divisor of degree p- 1. Moreover, in this latter case, the order of vanishing of O(u) at e equals i(D), the index of speciality of D. (By the Riemann- Roch theorem, i (D) equals the number of linearly independent meromorphic functions which are multiples of -D since the degree of D is p-1.) Moreover, O(u(D) + K)

= 0 whenever D

is an integral divisor of degree at most p- 1. Finally, if D is a canonical divisor, then u (D) = -2K (mod J(W)) .

IV. ARIBITRARY UNRAMIFIED COVERINGS. For this section assume

!:!. :

W1

-+

W0 is unramified but otherwise arbitrary. Continue the previous

notation and let Oj be the theta-divisor in J(Wj) for j

= 0, l.

LEMMA 4: There exists a half-period, e 1 , in J(W 1 ) so that (1)

(e 1 depends only on the canonical homology bases chosen.) PROOF: Let D 0 be an integral canonical divisor on W0 of degree 2p 0 - 2. Then D 0 - (2p 0 - 2) z 0 is a divisor of degree zero whose image under u 0 is u 0 (D 0 ) since u 0 (z 0 ) == 0. ~(-2K 0 )

~.Do

is canonical in W1 so we have

= !!_u 0 (D 0 ) = !!.u 0 (D 0 - (2p 0 - 2) z 0 ) = u 1 (!!.D 0 -(2p 0 -2)!!_z 0 )

= -2K 1 -(2p 0 -2)u 1 (!!_z 0 )

11

VANISHING PROPERTIES OF THETA FUNCTIONS

Dividing by two gives (2)

where 2e 1 ""' 0. If g 0 f 0 0 , then there is a divisor D 0 of degree p 0 -1 on W0 so that

Thus

or

By formula (2) it follows that

Since the degree of

~.Do

is n(p 0 -1)

= p 1 -1

we have

Thus

q.e.d. V. ABELIAN COVERS. 5 In this section assume ramified abelian cover; i.e., W0

= W1 I G

£:

W1 -. W0 is a possibly

where G is an abelian group of

automorphisms of W1 whose elements may have fixed points. Let R be the set of characters of G; i.e., the set of homomorphisms of G into the multiplicative group of complex numbers of modulus one. Since G is a finite group, R is isomorphic to G although there is no canonical isomor5 The author wishes to express his thanks to Professor M. S. Narasimhan for many valuable discussions concerning the material of this report, especially this section.

ROBERT D.

12

M.

ACCOLA

phism. The group A is, however, canonically isomorphic to R. For suppose f is meromorphic on W1 and fn is a function lifted from W0 • Then the divisor of f is invariant under G. Thus if T ( G, then f o T = xr(T) f where X£ is easily seen to be a character. If f and g yield the same character, then £/g gives the identity character and so lies in M 0 • Thus the map f --+ Xf is an isomorphism of A into R. That this map is onto is seen by examining, for each

x f R, the cyclic extension of

M0 given by the

fixed field for kernel X. This completes a proof of Lemma 1 when M 1 = MA. In the above situation where we have a function f whose divisor is invariant under G, we shall say that f is associated with the character Xf. Thus the field extension M1 over M0 can have as a vector space basis functions corresponding to each character of R. If f is an arbitrary function, then let fx= n- 1 ITfGx(T- 1 )foT. Then f= IXfRfX andthe fx's which are not zero are linearly independent since they are associated with different characters. VI. UNRAMIFIED ABELIAN COVERS. In this section assume 2_: W1 --+ W0 is an unramified abelian cover. In this context we give a proof of Lemma 2; that is, we now show that the kernel of .!!. : J(W 0 )--+ J(W 1 ) is isomorphic to

A. For the proof define a map from A into ker.!!. as follows. If f f

M;

and fn f M~, then the divisor of f, (f), is invariant and so (f) = .!!.Do where D0 is a divisor on W0 of degree zero. Let Ulf = u 0 (D 0 ). Then .!!_UJf

= .!!.uoCD 0 ) = u 1 (,!!.D 0 ) = 0, and so cuf f ker .!!.· The map f--+ cuf is

easily seen to be well-defined on A and one-to-one, To show the map is onto suppose .!!_cu = 0. Then w = u 0 (D 0 ) where D 0 is a divisor of degree zero on

w0 •

Since !!Do = 0, suppose (f)= ,!!_D 0 • Since ,!!_D 0 is invariant

under G, f o T = X{(T) f for some character X£. Consequently, fn f M0

•

This completes the proof of Lemma 2 in the unramified abelian case. Let us index ker .!!. by R rather than A; that is, ker .!!_ = lcuX I X f Rl where u>x is defined as follows. If u 0 (D 0 ) = cuX and (f)

= !!Do,

then f

13

VANISHING PROPERTIES OF THETA FUNCTIONS

is associated with X • The method of indexing, X ... wX is an isomorphism of groups. The reader is reminded that throughout this discussion canonical homology bases on W0 and W1 are fixed and all results are related to these bases. Thus the bases of analytic differentials are determined as are the theta functions and the half-period e 1 of Lemma 4. Base points are also fixed and so maps u 0 and u 1 are defined, but these are in a sense auxiliary devices and the base points do not enter into the statements of the final results. At this point let us introduce a convenient abuse of notation. If

e(u; B)

is a theta function and

T

=

7T

i h + Bg, where g and h are real

p- vectors, then there is an exponential function E( u) so that

e(u + B) = E( u) e[ ~ ]( u ; B) . In this context write e [r ](u) for the usual e[~ ]( u; B). T ;

This notation will

be extremely convenient and will lead to no confusion provided the canonical homology bases remain fixed. A main result of this report is the following. By Lemma 3, there is an exponential, E(u), so that E(u)e[e 1 ](~u; B 1 ) is an nth order theta function with respect to (rr i E, B 0)pOx 2Po. Then there is a constant C ,j 0 so that (3)

That this is a plausible result can be seen from Lemma 4. If wX

I'

ker !!._,

then ~ 0 at

~g 0 •

Then there is an integral divisor D 0 of degree p 0 on W0

and an integral divisor D 1 of degree p 1 - 1 on W1 so that (5) (6) ~

Applying

to formula (5) gives

.!!.go = !!uo 0.

Thus any

=1, 2, ... , s. Thus

N

:S

x

IJ=

for which N~ ,J 0 belongs to the

1 Nj .

q.e.d. Formula (3) now follows since we can find values of g 0 for which

¢ 0 for all X· Thus

()[wX](g 0 ; B 0 )

zero and so

e ,;,

(J[e 1 ](~u,

B 1 ) cannot be identically

0.

If G is a cyclic group, then e 1 of Lemma 4 can be determined. For n = 2 this has been done by Farkas, Rauch and Fay.

VII. APPLICATIONS.

Formula (3) for n

=2

seems to have been known to Riemann. 6 For n

=2

the formula is equivalent to the relations between () and Tf constants derived by Farkas [2], and these in turn, as shown by Farkas, yield the Schottky- Jung relations. Also, the vanishing properties discovered by Farkas [l] follow immediately from the formula and theorem. Thus the 6

See Riemann

[6], Nachtriige, p. 108.

17

VANISHING PROPERTIES OF THETA FUNCTIONS

theorem can be viewed as a generalization of the work of Farkas and Rauch. One can hope that applying the methods of Farkas to formula (3) for other cyclic covers (n

> 2) will yield further relations between more general

types of theta constants. Another type of application of the theorem is as follows. It is known that on the surface of genus p, p odd, the vanishing of the theta function to order (p + 1)/2 at some half- period is equivalent to the surface being hyperelliptic. 7 This is an immediate consequence of Riemann's solution of the Jacobi inversion problem and Clifford's theorem. We now use this face to prove the following. Suppose W0 is a Riemann surface of genus p0 , p 0 ;::: 4 and p 0 even. Suppose that the theta function vanishes at two distinct half-period to order p 0 :/2. Then W0 is hyperelliptic. 8 An outline of the proof is as follows. 9 Suppose that for a given canonical holology bases on W0 , 0 [a](u; B 0 ) and 0 [r](u; B 0 ) vanish to order pd/2 at u = 0. The half-period (ar) defines a smooth two sheeted cover, W1 , of W0 . By suitable choice of canonical homology bases on W0 and 000

0

000

0

W1 we can arrange that (ar) = ( 1 0 0 · · · 0 ) and ( e 1) = ( 1 0 0 · • • 0 ) where e 1 is the half-period of Lemma 4. By suitable relabeling, if necessary, it follows that [a]

= [ ~ ~ ,]

and [r]

= [ ~ ~ ,]

where [

~,] is a

p0 -1 theta character whose parity is that of p 0 /2. Then formula (3) applied to this situation yields (8)

0

0 ... 0

E(u)0[ 1 0 ... 0 H!!.u;B 1 )

7 For p = 3, the result is due to Riemann [6], Nachtri:ige, p. 54. 8 For p:::: 8, p even, Martens has proved that the vanishing of the theta function at one half-period to order p/2 suffices to insure hyperellipticity [5]. He also covers the p

=6

case. For p

=4

the result seems to be due to Weber [7].

9 For the remainder of this report all theta characteristics will be half-integer characteristics.

ROBERT D. M. ACCOLA

18

°

Setting u equal to the half-period whose character is ( 0 0 E

8[l

€

E' E,]

€ ,) €

shows that

(u 1 ; B1) vanishes to order Po at u 1 = 0. Since the genus of

W1 is p 1 = 2p 0 -1 we have Po = (p 1 + l)/2 and so W1 is hyperelliptic. But then W0 must also be hyperelliptic.

REFERENCES [l] Farkas, H. M., Automorphisms of compact Riemann surfaces and the vanishing of theta constants. Bulletin of the American Mathematical Society, Vol. 73 (1967), pp. 231-232. [2]

, On the Schottky relation and its generalization to arbitrary genus.

[3] Krazer, A., Lehrbuch der Thetafuncktionen. B. G. Teubner, Leipzig,

1903. [4] Lewittes, J., Riemann surfaces and the theta function. Acta Mathematica. Vol. 111 (1964), pp. 37-61.

[5] Martens, H. H., Varieties of special divisors on a curve. II. journal filr die reine und angewandt Mathematik. Vol. 233 (1968), pp. 89-100. [7] Riemann, B., Gesammelte mathematische Werke. Dover, 1953. [7] Weber, H., Ueber Gewisse in der Theorie der Abel'sehen Funktionen auftretende Ausnahmfalle. Mathematisd!e Annalen. Vol. 13 (1878), pp. 35-48.

REMARKS ON THE LIMIT POINT SET OF A FINITELY GENERATED KLEINIAN GROUP Lars V. Ahlfors

1. It is a conjecture of several years' standing that the limit point set of a finitely generated Kleinian group has areal measure zero at least under some restrictive assumptions. This question has proved to be very elusive and all attempts to find the answer have been abortive. The method I am going to describe is no exception. It leads deceptively close to a solution, but my efforts to push it through have again been futile. I am nevertheless using this opportunity to publish some of my ideas on the subject that I believe to be independently interesting. 2. We use conventional notations: [' is a Kleinian group, A the limit point set'

n

the set of discontinuity. dm denotes two- dimensional euclid-

ean measure, and dw is spherical area. The spherical (chordal) distance is denoted by [z 1 , z 2 ]. Let

s

be the Riemann sphere and let A

t ['

act on

s3 = s X s X s

by

A(z 1 , z 2 , z 3 )= (Az 1 , Az 2 , Az 3 ). We denote by S~ the subset of S3 on which all three coordinates are distinct. Then S~ is invariant under the Mobius group, and there are no fixed points. We observe that the measure

is invariant and can be identified with the Haar measure. It can also be written in the form

*

This research was partially supported by the Air Force Office of Scientific

Research under Contract No. F44620 69 C 0088.

19

LARS V. AHLFORS

20

THEOREM 1. f' is properly discontinuous on S~ in the sense that every point in S~ has a neighborhood U whose images AU, A

f [',

are

disjoint. Proof: We write Z= (z 1 ,z2 ,z3 ), ' = (,1 , ,2 , ,3 ) and d(z,Q

l~ [zk, 'k ]. For fixed z E S~, d(z, Az)

-+

0 only if A

-+

=

I. Therefore

dr(z) == infAEf'-{Il d(z, Az) is positive. It is lower semicontinuous and has therefore a positive minimum on any compact subset of S~. Given 'E S~ let d 0 be the shortest distance from ' to

s3 -

S~. Denote by K

the compact set in s~ whose points have distance 2: do:/2 from and let

o be the minimum of

whose diameter is then d (z, A- 1z) Hence U

n AU

=


y, f3+y > a, y+a > (3 and a+ {3 + y > 2. Then the series

LIMIT POINT SET OF A KLEINIAN GROUP

21

(4) converges almost everywhere on every compact subset of

s3,

everywhere on D 3 , and uniformly on

n3 .

Proof: Suppose that a, b, c

0 it is important that we have convergence almost everywhere even if one or more of the zk lie on A, and in particular, almost everywhere on A 3. 5. Let Saf3y be the subset of S3 on which [A '(z 1 )]a [A'(z 2)]f3 [A'(z 3)]Y

< 1 for all A(_ r- II}. THEOREM 3. Almost every (z 1 , z 2 , z 3)

f

S3 is equivalent under

r

to

exactly one point in saf3y • Proof: We use the convergence of (4). Except on a countable number

of hypersurfaces the series has a term that is strictly larger than all the others. If this term corresponds to A0

In other words (A 0 z 1 , A0 z 2 , A0 z 3 )

f

f

r

we have clearly

Saf3y. The uniqueness is trivial and

the theorem is proved. 6. To avoid special considerations for the point oo we have given preference to the spherical derivative [A'(z)]. On the other hand the ordinary derivative A'(z) has the advantage of having not only an absolute value but also an argument, while suffering from the drawback that it becomes infinite at A-loo . lin the following we shall assume that oo is not a limit point. Then A is bounded, and for z fA the derivatives IA•(z)l and [A'(z)] differ only

UMIT POINT SET OF A KLEINIAN GROUP

23

by a bounded factor. As a result

I

Afr

IA'(z 1 )1a IA'(z 2 )1f3 IA'(z 3 )\Y

converges almost everywhere on A 3 and we can define a fundamental set Ea{3y for A 3 by the condition \A'(z 1 )\a \A'(z 2 )1f3 \A'(z 3 )iY < 1 for all A fr-Ill. Naturally, this makes sense only if m(A) > 0 as we shall suppose from now on. 7. A complex-valued measurable function ll on A3 is called a Beltrami differential if it is bounded and satisfies

for all A Jl

f

L

00

(

f

r.

Because Ea{3y is a measurable fundamental set, every

Ea{3y) can be extended to a Beltrami differential on A 3 . This is

the only use we shall make of Ea{3y and we shall even replace it by an arbitrary measurable fundamental set E. Given a Beltrami ll we form the function

i= 1 j = 1

which is obviously holomorphic in all variables when at once that

for all A

f

r.

As a consequence the function

'j

f

0. One verifies

LARS V. AHLFORS

24

Fl1(() = ¢11((1, (2, (3, (4)

II ((. - ?.) 1 "'J 1Si

L 00 ('U, r)

is an isometric bijection. The proof is clear. Let 11 and 11 1 be elements of the open unit ball of L 00 (!l). We say that 11 and 11 1 are equivalent, and we write 11 - fJ.l if and only if h*fJ. -

h*fJ. 1 .

This definition does not depend on h. We recall now the definition of the union b (!l) of the set of ideal boundary curves of fl. Let

'U/r. of

R=

If

r

r

be as in Lemma 6 so that we may identify tl with

is of the second kind, that is, if the maximal open subset

R U {oo! on which

r

mr

acts discontinuously is not empty, one sets

,i = ('U u mr)/r' b (!l) = ,i- fl. If r is of the first kind, that is, if mr = 0, one sets b (!l) =

0, .-i =

LEMMA 7. Let w : tl

fl. -+

C and

w : tl

-->

Cbe quasiconformal homeomor-

phisms with Beltrami coefficients fJ. and 11, respectively. Then fJ. and only if there exists a conformal mapping f: ~ (!l) w- 1 o f o

~ (or rather its continuous extension to

J.,

-+

fJ.

if

w (!l) such that

which always exists)

is homotopic to the identity modulo b (ll). This is a known result in the theory of Teichmiiller spaces [2, 7, 8]. It can be restated as follows: two Beltrami coefficients, on the same domain

tl, are equivalent if and only if they belong to mappings which define the same point in the Teichmiiller space T(tl).

EXTREMAL QU AS! CONFORMAL MAPPINGS

LEMMA

31

8. Let w and w be as in Lemma 7. If there is a compact set

K C ~ with w I~- K

~I~- K, then 11 ,., ~.

=

This is an immediate corollary of Lemma 7.

~C

LEMMA 8. Let

C be a domain bounded by finitely many disjoint ~ .... C be a quasiconformal homeomorphism with

jordan curves. Let w:

Beltrami coefficient 11, and let ~ ,.:; 11· Then there exists a quasiconformal homeomorphism W : ~ .... w(~) and

w- 1 oW

C with Beltrami coefficient

~such that W(~)

=

has a continuous extension to the closure of ~ which

leaves every boundary point of Proof: Let ~: L\ ....

~

C be some

fixed. quasiconformal homeomorphism with

Beltrami coefficient ~. Apply Lemma 7, set W - f o ~, and note that, under the hypotheses of Lemma 9, we may identify boundary of

b(~)

with the set theoretical

~.

An element 11

L,,)~) will be called canonical if ,\~ ii. is holomorphic.

t:

Such elements form a closed linear subspace L c:n (~) C L,)~). If

r

is a

Fuchsian group which leaves R fixed, we set

LEMMA 10, If

r

is the covering group of h:

11

->

~. thet;J

is a bijection. The proof is a calculation. LEMMA

11: If 11 and 11 1 belong to the open unit ball in

L~an(~) and

11 ,., 111' then 11 = 11 1 •

This follows from Lemmas 2, 3 and 10. LEMMA 12. There exists a holomorphic mapping ll~ of the open unit

ball in Loo(~) into L~an(~) such that n~ (O) = 0, n~ (11) ,., 11 if IITI~ (11)11 < 1,

LIPMAN BERS

32

Proof: Set II~ = (h*)- 1

o

II

o

h*.

§3. Smoothing Theorem A configuration (D, G) is a pair consisting of an open non-empty set DCC, such that C - D contains more than two points, and of a discrete group G of Mobius transformations such that y (D) = D for y < G. We choose such a configuration once and for all. Note that G acts properly discontinuously on D. To simplify writing, we assume from now on that !O, l, oo! C C- D. The configuration (D, G) will be called symmetric if the conjugation z y ('U) =

'l1

t->

z takes

D into itself, and if

for all y < G.

If 11 < L oo (C) and ll11ll < l. we denote by z t-> wl1(z) the unique automorphism of C, with Beltrami coefficient 11, which keeps 0, l, oo fixed. We denote by L 00 (C, G) the set of 11 < L 00 (C) with 11[y(z)]y'(z)/y'(z)

=

11(z),

y < G .

It is a closed linear subspace of L 00 (C). LEMMA 13.

only if

11

f

The pair (wi1(D), wi1G(wP)- 1 ) is a configuration if and only

L 00 (C, G).

The proof is standard and easy. We call 11 < L 00 (C) symmetric if 11 (z) = 11 (z) sym (C). Such 11 form an R-linear closed subspace of L 00 (C), denoted by L 00 LEMMA 14. If (D, G)

is symmetric, 11 < L!,ym (C, G), and ll11ll < l then

(wi1(D), wi1G(wl1)- 1 is symmetric. The proof of this is obvious.

If 11 < L 00 (C), we denote by v

IID(I1) the element v < Loo(C) deter-

EXTREMAL QUASICONFORMAL MAPPINGS

33

mined by the conditions

v\C-D = 11\C-D, vi~= II~(ILI~)

for every component LEMMA

~

of D.

15. II 0 is a holomorphic mapping of the unit ball in L,,.,(C)

into L,,.,(C), with II 0 (0)

=

0 and

rr 0 a rrn

rr0

.

Also, if (D, G) is a configuration, then II 0 [L00 (C, G)] C L 00 (C, G) ,

and, if (D, 1) is symmetric, then

Proof: Only the second assertion needs verification. A direct calculation shows that if y is any Mobius transformation, and if we set, for any 1L c L 00 (C), (IL • y)(z) = 11 [y (z)] y '(z)/y '(z)

then

If (D, G) is a configuration, y c G and 11 c L 00 (C, C), then y(D) = D, 1L • y = 1L and therefore II 0 (1L) = II 0 (1L) · y, so that II 0 (11) c L 00 (C, G). Symmetry is proved similarly. LEMMA

\III 0 (1L)II

16. If IL ( Loo(C) and 1111 II < l, 1111 I Dl\


C,

with

= ITL'l (p. I 1'1) such that wi'l {1'1) = w 11-(1'1) and w1'1

has the boundary values wiL I ai'l on the boundary atl of 1'1. This follows from Lemmas 9 and 12. Hence there exists a topological automorphism W of C such that

for every component 1'1 of D, and

Since isolated analytic arcs are removable sets for quasiconformal mappings, the mapping W is quasiconformal and its Beltrami coefficient is v. Since W leaves 0, 1, oo fixed, we conclude that W = w v. This proves the theorem for the case of a tame D. In the general case, construct a sequence of tame open sets Dj such that D 1 C D2 C ... C D, D = D 1 U D 2 U ···, every component of Dj is of the form Dj n 1'1 where 1'1 is a component of D, and Dj n 1'1 is relatively

35

EXTREMAL QU AS! CONFORMAL MAPPINGS .

compact in ll. Set vJ

D =

II

. j

(11). Then llvJ II

< max(llllll, k0 ), by Lemma 16.

If ll is a fixed component of D and llj = Dj nll, then llj is a domain, and ¢j = .\~.(vj lllj) is holomorphic. Using Lemma 5 and compactness properJ

ties of holomorphic functions, we conclude that a subsequence of l¢j l converges normally. We conclude next that a subsequence of a. e. to an element p p Ill

f

L~an (ll)

f

L 00 (C) such that pIC- D

=

lv j l

converges

111 C-D and

for each component ll of D. By what was proved above V·

~

~

we know that w JIC-Dj = wlliC-Dj' Therefore, byLemma8, vj Ill - 11lll, so that, by Lemma 1, pIll -

11lll. We conclude, by Lemmas

11 and 12, that pIll = Illl(ll). Hence p = v. Thus the selection of a subsequence was unnecessary and we have that lim v j(z) there is a number k

=

v (z) a.e. Since

< l with llvj II ::; k, we have that lwvj(z)l converges

to wv(z). Thus wviC-D

=

wlliC-D,

q.e.d.

§4. Trivial Families

An element 11

f

L 00 (C) with 111111 < l will be called trivial if 11 - 0

w.r.t. D. The aim of this section is to construct one-parameter families of trivial Beltrami coefficients. An element 11

f

L 00 (C, G) will be called locally trivial (with respect

to D) if wEil(z)-z = o(lsi),E--> 0,

z

f

C-D,

f

C-D

that is, if wll(z)

0 for z

.

(It is known [3] that wE ll(z) is a holomorphic function of E at E = 0.) LEMMA

17. We have wll(z)

_ z(z-1) TT

!!

11(() d.; dry C (((-1)((-z)

36

LIPMAN BERS

This is a known formula [10]. A quadratic differential (automorphic form of weight - 4) for (D, G) is a holomorphic function ¢ (z), z < D, such that ¢(y(z))y'(z) 2

=

¢(z),

y 0.

If (D, G) and 11 are symmetric, so are all v j . Proof: Set v "' rrD(t/1). For small It I this is a holomorphic function of t with values in L~an( C, G), by Lemma 15; hence there is a convergent expansion v "' v 0 + tv 1 + t 2 v 2 + .... Clearly, v 0 == 0 and the vj, j 2: l, have the required properties. LEMMA

22. Under the hypothesis of Lemma 21, assume that 11 is local-

ly trivial. Then v 1

=

0.

Proof: For small It I we have that w v(z) is a holomorphic function of t; if z < C-D, then wv(z) = wtll(z). One computes easily that

. I

aw v(z) at

t= o

""

aw tv l(z) at

I

= w v l(z)

t= o

so that wvliC-D == 0 and v 1 is locally trivial. Hence, v 1 C-D== 0, 1

EXTREMAL QU AS! CONFORMAL MAPPINGS

by Theorem 2. Also, there exists a holomorphic function that v 1 1f.. =

t\i

2 ·

39

1/J (z) in D such

(!pI f..) for every component f.. of D. Since v 1 is local-

ly trivial, we conclude, by Theorem 2 and Lemmas 18 and 19 that, for every

¢

A(D, 1),

£

Jf

v 1 (z)¢(z) dxdy

=

D

Jj

v 1 (z)(El¢)(z)dxdy = 0 D/G

This implies that for every f.. and every holomorphic function f(z), z

£

f..

with

ff

lf(z)l dxdy
0

and

If (D, G) and 11 are symmetric, and t is real, r is symmetric. Proof: The v from Lemma 21 satisfies llvll = O(t 2), in view of Lemma

22. Define

(for small

v

£

L 00 (C)

by

It I). Then llv II

llv 11. Define a

£

L 00 (C) by

40

LIPMAN BERS

Then a is trivial. Also, since v and tfl belong to L00 (C, G) we have, for

y

f

G, y

so that, by Lemma 13, a

L 00 (C, G). Next we compute that

f

tfl + v" l + 1iiv

a=

where

v [wtfl (z)]awtfl(z)/az

~(z)

Thus, llvll

awt:p.(z)/a z

= llvll = O(lti 2 ) and r = (a-tfl)/t 2 has the required prop-

erties. We also note that if (D, G) and fl are symmetric and t real, then a and r are symmetric.

§5. Hamilton's Extremality Condition

A Beltrami coefficient fl

L 00 (C, G), llflll

f

< l will be called extremal

with respect to (D, G) if

THEOREM 4. Every flo

f

with llfloll < l, is equivalent,

Loo(C, G),

with respect to (D, G), to an extremal one. Proof: Set k = inf llf.lll where jJ.

f

L 00 (C, G), !IiLII ~ llfl 0 11 and jJ. -flo·

There is a sequence of Beltrami coe~ficients ljJ.j such that lim lliLjll

=

k. Set wj

=

l with the above properties

wflj • By known compactness properties

of quasiconformal mappings one may assume, selecting, if need by, a subsequence, that lim wj

=

wfl, where fl

vergence is normal. For every y such that wj oy owj- 1 = wfl oy o(wfl)- 1

y

f

= wflo

f

L00 (C, G), llflll < l, and the con-

G there is a Mobius transformation ji o y o(wfl0)- 1 . It follows that

= y, so that, by Lemma 13, we have

fl

f

L00 (C, G). By

the lower semicontinuity of the dilatation of a quasiconformal mapping with respect to normal convergence [6] we have llflll ~ k, so that llflll = k.

41

EXTREMAL QU AS! CONFORMAL MAPPINGS

Clearly, wll(z) =lim wj(z) = wp.O(z) for z k, then

and 11 does not satisfy the Hamilton condition. The reverse implication is proved similarly.

If (D, G) is symmetric, we call a quadratic differential ¢ (z) symmetric if ¢ (z) "" ¢ (z), antisymmetric if ¢ (z)

= -

7J; (z).

The symmetric integrable

quadratic differentials form the real Banach space A+ (D, G) C A (D, G). We also set D+ "" D

n 'U.

LIPMAN BERS

44

LEMMA 24. If (D, G) and 11 t L 00 (C, G) are symmetric, then 11 is lo-

cally trivial if and only if

J!

11 (z) ¢ (z) dx dy =

0

D+/G

forall ¢

t

A+(D,G), and 11IC-D

Proof: If ¢

t

¢+(z)

=

0.

A (D, G), set =

.!..

.!.. ¢

¢ (z) +

2

¢_ (z)

(z),

=

¢ (z) - ¢+(z) .

2

Then ¢+ < A+(D, G), ¢_ is antisymmetric, and ¢ = ¢+ if ¢ < A+(D, G).

If 11 is symmetric, then

Hence the lemma follows from Theorem 2. LEMMA 25. If (D, G) and 11

f

Loo(C, G) are symmetric, then 11 satisfies

the symmetric Hamilton condition if and only if 11111 D+ll

=

sup Re

J1

11(z)¢(z)dxdy

D+/G

where the supremum is taken over all¢

E

A+(D, G) with II¢ IIA

=

2.

The proof is analogous to that of Lemma 23 and need not be spelled out.

§6. Teichmiiller' s Condition

The connection between Hamilton's condition and the classical extremality condition of Teichmiiller is contained in the following observation, due ~o

Hamilton. LEMMA 26. If dim A(D, G)

ton condition, then either 11

=

< oo, and 11

E

L 00 (C, G) satisfies the Hamil-

0, or there' is a ¢

E

A (D, G) such that ¢

I-

0

EXTREMAL QU AS! CONFORMAL MAPPINGS

45

and 1\11 I D\1 l¢(z)\ = 11(z)¢(z)

a,e. in D. Proof: The hypothesis implies that the boundary of the unit ball in A (D, G) is compact. Hence, if 11 -1- 0 satisfies the Hamilton condition, then there is a ¢

f

A (D, G) with II¢ II A 1111 I Dll

=

JJ

= 1 such that

11 (z) ¢ (z) dx dy

DIG

or

!!

lii11IDI\ l¢(z)l-11(z)¢(z)ldxdy = 0 ,

D/G

and, since 1111 I Dll I ¢(z)l 2:. 111(z)¢(z)l a.e. in D, the conclusion follows. We say that a 11

f

L,)C, G) satisfies the Teichmiiller condition with

respect to (D, G) of either 11 = 0, or there is a ¢

f

and a non-negative locally constant function k (z), z

A (D, G), with ¢ -1- 0, f

D, such that

(i) for every component 11 of D with ¢ 111 -1- 0, we have k 111 > 0 and 11111 = (kl/1)1¢1111/(¢111) , and (ii) for every component 11 of D with ¢ 111

=

0, we have k \11

= 11

111

= 0. Note that 1111 \ 1111 = k 111 for every component /1. THEOREM 8. If dim A (D, G) < ""• and D is connected, then every

extremal Beltrami coefficient satisfies the Teichmiiller condition. Proof: The assertion follows from Theorem 5 and Lemma 26, since if ¢

f

A(D,G) and¢ -1- 0, then ¢(z) -1- 0 a.e. in D. THEOREM 9. If dim A (D, G) < ""• then every Beltrami coefficient is

equivalent to an extremal Beltrami coefficient which satisfies the Teichmiiller condition.

46

LIPMAN BERS

Proof: Let us call a union of components of D which is invariant under

G a part of (D, G). Let Dl' D2 , ••• be the minimal parts of (D, G). Then dim A(D,G)

=I dim A(Dj,G), so that dim A(Dj,G) ,J 0 only for finitely

many parts. Now let p. 0 < L,)C, G) with 1111 0 11 < l be given. By Theorem 3 there is an extremal p. 1 equivalent to p., by Theorem 4 this p. 1 satisfies the Hamilton condition. If p. 1 = 0, there is nothing to prove. Otherwise there is, by Lemma 26, a¢< A(D,G) with¢ .J, 0 and with k 1 1¢ I = (p.l D)¢ where k 1

=

11111 Dll > 0. Let D 1 be the largest part of (D, G) such that

¢Ill .J, 0 for every component of D 1 , and set ¢ 1

If D

=

=

¢I D 1 . Then

D 1 , the proof is completed. Otherwise, apply the same procedure

to the Beltrami coefficient p. 1 and the configuration (D- D 1 , G). This will yield a Beltrami coefficient

which is equivalent to p. 1 with respect to (D- D 1 , G), and therefore to p. 0 with respect to (D, G), which is extremal with respect to (D- D 1 , G) and therefore also with respect to (D, G), and {1 2

such htat either p. 2 1D- D 1 = 0, or there is a part D2 of (D- D 1 , G) and an element ¢ 2 < A (D 2 , G), ¢ 2 .J, 0, such that 11 2 1D2

=

k 2 1¢ 2 1/¢ 2

with k 2 = ll11 2 l D2 11 > 0. If D = D 1 U D 2 , the proof is completed. Otherwise, apply the same reasoning to p. 2 and to the configuration (D- D 1 - D 2 , G). After a finite number of steps, one obtains the desired result. LEMMA 27. If (D,G) is symmetric, dim A(D,G)
'U,

such that w G w- 1 is a Fuchsian group, is of the form w fl

where fl is a symmetric element of L 00 (C, G) with

l!flll
0

=

w0 • Set w 1

= ¢ 1.

One verifies that w 1 is

an automorphism.

=

For every y in G and every t, we have that g o Cl>t o y

Pt o goy

=

Pt o g = go Cl>t" Since Yt depends continuously on t, it does not depend on t at all. Since y 0 y

f

=

y by hypothesis we have w 1 o y

=

y o w 1 for all

G. B) LEMMA 14. Theorem 2" holds for a group G if it holds whenever

the mapping w :

UG-> UG

is locally quasiconformal. If

G

is finitely gen-

erated, then Theorem 2" holds if it holds for every quasiconformal w. Proof: Set S = U G/G, and let w induce the automorphism -n1 of S. It is known (cf. [ 4]) that m' is homotopic to locally quasiconformal auto-

morphism 115 1 . ( Cf. [ 4] ; "locally quasi conformal" means "quasi conformal on every relatively compact subdomain. ") In view of Lemma 13 the mapping

ISOMORPHISMS BETWEEN TEICHMULLER SPACES

ti1 1 is induced by an automorphism w 1 of U G such that y o w 1 for all y

f

71

= w1 o y

G. Since g is conformal and ti1 1 o g = g o w 1 , w 1 is locally

quasiconformal. If Theorem 2" holds for wl' "1'if 1 is homotopic to the identity and so in t6 . C) In order to prove the second half of the lemma, we show first that for a finitely generated G the automorphism 'li1 is homotopic to a quasiconformal one. Now S can be represented as S is a compact surface of genus P and

=

o1 , •.• ,oN

S- (o 1 U

S S;

•·• U oN) where

are disjoint continua on

the numbers P and N are subject to the sole restriction that 3P- 3 + N

~

0.

In order that 1i1 : S .... S be homotopic to a quasiconformal automorphism it is sufficient that for every sequence {Pv I C S with lim Pv

E

oi

and

lim 1!l"(pv) ( oj ' the continua oi and oj be either both points or both nondegenerate (cf. [4]). We can assert more: if

(6.1)

Indeed, to every

oi

there belong simple closed curves C on S with the

property: the complement of C in S has two components, one of which (called the interior of C) is simply connected and contains

oi .

Consider a

sequence cl, c2, c3, ... of curves belonging to oi ' such that en+ 1 lies in the interior of en and no sequence of points zn ( en, n

= 1, 2, 3, ... ,

converges in S. In S every such sequence converges to oi. To every oi there belongs a conjugacy class of maximal cyclic subgroups Dv

E

G such that for every curve C belonging to pi, each component of

g- 1 (C) is a curve invariant under some Dv, and, conversely every simple curve C on S with this property belongs to pi. Now, if a curve C 0 C UG is invariant under a subgroup D C G, then the curve w (C 0 ) is invariant under the same subgroup. It follows that if the curve C C S belongs to pi, so does the curve uf(C). This remark implies the validity of (6.1). The proof of Lemma 14 can now be completed by repeating the argument given in B).

72

LIPMAN BERS AND LEON GREENBERG

D) We proceed to prove Theorem 2' for a finitely generated group G. Since Theorems 2' and 2 "are equivalent, we may assume that the mapping w is quasiconformal (see Lemma 14). It is also easy to check that we loose no generality in assuming that the groups G, F, H (see §3, A)) are normalized. The bijection w : UG -> UG can be lifted to U: there is a topoligical selfmapping W0 of U such that (6.2)

hoW 0 = woh.

Since w commutes with every y h o W0 o ¢

f

G, we have, for ¢

f

F,

= w o h o ¢ = x oX(¢) o h = X(¢) ow o h X(¢)ohoW 0 = ho¢oW 0

=

so that (6.3) Since h is conformal and w is quasiconformal, W0 is quasiconformal. Also, if ¢

F is such that 0 is an attracting fixed point of ¢, then w o ¢n(z) =

f

¢now (z), so that, letting n--> oo, we have w (0) = 0. Similarly, w (1) = 1, w(oo)

= oo

and thus w

f

~*(G).

(Here we used the fact that G is normalized.)

Now let a be a real Mobius transformation such that a o W0 leaves 0,1, oo fixed. By (6.2) we have W0 h

oa- 1 :

f

~(G).

Hence W =

a

o W0

f

~*(G).

Also, h 1 =

U--> UG is a universal covering, and h 1 oW = w o h.

By Lemma 2 we have w = m(W). Since we already know that m : Ti'f(F)-> Ti'f(G) is injective, and since w is strongly G -equivalent to the identity, W is strongly F -equivalent to the identity (i.e., commutes with all ¢ (6.4)

f

F). Thus ..~.. w-1 ao WOO'f'O 0 oa -1

Together with (6.3) this yields a

f

a

..~..-1

O'f'

.d 1 for all ¢

=1

o¢ oa- 1 o¢- 1

f

f

H for¢

H by the Corollary to Lemma 10. Therefore h 1 = h

F . f

oa- 1 =

F, so that h, and

73

ISOMORPHISMS BETWEEN TEICHMOLLER SPACES

W= a

o

W0 has the properties asserted by Theorem 2 '.

E) Now let G be infinitely generated. Since G is countable, there

exists a sequence G 1 C G2 C G3 C •·· of finitely generated subgroups of G such that G

= G 1 U G 2 U G3 U

··· . For each j let hj: U ..... UGj be the

uniquely determined universal covering which satisfies

(6.5)

h. '(i) h '(i) J

>0 .

We construct the groups Hj and Fj as in Section 3, A), using hj and Gj instead of h and g. Thus there is an exact sequence Xj

1 --> HJ· C. F. - - G. --> 1

(6.6)

J

J

with h.

(6.7)

J

o

¢ = X. (¢ ) J

o

h.

J

for

¢

f

F1. .

LEMMA 15. We have

(6.8)

. lim h/z) = h (z) J ..... 00

normally

(that is uniformly on compact subsets of U). Also, for every ¢

f

F there

is an integer N > 0 and a sequence ¢j, j = N, N+1, ... with ¢j X/¢j)

f

Fj,

= X(¢), and .lim ¢j = ¢ . J-->oo

The proof is rather routine, but for the sake of completeness we shall carry it out. F) Proof of Lemma 15. First we assume that h/z) converges normally to some function k (z); we shall show that k (z) The function k is holomorphic and k (i)

= h (z).

= h (i).

Let ( be a point in

UG. We construct a simply connected domain DC UG containing ( and

h (i). There is a holomorphic mapping gj : D --> U such that gj (h (i)) = i and h/gj(z))

= z. Since gj(D) C U, Schwarz' lemma implies that for every

74

LIPMAN BERS AND LEON GREENBERG

D 0 C CD there is a U0 C CD (independent of j) with qj (D 0 ) C

u0 •

(Here

C C denotes relatively compact subsets.) Selecting, if need be, a subsequence, we may assume that lim q/z) = q(z) normally in D, and k (q (z)) = U- UG we have hj(z) f, ( 0 U and large j, we must have k (z) f, ( 0 (by Hurwitz' theorem).

z. Hence k (U) '-:; U G . Since for every ( 0 for z

f

f

Thus k(U) = UG. Next let z 0 be any given point in U. Let DC UG be an open disc centered at h (z 0 ), and let j be so large that hj (z 0 ) < D. For such a j there is a holomorphic function Pj(z) in D with Pj (D) C U, Pj (h (z 0 )) = z 0 and hj (pj (z)) = z. As before, a subsequence of {pj(z)l will converge to a holomorphic function p (z) with k (p (z)) = z. We have shown that k is an unbounded and unramified, and thus universal, ~ing of UG by U. Therefore k '(i) f, 0, so that, by (6.5), k '(i) h '(i) > 0. This implies that k = h. We remark next that, by Schwarz' lemma, every U0 C C U has the property that h/U 0 ) C U 1 C C U 0 , where U 1 does not depend on j. Therefore every increasing sequence of integers contains a subsequence {jn} such that {h. (z) I converges normally. But we have just shown that the limit Jn function of this convergent subsequence must be h (z). Hence the selection of a subsequence was unnecessary and (6.8) holds. G) It remains to prove the second statement in lemma 15. We note first

that Fj and F act freely on U, so that an element of these groups is determined by its action on a single point. Let ¢ f, id, an element of F, be given, and set y = X(¢), Z = ¢ (i) and ( = y(h(i)) = h(Z). There is a sequence {Zj

l

C U with lim Zj = Z

h (Z). There is a ¢. < F. with ¢ 1. (i) = z1. . We note that this J J J J ¢j is unique, that Xj(¢j) = y and hence hj o ¢j = yo hj. We may assume, and h. (Z.)

=

selecting if need be a subsequence, that there is a real Mobius transformation t/1 with lim ¢j = t/1. Then h ot/1 =yo h, so that tjJ

f

F. Since t/J(i) = Z,

we have tjJ = ¢ , and the selection of the subsequence was unnecessary.

ISOMORPHISMS BETWEEN TEICHMOLLER SPACES

75

H) Now we prove Theorem 2' for the group G, assuming that the mapping w is locally quasiconformal in UG. In view of Lemma 14 this involves no loss of generality. Since U- U G is discrete we may extend w, by continuity, to an automorphism of U; we denote the extended mapping by the same letter w. Note that w need not be locally quasiconformal in U. We use the notations introduced in E). Since Theorem 2' is already gen.~rated

established for finitely

groups, we know that for every j

=

1, 2, ... ,

there is an automorphism Wj of U such that hj

(6.9)

¢

(6.10)

o

Wj

o

= Wj

wj = w o

¢

o

hj

for all

¢

f

Fj .

We now establish LEMMA 16.

For every domain D C C U there are numbers N and K

such that wj I DisK quasiconformal for j > N. (We recall that a K quasiconformal mapping is a homeomorphic solution of a Beltrami equation

aw/az

=

ll(z)(aW/az) where lll(z)l _:; (K-1)/(K+ 1).)

Proof: The closure of h (D) is compact in h (U)

= U G and there are do-

mains ~ 0 and D0 such that h(D) C C ~ 0 C C UG, DCC D0 and h (D 0 ) C

~0 •

Let N be so large that hj(D) C h (D 0 ) for j > N; such an N Since hj is conformal everywhere and w I ~ 0 is

exists in view of (6.8).

quasiconformal, and therefore K quasiconformal for some K, we conclude from (6.9) that wj I D is K quasiconformal for some K. Now let DCC U be a circular disc and let N and K be as in the lemma. Then we can write, for j > N,

w.J I D = (.J

0

X·

J

where Xj is a K quasiconformal automorphism of D which leaves the center of D fixed, and (j : D

-->

U is a conformal mapping. Indeed, we simply

76

LIPMAN BERS AND LEON GREENBERG

choose for Xj that unique solution of the Beltrami equation satisfied by Wj which maps D and its center onto themselves. It is well known that all

> N, are equicontinuous. On the other hand the holomorphic functions ( j = ((j- i)/(( j + i) are uniformly bounded. Xj, j

Applying the preceding argument to a sequence of discs exhausting U, and selecting if need by a subsequence, we may assume that either (I) the sequence Wj converges normally to a function, or (II) the sequence diverges to

oo ,

again normally.

Next, case (II) cannot occur. For if it did, choose an element ¢

= (as+ b)/(cz +d),

c

¢j ~ ¢. Then one may write ¢j(z)

=

such that ¢ (z)

lim bj

I= 0,

and a sequence ¢j

f

f

F

Fj with

(ajz + bj)/(cjz + dj) with lim aj

=

a,

= b, lim cj = c, lim dj = d. Now, by (6.10), a. w. (z) +b. J J J C· W.(z) +d. J J J

The left hand side converges to a/ c and the right hand side diverges to

oo ;

this is absurd. It follows that .lim Wj(z) J ~ 00

= W(z) where W : U ~ U is a continuous

mapping. Since also hj o Wj 1 = w- 1 o hj, the same argument shows that W is a topological self-mapping of U. Clearly, h oW ,., w o h. It also follows from Lemma 13 that W commutes with all ¢

f

F.

§7. Conclusion Now we can complete the proof of the isomorphism theorem and discuss the effect of the canonical isomorphisms on the modular groups. We assume again that G and F are normalized. A) LEMMA 16. Let W f I*(F) and let w

= m(W) commute with allele-

ments of G. Then W commutes with all elements of F.

ISOMORPHISMS BETWEEN TEICHMtJLLER SPACES

Proof: By Theorem 2' there is a w

¢

f

f

77

I (F) which commutes with all

F and satisfies h ow = w o h. Since F is normalized, w leaves

0, 1,oo fixed. Thus w

f

I*(F), and w = m(w) by Lemma 2. Hence w

COROLLARY. If W,

Wf

= W.

I*(F) and w = m (W) is strongly G -equiva-

lent to ~ = m(W), then W is strongly F-equivalent to W. Proof: By hypothesis, ~ o w- 1 commutes with all elements of G 1 wGw- 1 . Hence, by Lemmas 3 and 16, of F 1

=

WFw- 1 •

Thus

w- 1

o

Wo w- 1

commutes with all elements

W commutes with

B) The corollary asserts that m : Ttl (F)

-+

all elements of F.

Til( G) is an injection. In

view of Lemmas 6 and 9 this establishes Theorem 1 '. C) Addition to Theorem 1 '. The canonical isomorphisms m induce iso-

morphisms of r(G) and rtt(G) into subgroups of r(F) and rtt(F), respectively. Proof: Let w

f

I*(G), w

formation such that a ow ow

f f

I 0 (G) and let a be a real Mobius transI*( G). There is a W f I*(F) with w =

m (W), so that we have the commutative diagram (4.2). Also, the mapping w : U G -+ UG can be lifted to a quasiconformal selfmapping

0 of U such

I 0 (F). Let -f3 be a real Ml)bius transformation such that f3 oW o 0 f I*(F), and set h 2 = a o h 1 o {3- 1 , G2 = a- 1 G 1 a. Then we have the commutative diagram that h o 0

= w o h. One verifies easily that 0

f

0- - u __ u-w__ u _ __;f3~- u h

h

h

and h 2 is a universal covering. By Lemma 2 we have that m(f3 oW o 0) a

oWoli.l.

=

LIPMAN BERS AND LEON GREENBERG

78

It follows that UJ* and 0*, considered either in the modular groups or

in the reduced modular groups, are connected by the relations m o 0* = UJ* om. In other words, m- 1 f' tt (G) m and m- 1 f' (G) m are subgroups of f'#(F) and f'(F), respectively. D) The subgroups considered above are, in general, proper subgroups. Example. Let S be a closed Riemann surface of genus 3 and let

p 1 , p 2 , p3 be 3 distinct points on S. Let G be a Fuchsian group such that Ud/G is conformally equivalent to S - {p 1 , p2 , p3

I.

Assume that G

contains elliptic elements of orders 2, 3 and 5, respectively. Then

r (F)

f'# (F) is isomorphic to the group of homotopy classes of quasiconformal selfmappings of S- {pl'p2 ,p3 l, but only selfmappings which leave each point Pv fixed induce elements of f' (G)

=

f' tf (G).

E) It can be shown that for a finitely generated group G the index of m- 1 f' tf (G) m

in f' #(F) is finite.

Department of Mathem·atics, Columbia University Miller Institute for Basic Research, University of California at Berkeley Department of Mathematics University of Maryland

ISOMORPHISMS BETWEEN TEICHMULLER SPACES

79

REFERENCES [1]

L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. VanNostrand Co., 1966.

[2]

_ _ and L. Bers, "Riemann's mapping theorem for variable metrics," Ann. of Math. 72, 385 -404 (1960).

[3]

L. Bers, "Quasiconformal mappings and Teichmiiller:s theorem," in Analytic Functions, 89-119, Princeton University Press, 1960.

[4]

___"Uniformization by Beltrami equation," Comm. Pure Appl. Math. 14, 215-228 (1961).

[5]

_ _ On Moduli of Riemann Surfaces (mimeographed lecture notes), Eidgenossische Technische Hochschule, Zurich, 1964.

[6]

_ _ "Automorphic forms and general Teichmiiller spaces," Proc. Coni. Compl. Anal. (Minneapolis 1964), 109-113, Springer-Verlag,

1965. [7]

_ _"Universal Teichmuller space," in Analytic Methods in Mathematical Physics, 65-83, Gordon and Breach Science Publishers, 1970.

[8]

C. J. Earle, "Reduced Teichmiiller spaces," Trans. Amer. Math. Soc. 126, 54-63 (1967).

[9]

R. Fricke and F. Klein, Vorlesungen "iJber die theorie der automorpher funktionen, Vol. 2, B. G. Teubner, 1926.

[10]

L. Greenberg, to appear.

[11]

S. Kravetz, "On the geometry of Teichmiiller spaces and the structure of their modular groups," Ann. Acad. Sci. Fenn. Ser. AI No. 278, 1-35

(1959). [12]

L. Keen, "Intrinsic moduli on ~iemann surfaces," Ann. of Math. 84,

404 -420 (1966). [13]

A. Marden, "On homotopic mappings of Riemann surfaces," Ann. of Math. 90, 1-8 (1969).

ON THE MAPPING CLASS GROUPS OF CLOSED SURF ACES AS COVERING SPACES by Joan S. Birman* and Hugh M. Hilden

§l. Introduction Let T g be a closed, orientable surface of genus g, and let H(T g) be the gro:.~p of all orientation-preserving homeomorphisms of T g-> T g· H(T g) contains a subgroup :D(T g) consisting of all homeomorphisms which are isotopic to the identity. The mapping class group M(T g) of T g is defined to be the quotient group H(T g)/:D(T g). Alternatively, M(T g) is known as the Teichmiiller modular group, and also as the homeotopy group of T g (although the latter term usually includes the orientation-reversing homeomorphisms of T g -> T g). M(T g) can also be characterized algebraically as the group of all classes of "proper" automorphisms of the fundamental group rr 1 T g of the surface, where the inner automorphisms constitute the class of the identity. (A proper automorphism is one which maps the single defining relation in any one-relator presentation of

7T 1 T g

into a conjugate

of itself. If one allows orientation-reversing. homeomorphisms, the corresponding algebraic characterization would be the totality of automorphism classes of rr 1 T g' where one admits automorphisms which map the basic relator into either a conjugate or itself or of its inverse.) It is well-known that every homeomorphism of T g -> T g is isotopic to a

sequence of special types of homeomorphisms known as twists, and that the 3g- 1 classes which are represented by the twists about the closed paths u 1 , ••• , ug, z 1 , .•• , zg_ 1 , y 1 , ..• , y g in Figure 1 generate M(T g)

*

The work of the first author was supported in part by a summer research grant from Stevens Institute of Technology.

81

JOAN S. BIRMAN AND HUGH M. HILDEN

82

[7]. A complete set of defining relations for the group M (T g), however, has not been obtained for g > l. Our main result is to obtain defining relations for M(T 2 ) and, for g > 2, for a sequence of subgroups Ms. (T g), i

=

1

0, 1, ... , g-1, which, taken together, generate M(T ), although each Ms (T ) g i g is a proper subgroup of M(T 12 ) if g > 2. The derivation, being based on the fact that T g can be regarded as a ramified covering of a sphere with an appropriate set of branch points, also casts some light on the relationship between mapping class groups of surfaces and Artin's braid group.

Yg

4

FIG. I

MAPPING CLASS GROUP OF CLOSED SURF ACES

83

To relate our work to known results, we remark that Siegel's modular group is known to be a quotient group of M(T g). Algebraically, Siegel's modular group can be characterized as the group of proper automorphisms of the abelianized fundamental group of T g• or alternatively as the group Sp(2g, Z) of 2g x 2g proper symplectic matrices with integral entries [15, page 178]. Defining relations for Sp(2g, Z) are known for arbitrary g [13], but the kernel of the homomorphism from M(T g) to Sp (2g, Z) is not known for g > l. For g = l the groups M(T 1 ) and Sp (2, Z) coincide because TT 1 T 1

is abelian, M(T 1 ) being the group of 2 x 2 matrices with integral entries and determinant +l [6, page 85]. This group has been studied exten-

sively in the literature. The group M(T 2 ) was investigated by Bergau and Mennicke [2], who proposed a system of defining relations; however, a gap in their proof (Lemma 5 on p. 430 of [2]) has left open the question of completeness, which is now confirmed by our proof. For g > 2 little is known about defining relations in M(T g), and our results are new. We begin in Section 2 by investigating the relationship between mapping class groups of T g and of a sphere with 2g + 2 points removed, denoted T0 , 2 g+ 2 , which arises from the fact that T g can be regarded as a 2-sheeted covering of T0 with 2g+2 branch points. We show that an isomorphism exists between M(T01 , 2 1i: + 2 ) and S(T g), where ~ (T g) is a quotient group of a special mapping class group S(T g), in which admissible maps are restricted to those homeomorphisms of T g covering space projection from T

-->

T g which preserve fibres in the

• Using known generators and T0 2 • i:+ 2 defining relations for M(T0 ., 2 g+ 2 ), as determined in ll4], we thus obtain a

presentation for

S(T g)

g

-->

which arises in a natural geometric manner.

In Section 3 the relationship between S(T g) and M(T g) is investigated. It is shown that S(Tg) is isomorphic to each of the subgroups M

S·

i

(T ), g

= 0, 1, •.. , g-1, where Msi(Tg) is defined to be the subgroup olM(Tg)

generated by the 2g+ 1 mapping classes represented by the twists about theclosedpaths u 1 , ••• ,ug,z 1 +., •.. ,z

., y 1 .,y . inFigure1 1 g- 1 + 1 + 1 g+ 1 ' where all subscripts are modulo g. We will use the symbol M (T ) to des g

JOAN S. BIRMAN AND HUGH M. HILDEN

84

note any one of these subgroups, e.g., Ms (T g). The establishment of this 0

isomorphism is the central idea in our development. It rests on a proof of the fact that if a homeomorphism

o~

T g .... T g preserves fibres in a particular

2- sheeted covering of T 0 , and is deformable to the identity map, then in fact it is deformable to the identity map via an isotopy which is fibrepreserving. It then follows (Section 4) that defining relations for S(T g) can be used to obtain defining relations for S (T g) and Ms(T g).

Section 5 contains some partial results on the relationship between Ms(T g) and th•:! full mapping class group M(T g) for g 2: 3. In Section 6 some connections with other work are discussed briefly. Since the group M(T 1 ) is well-known, and also because some of our results (noteably Lemma 4.1 and Th. 6) require special treatment for the case g = l, we restrict our attention for the most part to g 2: 2. §2. The special mapping class group

Let T g be a surface of genus g, embedded in E 3 in the manner illustrated in Figure 2. The embedding is chosen in such a way (see Figure 2) that:

z

i

..X

FIG. 2

85

MAPPING CLASS GROUP OF CLOSED SURFACES

(a) T g is invariant under reflections in the xz and xy planes.

n !(x, y, z)/ -1 -< y -< ll is a set of g right circular cylinders g of radius 1 with axes parallel to the y axis. Let cl' c 3 , ... , c 2 g+ 1

(b) T

denote the intersections of T g with xy plane, and c 2 , c 4 , ... , c 2 g the intersections of T g with the xz plane. Each ci is a circle. Define an equivalence relation on T g by (x, y, z) ,...., (+X, -y, -z). The natural projection rr: T g-+ T gl,..,

is easily seen to be 2-to-1 except for

2g+ 2 points on the x axis which we shall call P 1 , ... , P 2 g+ 2 in order of increasing x coordinates. The surface T g/,...., is homeomorphic to a sphere T 0 , the images of the exceptional points P 1 , ... , P 2 g + 2 under set of exceptional points Q1 , ... , Q2 g+ 2 on T 0 Let

t :

T g -+ T g be defined by

t

(x, y, z)

=

'Tl

being a

•

(x, -y, -z). (Thus

t

inter-

changes the points in each fibre.) We say that a homeomorphism H: T g

-+

Tg

is "symmetric" if it has the property: H (t(z))

(1)

= t (H (z))

where z now denotes an arbitrary point on the surface T g' We remark that the involution

t

is a symmetric homeomorphism. Also, that every symmetric

homeomorphism keeps the special points P 1 , ... , P 2 g+ 2 fixed as a set, although not necessarily individually fixed. Of particular interest will be the fact that certain of the twists which generate M(T g) can be represented by symmetric homeomorphisms (for genus 1 and 2 every homeomorphism is isotopic to a symmetric homeomorphism), as will be demonstrated in Section 2. Let S(T g) be the "symmetric" mapping class group of T g' that is the group

S(T g)

of all orientation-preserving symmetric homeomorphisms modulo

S:D (T g)

~

S(T g) of those symmetric homeomorphisms which are isotopic to the identity map within the group S (T g). (We will describe the subgroup

this situation by saying that a map is "symmetrically isotopic to 1"). Let ~

£

S(T g) denote the element represented by the involution

order 2 and lies in the center of S (T g). Define

t •

Then g has

S(T g) to be the factor group

86

JOAN S. BIRMAN AND HUGH M. HILDEN

of S (T g) modulo the cyclic subgroup of order 2 generated by

S(T g)

THEOREM 1:

g,

We claim:

is isomorphic to the mapping class group

M (T0 , 2 ~+ 2) of the sphere with 2g + 2 points removed.

Proof: First we show that M (T0 2 + 2 ) is a homomorphic image of ' ~ S(T g); second that the homomorphism has an inverse. Let H : T g

4

T g be symmetric" Consider any point ~

TT- 1 (~) will be a pair of points z,

t

points H (z), H( t (z)) are mapped by

(z) TT

f

f

T 0 • Then

T g' Since H is symmetric, the

into a single point on T0 which we

denote H* (~). Thus every symmetric homeomorphism H induces a homeomorphism H*: T 0

4

cial points P 1 , ... ,

T0 defined by H* =

P 2 ~+ 2

TT

H TT- 1 • Since H keeps the spe-

fixed as a set, and TT: Pi

4

Qi for each i =

1, ".,2g+2, it follows that H* will keep Ql' ... ,Q 2 i;>;+ 2 fixed as a set. Moreover, if H is symmetrically isotopic to 1, then the isotopy Ht will induce an isotopy Ht defined by Ht =

TT

Ht TT-l

o

Thus the projection

TT

induces a homomorphism from S(Tg) to M(T 0 , 2 i:+ 2 ), and also from S(Tg) to M(T 0 , 2 ~+ 2 ), since

TTHTT-l =

TTiHTT- 1 •

is any simple closed curve on T0 , 2 ~+ 2 ' then r lifts to a closed curve if and only if r encloses an even number of special points. Now, if

r

The property of enclosing an even number of special points is preserved under homeomorphisms, the property of a closed curve lifting to a closed curve is preserved under homotopy, and TT 1 (T0 ,, 2 i:+ 2 ) is generated by equivalence classes of simple closed curves. This implies that a closed curve

r

lifts to a closed curve if and only if its image under an arbitrary homeo-

morphism of T0 2 + 2 lifts to a closed curve. '

g

Now we claim every homeomorphism H*: T0 , 2 i:+ 2 4 T0 , 2 i:+ 2 can be lifted to a homeomorphism H : T g 4 T g which represents a unique element of S (T g). Let H* be such a map. Choose any particular point z 0

f

T ~~;, 2 ~+ 2 ,

where Ti:, 2 ~+ 2 is the surface Tg with the points P 1 , ... ,P2 i:+ 2 removed. We note that the pair (TT, T ~~;, 2 i:+ 2 ) are a covering space over T0 , 2 i:+ 2 • If ~0

=

TT(z 0 ), then

TT-l

(H*(~0 )) will consist of two points, and we pick one

arbitrarily, denoting it H(z 0 ). Now choose any y

f

T~. 2 ~+

2 ' and let y be

MAPPING CLASS GROUP OF CLOSED SURFACES

87

a curve from z 0 to y. Then 77(y) is a curve from ( 0 to 77(y), and H*(77(y)) is a curve from H*((0 ) to H*(77(y)). We define H(y) to be the endpoint of the unique lift of H*(77(y)) whose initial point is H(z,). It is easily checked that H is well-defined and a homeomorphism. Now, suppose H* is isotopic to the identity, v.i.a an isotopy which we denote by H'~. For any particular point (

f

T0 , 2 ~+ 2 the isotopy Ht will

define a path on T0 , 2 it+ 2 joining H* (() to (. By the path lifting property for covering spaces, this path can be lifted to a path joining one of the points in the set 17- 1(H*(()) to one of the points in the set 77- 1(() which is unique modulo the choice of the initial lift

17- 1(H* (( 0 )).

However, where-

as Hl pulls each point ( back to the identity, the induced isotopy Ht might pull each point H (z) back to

t

(z) instead of z. Thus a homomor-

phism exists from M(T0 ,, 2 it+ 2) onto

S(T g),

but not necessarily onto S (T g).

Since the homomorphism from M (T0 , 2 g+ 2) onto follows that

S(T g)

S(T g)

is invertible, it

and M (T0 , 2 g+ 2) are isomorphic groups, which com-

pletes the proof of Theorem 1. Now consider the mapping class group M(T0 , 2 g+ 2). This group was studied by W. Magnus [14], who showed that M(T0 , 2g+ 2) admits the presentation*:

generators: a 1 , ... ,a2 g+ 1 defining relations: (2.1)

1 ~ i ~ 2g-1

\i- j\ ;::: 2

(2.2)

ai ai+ 1 ai = ai+ 1 ai ai+ 1

1.~ i ~ 2g

(2.3)

(a1a2···a2g+1 ) 2g+2

(2.4)

a1 a2 ••· a2ga2g+ 1 a2g+ 1 a2g ·•• a2 a1 = 1

The generators a-, 1 I

=1

D by h(r,(J)"' (r,0-77r). Let hi beanembedding

88

JOAN S. BIRMAN AND HUGH M. HILDEN

of D into To, 2g+ 2 , where h/D) includes Qi and Qi+ 1 but is disjoint from all QJ·, j .;,. i, i + 1. Suppose also that h-(1, 0) = Q·1 and h.(1,rr) = 1 1 Qi+

1•

Then hi h hi- 1 extended to the identity map outside hi(D) defines a

self-homeomorphism of T 0 , 2 g+ 2 which represents ai. The effect of this map is to interchange of the points Qi and Qi + 1 , while leaving the other points Qj fixed. Our next task is to determine what types of self-homeomorphisms of T g are defined by the proceedure we described in the proof of Theorem 1 for lifting self-homeomorphisms of T0 , 2 g + 2 to self-homeomorphisms of T g? In fact, it will turn out that the representatives of the ai lift to precisely the types of twist maps which are known to generate Ms(T g). A twist rc about a simple closed curve c on the surface T g is defined by considering a neighborhood of c on T g which is homeomorphic to a cylinder. Cut the surface T g along one base of this cylinder, twist the free end of the cylinder through 2rr, and then glue it back together again. The resulting map of T g

~

T g is the twist r c .

In order to study how the maps which represent the ai lift to T g' we go back to the representation of T g illustrated in Figure 2, and show that a twist about the curve ci on T g projects to a representative of ai on T0,2g+ 2· We note that in Figure 2, if i is even the curve ci is the unit circle about (2i-1, 0, 0) in the xz plane. We recall that ci lies on a right circular cylinder of radius 1 whose axis coincides with the y axis. For convenience, introduce coordinates (y,()) on this cylinder. A twist rc· about 1

ci is given by the identity map outside the cylinder, with r c. (y, ()) = (y, () + rr (y + l)). It is easy to check that

r c.

1

preserves fibres with respect

1

to the projection

rr.

Also that r c. maps Pk ~ Pk if k /. i, i+ l, white 1

Pi ->Pi+ 1 and Pi+ 1 ~Pi, and that r ci projects to the elementary twist on TO, 2 g+ 2 interchanging Qi and Qi+ 1 • A similar argument holds for each curve c 1 , •.• , c 2 g + 1 • Thus we have:

89

MAPPING CLASS GROUP OF CLOSED SURFACES

THEOREM

2. The maps which represent the generators al' ... , a 2 g+

1

of M (T0 2 + 2 ) lift to twists rc , ... , rc

about the curves c 1 , ... , c 2 1 ' g 1 2g+ 1 ~ g+ on Tg, and these twists represent elements G- 1, ... , 2g + 1 of S(T g ) •

a

Theorems 1 and 2 then combine to give: THEOREM

3.

S(T g) admits a presentation:

generators: a1, ... ,a2g+ 1 ~

...

~

1::; i::; 2g- 1,

(3.1)

a 1 ->ai

(3.2)

aiai+1

(3.3)

~ ~ ~ ~ )2g+ 2 = l ( v1 v2 "' v2gv2g+ 1

(3.4)

~ ~ ~ ~2 - al a2 ••. a2ga2g+ 1 a2g ... a2 a1

d\aiai+1

l::;

i::;

1i -

j\ > 2

2g

l

§3. The relationship between M (T g) and S(T g) Putting aside the group

S(T g)

for the moment, we fix our attention on

the relationship between S (T g) and M (T g). Since any self-homeomorphism of T g which represents an element of S (T g) clearly also represents an element of M (T g), and since maps which are symmetrically isotopic to 1 and thus represent the identity in S (T g) also represent the identity in M (T g), therefore a natural homomorphism exists from S (T g) to M (T g). The question we now consider is the extent to which the converse of this statement might be true.

If we restrict our attention to the subgroup Ms(T g) generated by the mapping classes represented by rc , ... , rc 1

we note at the outset that 2g+ 1

every element in this subgroup can be represented by a symmetric phism which is an appropriate product of rc , ... , rc 1

homeomor~

• Much more subtle 2g+ 1

is the following theorem, which we will develop via a sequence of pretheorems in the following pages: THEOREM

7. If G : T g -> T g is a. symmetric homeomorphism which is

isotopic to 1, then G is symmetrically isotopic to 1.

JOAN S. BIRMAN AND HUGH M. HILDEN

90

As a Corollary to Theorem 7 we will establish:

The proof of Theorem 7 begins with: THEOREM 4: Let G be a symmetric homeomorphism of T g' isotopic to Id, such that G leaves P 1 , ... , P 2 g + 2 fixed. There is a second isotopy

between G and Id such that the entire isotopy leaves P 1 , ••• , P 2 g+ 2 fixed. Proof: Let G1 be the original isotopy. Then p/t) = Gt(Pj) is in general not constant, although it is a closed loop since G (Pj)

=

Pj. The next

two lemmas allow us to construct a sequence of isotopics Kf, ... , K;g+ 2 of G with Id, each one leaving one more special point fixed than its predecessor. LEMMA 4.1. Let Gt be an isotopy such that G0 = Id, G 1 = G, where G

E

S(Tg) and G(Pj)

= Pj, j = 1, ... ,2g+2. Then the orbit of each spe-

cial point P/t) = Gt(P j) is homotopic to the constant curve Pj" Furthermore if Gt(Pk) = Pk for 0

s t s 1,

k < j, a homotopy hs(t) may

be chosen so that h 1 (t) = pj(t ), h 0 (t) = P j and hs(t) k

# Pk, 0 s s

s 1,

< j. Proof: The idea of the proof is to show that pj(t) is homotopic to the

product of two curves r1 and r2 such that r 1 lies on cj_ 1 and r 2 lies on cj" It will then be shown that as a consequence of the symmetry of G, the curves p.(t) and p.(i (t )) are isotopic, which is impossible for curves of J J type r 1 r2 unless p/t) ~ 0. In the following construction refer to Figure 3.

1. To make the technical argument simpler we shall assume that T g is a polyhedron in R 3 such that c 2 j _ 1 lies in the xy plane and c 2 j in the xz plane for each j. Let j be any integer, 1 j ,S 2g+ 1. Let yj(O) be a

s

parametrization of cj, where yiO)

= yil) = Pj, yfh) = Pj+ 1 •

2. Let C 1 be a right circular cylinder in R 3 parametrized by (t, 8),

MAPPING CLASS GROUP OF CLOSED SURFACES

0

< t < 1 and let

f:

c 1 -> Tg

91

be define~ by f(t, ()) '""Gt(Yj_ 1(())). We need

to approximate f by another function f so that a certain set is a 1-manifold. This will become clear in 4.

(.~··J

{"

___ .. '

' ~

,"' ~-_,', ....

'

~--. , ,r• . . ,,

'

,,-

; ...... ,

'\

.. ---

.

P·J

t

-----P·

J

p.(t) J

P· J

FIG. 3

3. Let

E

> 0 be such that Iq (s)- r (s)l
2, If g

however, Ms(T g) is a proper, non-normal subgroup of M(T g), and our results are more limited. Using the representation of T g given in Figure 1, we note [12] that M(Tg) is generated by elements Ul' ... ,Ug,Z 1 , ... ,Zg,Y 1 , ... ,Yg which arerepresentedbythetwists r , ••• ,r ,r , ... ,r ;r , ... ,r , where ug z 1 zg y 1 yg u1 any one of the Z-type generators can be omitted. In this representation the subgroup Ms(Tg) is generated by Ul' ... ,Ug,ZJ, ... ,zg.-l'yl'yg or g-1 other set.: of generators obtained from these by a cyclic permutation of the handles. All such relations are easily obtained from one such set by introducing a cyclic permutation of the handles. The expression for the element

112

JOAN S. BIRMAN AND HUGH M. HILDEN

e which accomplishes (17)

e= X1·

=

this, as a function of the generators of M(T g), is:

(Y1U1Y1)-4x1x2 . . . xg-1 (Y.u.Y.)(Y.u.z.u. 1Y. 1)\Y.u.y.)- 1 1 1 1 1 1 1 1+ 1+ 1 1 1 1~i~g-1

The fact that the right-hand side of Equation (17) is indeed a cyclic permutation of the handles can be verified by calculation, again using the representation of M(T g) as a group of automorphism classes, as given by Equations (12)-(14). Of course:

(18) The generators of M (T g) will be related by the equations: (19)

ui+ 1 yi+ 1 2 i+ 1

e- 1u.e 1

e- 1 y.e1 e- 1 z.e 1

where all subscripts are modulo g. Equations (19), in combination with the relations obtained by replacing

Tl' ... , T2g+

1 by y 1' u1, 21, U2, 22, ...

Ug_ 1 , Zg_ 1, Ug, Yg respectively in Equation (15.1){15.4) then give all relations in M (T g) which arise by regarding T g as a 2-sheeted covering of T 0 • We conjecture that these relations, plus relations (17) and (18), are a complete set of defining relations for M(T g).

MAPPING CLASS GROUP OF CLOSED SURFACES

113

§6. Additional results An interesting tie-in with known work concerns a matrix representation for the braid group Bn originally discovered by Burau, and recently investigated anew by W. Magnus and A. Pelluso [17]. It is shown by Magnus and Pelluso that the Burau representation for B 2 g+ 2 induces a representation for B 2 g+ 2 in terms of matrices which generate Siegel's modular group Sp (2g, Z) for g = 1 and 2, but a proper subgroup of Sp (2g, Z) for g ~ 3. Now, it follows from the present analysis that B 2 g+ 2 can be mapped homomorphically to M (T g), the mapping being into for g ~ 3. Since Sp (2g, Z) is known to be a homomorphic image of M(T g) it follows that a homomorphism exists from B 2 g+ 2 to Sp (2g, Z) which is into if g ~ 3. Thus one might expect exactly the type of representation which is established in [17], and indeed the proof given there is motivated by the same underlying geometric situation in terms of covering spaces which was considered here. We point out also that the analysis presented here offers both a generalization and a geometric interpretat:.on of the well-known but previously unexplained fact that the modular group, i.e., M (T 1 ), is a homomorphic image of Artin' s braid group B 4 • Since M(T 0 , 2 g + 2 ) is a homomorphic image of B 2 g+ 2 (the geometric basis for this homomorphism is established in [14, also 4], and since Ms(T g) is now known to be a homomorphic image of M(T 0 , 2 g+ 2 ), it follows that Ms(T g) is a homomorphic image of B 2 g+ 2 for every g ;::: l. As a final remark, we note that the isomorphism established in Theorem 7 between Ms(T g) and S (T g) has a rather unexpected implication, namely that the mapping class group of the (2g + 2)- punctured surface of genus g has a subgroup which is isomorphic to the subgroup Ms(T g) of the mapping class group of the closed surface. Since M(T g) is a homomorphic image of M (T g- n points) for every g and n, one would really not expect this to be

114

JOAN S. BIRMAN AND HUGH M. HILDEN

the case. For example, for g = 2 we obtain that M(T 2 ) is both a homomorphic image of M(T 2 -6 points) and a subgroup of M(T 2 -6 points), where the former property is easily understood but the latter much more subtle. Stevens Institute of Technology

Acknowledgment The authors wish to express their thanks to Professor Wilhelm Magnus, who stimulated their original interest in this problem, and is never too busy to listen to or discuss Mathematics. Also, to Professor Seymore Lipschutz, who pointed out the relevance of reference 11 to the proof of Lemma 4.1. Finally, to Professor

J. Mennicke, for helpful comments,

REFERENCES [1]

J. W. Alexander, "On the deformation of ann-cell," Proc. N.A.S. 9 (1923), 406-7.

[2]

P. Bergau and~ Mennicke, "Uber topologische Abbildungen der Brezelflache vom Geschlecht 2," Math. Zeitschr. 74 (1960), 414-435.

[3]

J.

Birman, "On Braid Groups," Comm. on Pure and App. Math., XXII

(1969), 41-72. [ 4]

, "Mapping Class Groups and Their Relationship to Braid Groups," Com. on Pure and App. Math. 22 (March, 1969), 213-238. , "Automorphisms of the fundamental group of a closed,

[5]

orientable 2-manifold," Proc, of Amer. Math Soc., 21, No. 2 (1969), 351-4. [6]

H. Coxeter and W. Moser, Generators and Relations for Discrete

[7]

M. Dehn, "Die Gruppe der Abbildungsklassen," Acta. Math., 69

Groups, Springer-Verlag (1965). (1938), 135-206.

MAPPING CLASS GROUP OF CLOSED SURFACES

115

[8] G. M. Fisher, "On the group of all homeomorphisms of a manifold,"

Transactions of the A.M.S., 97 (1%0), 193-212. [9] R. Gillette and J. Van Buskirk, "The word problem and its consequences for the braid groups and mapping class groups of the 2-sphere,"

Trans. Amer. Math. Soc., 131, No. 2 (May, 1%8), 277-296. [10] L. Goeritz, "Die Abbildungen der Brezelflachen und der Vollbrezel vom Geschlecht 2," Hamb. Abh., 9 (1933), 244-259. [11] M. Greendlinger, "Dehn's algorithm for the word problem," Comm. Pure and Appl. Math., 13 (1960), 67-83. [12] M. Hall, Jr., The Theory of Groups, MacMillan Co., (1964), Section 15.4. [13] H. Klingen, "Charakterisierung der Siegelschen Modelgruppe durch ein endliches System definierender Relationen," Math. Ann., 144 (1961), 64-82. [14] W. Magnus, "Uber Automorphismen von Fundamentalgruppen berandeter Flachen," Math. Ann., 109 (1934), 617-646. [15] W. Magnus, A. Karass and D. Solitar, Combinatorial Group Theory, Interscience, New York, 1%6. [16] W. Magnus and A. Pelluso, "On knot groups," Comm. on Pure and

Appl. Math., XX (1967), 749-770. [17] _ _ _ ,"On a theorem of V.I. Arnold," Comm. Pare and Appl. Math. XXII, (1969), 683-692. [18] H. L. Smith, "On the continuous representations of a square upon itself," Ann. Math., 19 (1918-19), 137-41. FOOTNOTE

* The presentation for M(T0 , 2 ~+ i} in equation (2) is not quite the same as that given in the original reference l14], however it follows from the latter by a sequence of calculations which can be described briefly as follows. Denoting equation numbers from reference [14] by a bar, we note that (2.1) and (2.2) are the same as (13a), and (2.3) is the same as the last equation in the set (19). Now, equation (14) in conjunction with (2.1), (2.2) and (2.3). imply equation (2.4). The remaining equations in the set (19) are redundant, since they are a consequence of (2.1), (2.2), (2.3) and (2.4). We omit details of these calculations because we believe they are well known [e.g., see 9] and present no real difficulty.

SCHWARZIAN DERIVATIVES AND MAPPINGS ONTO JORDAN DOMAINS Peter L. Duren

Abstract

Some years ago,

z.

Nehari proved that if f(z) is analytic in Izl

1. The function E{r, s) is known as the Eisenstein series. It has a meromorphic continuation to the whole complex s plane. Moreover E(r, s) satisfies the functional equation (13)

E{r, 1-s)

= x(s)

E{r, s) .

Here x(s) has a simple expression in terms of the r function. What is important for us here is that E(r, s) gives the continuous part of the spectrum of D on r\H. More precisely,

LEON EHRENPREIS

136

THEOREM 1 (Selberg, Roelke). The spectral representation of a func-

tion f on r\H takes the form (14)

f(r)

=1

F(s)E(r,s)ds+I Fj u(r, sj).

Rs= 'h

(15)

F(s)

r

-

dxdy f(r)E(r, s ) 1\H y2

= }._

Problem 2. What are the values sj which occur in (14)? Any information on Problem 2 is probably very important for number theory. For the general group

r

with h cusps in the fundamental domain, we

can construct h Eisenstein series by transforming the cusps successively to infinity and then applying the same construction as for the modular group. Selberg has shown that these Eisenstein series have analytic continuations and give the continuous spectrum on f'\H.

Part II. The three dimensional picture. We now regard G as the three dimensional proper Lorentz group SL(1, 2), that is, the group of 3 by 3 real matrices which leave invariant the quadratic form t 2 - x 2 - y 2 and have determinant + 1. There are three types of orbits of G on R 3 • A. One sheet of hyperboloid of two sheets. As in the case of the upper half-plane H, we can identify this sheet with the quotient of G by the subgroup leaving a point fixed. In particular we see easily that the subgroup leaving (1, 0, 0) fixed is just the group of rotations about the t axis which is exactly the group K. B. The part of the light cone with t > 0 or t fied with G/B where

< 0. This can be identi-

AN EIGENVALUE PROBLEM FOR RIEMANN SURFACES

137

1 b

B=HotH, C. Hyperboloid of one sheet. This can be identified with G/ A where A

= I(

a

0

0 a-1)} .

The differential operator which is invariant under G is the wave opera tor

The relation between [ l and D is separation of variables. The general theory of separation of variables is rather complicated (see [1]). For our purposes we can think of it as follows: Let h be a solution of the wave equation (16)

0h = 0

in the interior of the forward light cone. Suppose that h is also homogeneous, that is h(rt, rx, ry)

(17)

Thus h can be thought of as a function ( 1, 0, 0), that is, (18)

= rsh(t, x, y) . l:i on the sheet of hyperbola through

l:i is a function on G/K which is the upper half-plane H. Dh =

- s (1- s) l:i .

The relation just described between [] and D is the analog of the classical relation between homogeneous harmonic polynomials in Rn and spherical harmonics on the unit sphere in Rn. It is important to note the following: If h satisfies (16) in the forward

light cone and is small at infinity (in a suitable sense) then we can decompose h into homogeneous parts by use of Mellin transform, that is, (19)

LEON EHRENPREIS

138

where the hs satisfy (17) and are given by

(20)

If

hs(t, x, y) =

0

h

= 0 then

D

hs

J

h (rt, rx, ry) r-s d;

= 0 since [l commutes with scalar multiplication.

We can think of h as a generating function for the hs. Now, let 1 be a discrete group as in Part I. If h is 1 invariant then so are all the functions hs because 1 commutes with scalar multiplication. We wish to show how to construct 1 invariant solutions of the wave equation. Let us take the case where 1

=

SL (2, Z) is the modular group. When

we think of G as SL (1, 2), 1 becomes a group of 3 by 3 matrices whose entries are integers. We denote this subgroup of SL (1, 2) again by 1. The orbit under 1 of the point (1, 1, 0) on the light cone is a discrete set of points 1 (1, 1, 0) on the light cone, which is contained in the set of integer points. [1 ( 1, 1, 0) is the set of all lattice points on the forward light cone which satisfy a certain "relatively prime" condition.] We can also write

1(1, 0, 0) in the form

1(1, 1, O)

=

{y (1, 1, O)!Y< 1110

Here 1 0 is the subgroup of 1 leaving the point (1, 1, 0) fixed; 1 0 is, as in Part I, the group

We denote by d =

!y f 111

:y

(1, l, O)

the measure formed by unit masses

at each of the points y (1, 1, 0). We can use d to construct 1 invariant solutions h of

n

h

=

0 in two ways.

Method a. Regard d as a measure in R 3 and form its 3-dimensional

Fourier transform. More precisely, form (21)

d (t,

x, y) =

! exp[(t, x, y) .

y (1, 1, O)]

139

AN EIGENVALUE PROBLEM FOR RIEMANN SURFACES

Since d is

r

invariant, so is d . Since the support of d is contained in

the light cone,

0

~

d

= 0.

Method b. Regard d as a measure on the forward light cone and solve the Dirichlet problem (see [3]) for

D

in the interior of the forward light

cone with Dirichlet data equal to d. We thus construct a function d 1 satisfying (22)

0

in forward light cone

(23)

d

on boundary.

Of course, (23) has to be taken in a suitable limit sense. Since

r

commutes with

D . it is easily seen that

dl is y invariant.

We can now state THEOREM

2. (Poisson summation formula). d 1 = d.

The reason for calling Theorem 2 a Poisson summation formula is as follows: the classiaal Poisson summation formula states that the sum of unit masses at all lattice points is its own Fourier transform. On the other hand, d is the Fourier transform of the sum of unit masses placed at some ~

lattice points. Certainly d ,J, d because

0

~

~

d = 0 so the support of d can~

not be such a small set as a set of lattice points. Instead of d = d we ~

have d

~

= d 1 which means that d is the Dirichlet data of d .

We can use the explicit solution to the Dirichlet problem to find an explicit formula for d 1 • We can then construct the functions d 1 , s by means of (20). This was carried out by the author and Mautner. The result is

cf1 ,s (r) = E(r, s)

(24) ~

We can also form ds (r ). We find (25)

Thus our Poisson summation formula gives the functional equation (13) for the Eisenstein series just as the ordinary Poisson summation formula gives the functional equation for the Riemann zeta function.

140

LEON EHRENPREIS

An analog of Theorem 2 holds for all groups of Riemann surfaces with

r

r

which are the fundamental

\G of finite measure. There are also

analogs for general semi -simple Lie groups. The details of Part II will appear in a book [l] being prepared by the author. BIBLIOGRAPHY

[l] L. Ehrenpreis, Representations of Semi-Simple Lie Groups, in preparation. [2] I. M. Gelfand, "Automorphic functions and the theory of representations,"

International Congress, Stockholm, 1962. [3]

J.

Hadamard, Lectures on Cauchy's Problem, Dover, New York, 1952.

[4] A. Selberg, "Harmonic analysis and discontinuous groups," Jour. Indian Math. Soc., Vol. 20 (1956), pp. 47-87.

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS Hershel M. Farkas* Introduction

If S is a compact Riemann surface of genus g, g?: 2, then the vector space of holomorphic quadratic differentials on S is a 3g- 3 dimensional space. We shall denote this space by A2 • If S is not hyperelliptic, then by Noether's theorem [6], one can choose a basis for A2 from the g(g+l)/2 quadratic differentials (i (j , i?: j = 1, ••. , g where ( 1 , ... ,(g is a basis for the holomorphic abelian differentials on S. Choosing such a basis we can then express each (i (j as a linear combination of the basis elements. Our problem is to determine the constants in these expressions. In the case g = 2 dim A2 the case g

= 3 dim A2 = 6

=3

and

(f ,(1( 2 ,(i is a basis for A2 •

In

and there are two cases to consider: case (i)

S is not hyperelliptic. In this case (; ,(1(

2 2 ,( 1 ( 3 , ( 2 ,(2 ( 3

,(i

is a

basis for A2 • Case (ii) S is hyperelliptic. In this case (; ,(1 ( 2 ,(1 ( 3

,(i ,( ,(i 2( 3

is not a basis for A2 and we can exhibit a

linear relation between these six quantities with the coefficients of the relation appearing explicitly as derivatives of a certain theta null. In this paper we shall be primarily concerned with the case g = 4 and consider the two situations of no even theta null vanishing on S, and precisely one even theta null vanishing on S. The methods used however are quite general and apply to any genus g?: 2. We consider the particular case g = 4 in order to show how the Schottky relation arises from the relations obtained. In part I of this paper I shall briefly describe the key result which is *Research partially supported by NSF Grant GP-12467.

141

HERSHEL M. FARKAS

142

needed to establish the desired relations. Details will be appearing elsewhere [2]. The results have already for the most part been announced in

[1,3]. The main result is that one can attach 2 sets of theta functions to a compact Riemann surface of genus g and that these two sets of functions are related in a very special way. In part II, I shall show how this result enables one to write down many relations between quadratic differentials. I.

Let (S,

r ,ll)

denote a compact Riemann surface of genus g;::: 2 together

with a cannonical homology basis

r

=

y1 ' ... ,y g

Ll

=

01 ' ... ,o g and let

,(g denote the basis for the space of holomorphic abelian differentials

( 1 , ...

dual to the cannonical homology basis

r0. 'i

'=

r ,ll.

Then

I Yj ,.1

=

0.. and IJ

rrij . It is well known that the g X g matrix II = (IIij) is a com-

pldx symmetric matrix with positive definite imaginary part. We shall call the g

X

2g matrix (I II) the period matrix for (S,

r ,ll)

where I denotes the

g x g identity matrix. We define a map¢: S -+ Cg as p-+ (¢1(p), ... ,¢g(p)) where ¢j(p) =

(. P (. and p0 is a fixed point on S. ¢ 1. is a finite regular multivalued Po J function on S, the multivaluedness arising from the dependence on the path of integration. In order to make the map ¢ single valued we simply identify all possible images of a point p. Since any two images of p can differ only by an integral linear combination of the columns of the g x 2g period matrix for (S,

r, ll)

we identify all such points in Cg. Cg under this iden-

tification is a compact commutative complex Lie group of complex dimension g called the Jacobi variety of S denoted by J(S). Definition 1. A meromorphic multivalued differential V on (S,

r

,ll) is

said to be multiplicative (or a Prym-differential) with characteristic [ E(] = [ E 1 , ... ' E g ] providing continuation of V along y 1.(8 1.) carries V E, E 1 , ... , E-g', into (-1) Ej V ((-1) EN)

where the q

and E[

are elements of Z/(2).

PROPOSITION 1. Let [ E,] be any non -zero characteristic. Then the E

set of everywhere finite multiplicative differentials with characteristic

143

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS

[ £,] on (S, £

r , ~)

is a (g- 1) dimensional vector space.

In this discussion we shall be concerned only with the vector space of everywhere finite multiplicative differentials with characteristic

(0, ... ,0)

£=

£'= (1,0,~u.O).

t/J = [ ~,] ,

If V 1 , ••• ,vg_ 1 isanybasisforthisspace

we can form the (g- 1) x 2g period matrix (AB) where i= 1, ... ,g-1 j = 0, ... ,g-1

A= (A .. )A .. = lJ lJ

and i = 1 , ... ,g-1 j = 0, ... ,g-1

B = (B .. ) where B .. lJ

lJ

It is then possible to choose a new basis W1 , ••• , Wg- 1 so the period matrix

will have the form

1 0 .••••• 0

0

1

T 10

T 11

• • • • • •

T 1g-1

T21

0 •....

0 1

•T

Tg-10Tg-12

g-1 g-1

We shall refer to W1 , ••• , Wg- 1 as the normal basis of everywhere finite multiplicative differentials with characteristic PROPOSITION

t/J

on (S,

r , A).

2. The matrix r = (rij) i, j = 1, ... , g-1 is a complex

symmetric matrix and has positive definite imaginary part. We recall the definition of

~

g• the Siegel upper half plane of degree g.

It is the set of g x g complex symmetric matrices with positive definite

imaginary aprt. Hence what we have shown till now is that we can attach to each (S,

r, ~)

an element of ~ g and an element of ~ g- 1 •

HERSHEL M. FARKAS

144

Definition 2. Let Z

=

(Z 1 ,

••• ,

Zg) be a complex g-vector and T = (tij)

an element of 6 g· The first order theta function with g -characteristic [ :,] = [

E~

···

E~]

E 1' .. E.g

where Ei and Ei are elements of Z, 0[ :,](Z, T),

is defined by the following series which converges absolutely and uniformly on compact subsets of Cg x @5 g.

where N runs over all integer vectors in Cg and all operations are matrix products. The theta constant or theta null with g -characteristic [ E,] at T E E E is 0[ E'] = O[ E,](O,T). It can be shown quite easily that the theta function

e[ EE,] (ZIT)

is an

even or odd function of Z depending on whether E · E' is even or odd. If E . E, is even (odd)

e[ :, ](z, T)

z.

is an even (odd) function of

Hence

we see immediately that all odd theta constants vanish.

Definition 3. The Riemann theta function with characteristic [ E

E

, ] asso-

E

r' 8) :s e[ E,] (Z, II) and the Schottky theta function as(S, r, fi) is Tf [ :, ] (Z, r), that is the theta function with

ciated with (S, sociated with T = r.

Note: The Z, II variables in the definition of Riemann theta function are an element of Cg x ®g while the Z, r variables in the definition of Schottky theta function are an element of C g- 1 x ® g- 1 For a fixed II the Riemann theta function is a function on Cg. It is not

J (S) since Z 1 = Z 2 does not imply that ~,](z 2 , TI); however it is true that 0[ ~,](z 1 ,1T)

a well defined function on 0[

;,](z 1 ,

TI) = 0[

=

EO[~ ,](z 2 , II) where E is some exponential multiplier. Therefore, the zeros of the theta function are well defined on

cp : S -+ J (S)

J (S).

Utilizing the map

described earlier we can view the theta function

cp [ :, ] (¢(p), II)

as a multivalued function on S with well defined zeros. The properties of this function are discussed in [5].

145

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS

Returning now to definition and proposition 1 we can construct an

(S, f,

~) where

S is a

smooth two sheeted cover of S and where

f, ~

is a cannonical homology basis for S constructed in a natural way from

r,

l'l.. The properties of

S are summarized in the following proposition.

S is a smooth two sheeted cover of

PROPOSITION 3.

S, therefore of

genus 2g-l, on which all multiplicative differentials with characteristic tjJ are single valued. If ~1

, ••• '~g;

s

of holomorphic differentials on

W1 , .•• , Wg_ 1 are the respective bases

dual to

r' ().

and normal basis of every-

r, l'l.)

where finite multiplicative differentials with characteristic tjJ on (S, then denoting the lift of ~i by ?i and the left of Wi by

W1 .•. Wg-l entials on

Wi' ~ 1' •.. , ~ g,

is a basis for the vector space of holomorphic abelian differ-

S. S permits a conformal fixed point free involution

T, the

sheet interchange, and one can choose a canonical homology basis for

S in such a way that

T(y i)

basis

T

= Yg+i- 1 i = 2 •.. g to f,~ is ~ 1 = ?1

t + w.

1

1 1-------r-

i

G1)

is homologous to

and T(Bi)

= 2, ... ,g

2

y1 , T (B 1) = § 1 .

= §g+i- 1 i = 2 .•. g.

ug+i-1

=·

f, ~

~-1

w.

1- 1

The dual

i = 2 , ... ,g •

2

IT) is the period matrix for (S, f .~) and if we denote the Riemann theta function associated with (S, f, ~) by 0 then at least 2g-Z (2g- 1 -1)

(1,

~

E

even theta constants () [ E,] vanish. One of the nice features of

S is

that aside from its theta function

0

we may also view the Riemann and Schottky theta functions associated with (S,

r,

~

l'l.) as multi valued functions on S. This is done via the maps p

....

(1

p w1, ..••

Po

The natural question to ask is what is the relation between This is partially answered by the following proposition.

0, ()

f

p wg-1)

Po

and TJ ?

HERSHEL M. FARKAS

146 PROPOSITION

~

Pf

F g-1

4.

(8-p).o' [ P glo-p (-1) 7J ](t 1 , ••• ,tg-1'2r)O[h 0 ](s 1, ... ,sg,2IT) 0

1

where F = Z/(2) , [ g1 ] is a 1 ~haracteristic [ ~,] is a g- 1 characterh1 u istic and

~ ~ g 0 0 In particular as functions on S we have 8 [ h1 0 ,0 ,]

1

We have already mentioned in proposition 3 that at least 2g- 2 (2g- 1 -1) even theta constants associated with (S,

f,.:i)

vanish. The next proposition

tells us which theta constants vanish. The proof of the next proposition was obtained jointly with H. E. Rauch [3]. ,....

QE

E

A

"""

8 [ 1 E, E,] (¢,IT) = 0 on (S, f' ,Ll) for all choices of base point p0 where [ ~,] ranges over all the 2g- 2 (2g- 1 - 1) odd g -1 characteristics. PROPOSITION 5.

From propositions 4 and 5 and a little bit of algebra we can deduce

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS

PROPOSITION

6. 17 [ ~ 0[

for any vectors

147

€.,

(j

Hcf>w, 2r)

77 [

o Hcf>w • 2r)

~ ~](¢,,2ll)

8

f

Fg-l and any base point p0 with F = Z/(2).

It now follows from proposition 6 and known formulae [ 4 p. 233] relating

theta functions with parameters

n

and r to theta functions with parameters

2TI and 2r that PROPOSITION

7.

S and any even

for any choice of base point p 0 on €.

g- 1 characteristics

(j

[E,],[(j,]. The preceding formula is the key to our method of obtaining relations between quadratic differentials on S.

II. We now consider the following expression: 77 [

€. E' ]

8

€.

77 [ s '] (cf>w • r )

8

11 [ 8,] 11 [ 8, Hcf>w , r >

HERSHEL M. FARKAS

148

Q(P)

and observe that in a neighborhood of the point p 0 , the base point of the maps ¢(, and ¢w, this defines a holomorphic function on §. (1) Letting Z be a local parameter about p 0 we can write out the Taylor expansion of Q. By virtue of proposition 7 however, all the coefficients must be zero since Q = 0 on

S.

We shall here be concerned only with the first three

coefficients of the Taylor expansion. Our first lemma is the key to deriving the Schottky relation and Schottky type relations for g > 4 in [2 ,31 LEMMA

1. For any two even g-1 characteristics [

~,]

, [

~,]

we

have OE. OE. 0[ 0 E.'] 0[ 1 E.')

=

o8 o8 O[o8'10[18'1

Proof: This is simply the statement that the constant term in the expansion of Q is zero. We have defined ¢t;!-p)

= ((

p

Po

~

(, 1, ... ,

(

( ( Pw 1 , ... , fp Wg_ 1). Weshalldenote Po Po ti. Computing now Q (p) we find

fz

d -Q(z(p)) dz

i

a o[

1=

o[

· 1 dsi

~, 1

a

8 71[ 8'1

atk

g- 1 11 [ =

k 0 E.

I

=1

p 5'

Po

£,g) and ¢w (p)

f/~i

by si and

(:Wi

o 11 [

o

~, H¢w, r)

8 11[ 8'H¢w, r)

0 E. 0 E. E.,Jo[ 1 E. -H¢c;,.m + o[ 1 E.'1 2

183 we have

we have

~ ] = 04 [ ~ ] + 04 [ ~ ].

151

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS

e( OOO]O( OOO]O( OOO]O( 000] 000

100

010

110

e( o o o] e( o o o] e( o o o] e( o o o] + e( o o 1 Je( o o 1 Je( o o 1 Je[ o o 1 ] 001

101

011

111

000

100

010

110

Proof: The first half of lemma 3 follows quite simply from lemma 1 at

least in the case where II is a period matrix. In this case the theta constants are Riemann theta constants and by lemma 1 they are proportional to the Schottky theta constants which are genus 1 thetas. Hence we get the first half of lemma 3. The second half, again in the case where II

f

~3

is a

period matrix, follows from the first half of lemma 3 and lemma 1. The cases where the IT's are period matrices in ~ 2 and ~ 3 have the dimension of ~ 2 .~3 respectively and hence the result must hold for the whole of s2 and

~3" We now rewrite the second identity in lemma 3 in the following form 7J[ 000] 001 77[000] 000

77[000] 101 77[000] 100

77 [ooo] 0 11

77[000] 111

77[000] 010

000] 77[110

+

77[ 001] 000

7J[ 001] 7J( 001] 77[ 001] 100

010

77[ 000] 77[ 000] 77[ 000] [ 000) 110 ooo 010 11 too

"' 1 and observe that since it is an identity in ~ 3 its gradient must vanish identically on ~3 • In particular these remarks hold for Schottky theta constants on a surface of genus 4. In the above formula we have eight quotients of Schottky theta constants. Denoting them by q 1 , ••• , ~ and the corresponding square roots of quotients of products of Riemann theta constants via lemma 1 by r 1 , ••• , r8 we clearly have LEMMA 4. For any rkf

kf "' l, ••• ,3

110

HERSHEL M. FARKAS

152

Proof: This is simply the statement that the gradient of ql q2q3q4

+ q5q6q7q8- 1 vanishes on s3.

THEOREM

2. Let S be a compact Riemann surface of genus 4 together

with a cannonical homology basis

r

,Ll. Let ( 1 , ••• ,(4 be the basis of the

holomorphic abelian differentials on S dual to

r

,Ll. Then

0 . Proof: By virtue of lemma 2 we have the eight relations

Multiplying the left hand side of the first relation by q 2 q 3 q 4 and the right hand side by r2 r 3 r4 which is clearly permissable by lemma 1, and continuing

RELATIONS BETWEEN QUADRATIC DIFFERENTIALS

153

in this fashion finally multiplying the left hand side of the eighth relation by q5 Bq(f'I1,U11) and, hence, we obtain a mapping of each Riemann surface sl1 into the same projective space P (Bq(f', U)*). It will turn out that the mapping N11q will depend only on the Teichmiiller class of 11· A similar mapping is also supplied by the theory of moduli. This

Qi

mapping will be in some sense natural, but will depend on 11 and not just the Teichmiiller class of 11· This paper investigates how the embeddings induced by

N:

and

Q:

vary as 11 varies. In Section 1 we review well-known facts about the mapping «q. In Section 2 we define the mapping

Ni

and, following Bers [5], we

define a fibre space F over Teichmiiller space, each of whose fibres is a surface of fixed genus with a complex structure determined by the image point in Teichmiiller space. Then we show how N q induces a natural 11

mapping 'P q from F to P(Bq(f', U)*). In Section 3, we show that 'P 2

:

set in P(B 2(f',U)*). In Section 4, we define the map

F -> P(B 2 (f', U)*) maps onto an open

Qi.

As far as we know, this mapping

really depends on 11 and not just the Teichmiiller class of 11· The only excuse we give for trying to give an analysis of it is that the analysis itself is interesting. In Section 5 we show that

Q:

induces embeddings of the surfaces

s11

in such a way that they fill out an open set containing the image of S in P (B 2 (f' , U)*). Many of the computational devices employed here are taken from Ahlfors' paper [l] and the basic mappings used are modifications of ones first explicitly written down by Bers [ 4, page 129 ]. §1. Facts about the map «1>. In this section we present some facts

about the map «q: S-> P(Bq(f' ,U)*) where S = U/f'. All of them are classical and we do not give proofs. First of all, note that the fact that there always is a non-vanishing q-differential (for q ~ l) at any point pI' U/f' means that « is defined everywhere on U/f'.

159

DEFORMATIONS OF EMBEDDINGS OF RIEMANN SURFACES Pick a point p

E

of p such that z(p)

S

= U/r

= 0.

and a local parameter z in a neighborhood

l¢>1' ••• ,¢>nl is called a basis of Bq(r ;u)

adapted to p if it is a basis of Bq(r, U) and the power series developments of the ¢>i's have the form ¢>i(z) = z Vi+ ali z Vi+ 1 + ... where 0

=

v1

< v 2 < ··· < vn.

for l

S iS

n and

The numbers vi are functions of p and do

not depend on the local parameter z. DEFINITION. Unless each vi(p)

= i- 1

for 1 .S i

.S n,

p is called a

q-Weierstrass point. THEOREM (Weierstrass). Let W(p)

= Ir= 1 (vi(p)-(i-1)). Let the

genus of S = g. Then

I

W(p)

= (2q -1) 2 (g-1) 2 g

if q > 1

pES and

I

W(p) = (g + 1) g(g-1)

if q = 1 .

pES In particular, the number of q -Weierstrass points for fixed q is finite. DEFINITION. S is hyperelliptic if for q

=

1 there is a point p

E

S

such that v 2 (p) > 1: LEMMA. If S is not hyperelliptic, then for all q vip)

~

1 and for all p

E

S,

= l.

THEOREM. If S isnothyperelliptic, then the map q: s~ P(Bq(r,U)*)

has non-vanishing derivative for all q ~ l and all pES. In fact, ~ vanishes precisely at those points where vip) > l. THEOREM. The map 1 : S ~ P(B 1(r ,U)*) is an injection unless S

is hyperelliptic and in this case 1 is two- to-one. The map q: S-. P(Bq(r, U)*) for q > 1 is an injection unless q = g

= 2 and, in this case, all surfaces are hyperelliptic and the mapping is

two-to-one.

160

FREDERICK GARDINER

§2. The fibre space of Teichmiiller space and the natura/linear map N11q. In all of this section we are summarizing notation and results given in Bers' Zurich notes [5]. First, let us define Teichmiiller space. Let r be a Fuchsian group operating on the upper half plane U. Much of what we say will apply in this setting but, of course, if we want the results of Section 1 to apply as stated, then we must assume S = U/r is a compact Riemann surface of genus g. Let M(r)

= Ill ill

is bounded measurable complex valued func-

< l, IJ.(Z)

=

rl. Let l*(r) = !w/L ill

f

tion on C, 111111 00

;tZ'). IJ.(y)y'(z)y'(z)- 1 = IL(z) forall M(r),

J w11

and w11 (0)

= 0,

w/1)

y

f

=

l, w/L(oo) = ool. The existence and uniqueness theorem for Beltrami

equation tells us that M (r)

J

f->

IL

w11

mal subgroup l*(r) by lJr) = lw11

= /La w/L

l*(r) is a bijection. Define a nor-

f

l*(r) I w11(x) = x for all x

f

f

Rl.

Teichmiiller space is defined to be T(r) = l*(r)/l*(r). It is well

-

-

known that if we take l*(r) (respectively, lJr)) to be those elements of l*(r) (respectively, lJr)) which are smooth, (Ck), in the interior of

-

-

U then T (r) is also given by l*(r)/l Jr), at least in the case when u/r is compact. We will need this fact later. Define w IL to be the unique quasiconformal homeomorphism of C which is normalized by wll(O) = 0, wll(l) = l, wll(oo) = oo and which satisfies

awll = ~a wll

where ll (z)

; (z) = { 0 Fix the following notations: U IL

for

z

f

U

for

z

f

L

= w ll( U), Lll = w ll( L) , C IL = w ll( R) ,

and if 11 ( M(r), rll = wllrwll- 1 • It is well known (see [2]) that Ull, C ll, and Lll depend only on the class of ll in l*(r)/lJr). In [5], Bers has shown that a system of complex analytic local charts for T(r) can be obtained in the following way. Let B 2 (rll ,Lil) = the holomorphic quadratic differentials on Lll with respect to rll. Suppose [w

flo

]

f

l*(r)/l*(r) and a

f

M(r). Define v by the relation wa = wvow 11 0.

161

DEFORMATIONS OF EMBEDDINGS OF RIEMANN SURFACES

Clearly v will have support in U IJ.o U C IJ.o and w /J. will be holomorphic in L11 0. Let be the Schwarzian derivative of wv in L11 o. Then for a sufficiently small neighborhood of [ w

IJ.o

[w)

-+

] in T(['), the mapping

B 2 (r IJ.o, L/J.O) is a chart. And, in fact, the tangent space

E

to T ([') at the point [ w

IJ.o

] can be identified with B 2(r 11 o, L/J.O).

Now we can define a fibre bundle F over T ([') in such a way that the fibre over each point [ w ] is a Riemann surface. -

F

=

{(z, [w ]) I z 11

/J.

E

UIJ. and [w ]

wu

f

E

•

C I I z-z 0 1
0 such that if llv lloo BE(z 0 ) = I z

U /J. /r 11. Let

=

/J. (z 0 , [1J. 0 ]) E F.

proceed as follows. Let boundary of U IJ.o be 2 E

E

S /J.

]) in

{[w 11 wu = wv. w 11 o, ~

F given by

]). In this neighborhood a chart is given by

(z, [wu])-+ (z, ). The choice of 8 and E that we have made assures that if z

f

BE (z 0 ), then z

one: ;(z, [ w/J.])

f

U u. The mapping ; :

F -+ T([')

is the obvious

= [w11 ]. There is a complex analytic action of r on

F

with respect to which the above system of charts are automorphic. This action is given by y(z, [w ]) _/J.

=

(yl1(z), [w ]), where yll /J.

=

wl1ywiJ.-l

f

r/J..

Hence we can form F = F /r and we obtain a fibre bundle rr : F -+ T([') which has holomorphic structure. Now it is time to define the mappings NJ: Bq(r,u)-. Bq(rl1, Ul1). We will simply write down the formula and from the fact that the behaviour of wll in L depends only on the Teichmiiller class of /J. it will follow that Nq 11

depends only on the Teichmiiller class of /J.·

FREDERICK GARDINER

162

Here z is an element of U~-' and A. is the Poincare metric for L: A.(() ... 1!1(-~\. By Theorem 2 of [6) one can see that in the case q = 2 this

formula is nearly the same as the derivative of right translation by [ w- 1] IJ.

at the point [ wIJ.] if representations of the fibre of the tangent space to T(r) at [w) and 0 are chosen properly. We do not need this result for what follows so we shall not attempt to explain it. From Bers [5] we know that NJ: Bq(r ,U)-+ Bq(rll, U~-') is a bijective linear mapping. We can now proceed to study an induced family of mappings of the surfaces

S~-'

in P(Bq(r, U)*). These mappings will be embeddings in almost

all cases. Section 1 describes the exceptions when U/r is compact of genus g. The family of mappings will vary holomorphically. Define

= the

'II q: F-+ P(Bq(r, U)*) by 'II q(z, [ wiJ.])

triviallinear functional fz, where fz (cp) ,.,.

class of fz

,IJ.

•IL

projective class of the non-

= (NILqcp)(z).

The projective

depends only on ( w ] and the images of z in U IL /r IL so IJ.

the mapping 'II q is well defined on F. It is clear that 'II q restricted to TT-l ([ wIJ.])

= U IL /r IJ.

has all the same

properties as the q, q described in Section 1 because it differs from it only by composition with the complex linear bijective map NJ. §3. The variation of the map 'Pq: F-+ P(Bq(r ,U)*). From the definitions made in Section 2 it is clear that 'II q lifts to a mapping Wq: F-+ Bq

=

f

M which

(~(z 0 ), [wal). Note that it suf-

(projective claSS of fwV(zo), a) has a SUr-

jective derivative at v = 0. We do not change the projective class of ewV(zo), a if we multiply it by

a~(zo) 2

and we can assume that

awv(zo)

is some well-defined number because we can assume that 11 is smooth in

rfo. By an obvious manipulation of the formula used in Lemma 1, we see that

the above problem will be solved if we show the following lemma.

2. Let ~ .Zo be the conjugate complex linear map on Bir llo ,LILO) defined by LEMMA

~ ' z o(t/1) = Then as

!

((,)awv((,)2 wv(zo)2 d(, d(,

0

Lllo

v varies in lv I llv I

v(yll(z))yll,(z)

a

t/1 ((,)\

(wv((,) -wv(zo))4

00

= yll'(z)v(z)

< 0'

supp v -c ullo u cllo and

forall yll (riLl,~ z '

hood of h 0 z '

in the conjugate dual space to 0

fillsoutaneighbor0

Bir llo,

Lllo).

To prove Lemma 2 we will use several ideas of Ahlfors in [l]. Let K(z,O = 1/(z-(,) 2 and KIL(z,(,) = K(wiL(z),wiL((,))awiL(z)awiL((,). Then

165

DEFORMATIONS OF EMBEDDINGS OF RIEMANN SURFACES

LEMMA 3. Let supp J1. ~ ullo

u cllo. Let

'f

LJl.o and Zo

f

ullo.

Then K ll(t;;, z 0 ) is a complex holomorphic function of J1. and

where t is a complex variable. This formula can be viewed as a consequence of one derived by Ahlfors in [1, page 170, (7.3)]. Here we shall derive it using the power series technique. There is no loss of generality in assuming that Jl has compact support because the formula is invariant under application of linear fractional mp11pings A if we establish the convention that K(A(t;;) ,A(TJ)) A '(t;;) A '(TJ) = K0 (t;;,TJ) where A(D) = U. And we can arrange for D to be the interior of the unit circle by picking A to be defined by A(z) = - i (z+ i)/(z- i) . Define (as in [2]) for v with compact support Pv(z) = - 12rri

f

C

and Tv(z)

_!_ p.v. 2rri

v(t;;)dt;;dt;; (t;;- z)

f C

v (t;;) dt;; d( (t;; _ z)2

Then it is proved in [3] that wv(z)

= z+Pv(z)+PvTv(z)+··· 1+Pv1+Pv1+···

It is easy to see that the denominator in this expression will cancel when

we differentiate Ktv at t

= 0. In fact,

166

FREDERICK GARDINER

t K v(z, ()- K(z, ()

1

(1 + tTv(z) + o(t)(1 +Tv(()+ o(t)

= ----------(z-( + t(Pv(z)-Pv(()) + o(t)) 2 1

Pvz-Pv(

(z-() 2

z- (

----= [ Tvz +Tv (- 2

+ o(1)] .

Hence, d

tv

1

dfK (z()l t= 0

(z-() 2

[Tv z + Tv ( - 2 Pv z - Pv ( ] z- (

If we collect the integrals P and T all into one integral in this expression, a simple calculation shows that the resulting kernel is

This shows that

and completes the proof of Lemma 3 because supp v C U 11 o U C Jlo. To prove Lemma 2 we start by differentiating the mapping v

->

h z at v, 0

v = 0. One finds that

-.ih dt tv,zo I

(I/1)

t= 0

In this expression TJ varies in u 11 o, ( varies in L11 0, and z 0 is a fixed point in U llo. What we want to show is that as v varies in L ([' Jlo, U 11 0) IP!supppcu 11 ou

c 11 o,p(yll(z))yll'(z) = yll,(z)p(z),IIPIIoo ~

ool, then

=

167

DEFORMATIONS OF EMBEDDINGS OF RIEMANN SURFACES

I

d h tv,zo t=O dt yields all conjugate linear functionals on B 2 (rllo ,LILO). Hence, we must show that if

for all such v then

t/J

= 0. After changing the order of integration (which

we will justify later) one sees that this reduces to showing that if

(*)

then

t/J

= 0.

Define a mapping ~: B 2 (r1Lo, LIJ.O) .... B 2 (r!LO, UIJ.O) by

We know from [4] and [5] that ~ is a bijection. Hence, to show suffices to show ~t/J

=

t/J =

0 it

0. The formula (*) evaluated at 7J = z 0 tells us

that ~ t/J (z 0 ) = 0. If we differentiate (*) k times with respect to 7J and evaluate at 7J = z 0 we find that (*) tells us that

Hence, ~t/J = 0 in ullo. To justify the change in order of integration and the application of duality between L 1 and L 00 we restrict attention to those elements v of L (rllo, ullo) such that supp v C ullo U cllo- B (z 0 ) where a oo

-

a

>

0.

Then the whole integrand is absolutely integrable as a function of ( and 7J over LIJ.o x U 1-Lo. Since a is arbitrary we easily see that (*) is valid. This completes the proof of Lemma 2 because of the inverse function theorem. Therefore, Theorem 1 also is proved.

FREDERICK GARDINER

168

Summary. The image of 'I' 2 : F that TT: F

-+

-+

P(B 2 (r, U) *) is an open set. Recall

T(r) . If genus (U/r) > 2 then 'I' 2 restricted to rr - 1([ w

])

llo

is an embedding for every [ w

llo

]

T(r). Moreover, 'P 2 is a holomorphic

f

mapping. §4. The mapping Q~. Just as with Nq, the idea for constructing Qq ~

ll

ll

is motivated by computing the derivative of the right translation map between T(r) and T(rll). Let r be a covering group for a compact surface S of genus g operating on L. So S = L/r. Let [w ]

f

is a translation map Rll: T(r )

T(r) defined by Rll([ w~) = [ wa o w ] .

ll

If we let

-+

T(r) and r

ll

=

w rw- 1 . Then there ll

ll

ll ll ll be of the form A- 2 (t;';) tjJ(t;';) for some quadratic differential tjJ,

then it is an easy computation to show that

Clearly this formula still has meaning even if we allow ll to be an arbitrary element of M(r). We define

Q;t:

Qllq(,~.. '+' )(z) =

Bq(r, L)

f

-+

Bq(rll, L) by

A2 - 2 0 and V(g, R)

< oo

•

Proof: From [7] and [18] we see that (11) holds for an arbitrary ring R if the supremum is taken over the larger class of functions g which are Borel measurable in R with A(g, R) and V(g, R) not simultaneously 0 or ""· Hence in order to establish Lemma 5, it is sufficient to show that when

> 0 there exists a function g which is nonnegative and continuous in R with A(g, R) > 0 and V(g, R)

R is approximable from outside, for each

< oo

E

such that

(12)

capn(R) :;;

A(g,R)n

+

E •

V(g,R)n-1 Fix

E

> 0. Then (10) implies that there exists a ring R' containing

R such that R

separates the components of C(R ') and

(13) Since

RC

R ', the components of C(R) are nondegenerate, capn(R) > 0

by [10], and thus we may assume also that capn(R') > 0. This implies that the components of C(R ') are nondegenerate and hence that there exists a unique extremal admissible function v for R' such that

F. W. GEHRING

182

(14)

(See [6] or[l2].) Choose r so that 0 < r < dist(R, C(R')), and let g(x) = (

_l_ mn(U)

where U =

! IV

1

v(x+ y)[ n-1 dmn(y)\ n-1

U

')

Iy: IYl < rl. Then g is continuous and nonnegative in R,

and from Minkowski's inequality and (14) it follows that (15)

V(g, R) S capn(R ')
0. Then for each E > 0 there exists an open polyhedron G '::> f (E) such that each point of G' 1i es within distance E

of f(E) and n-1 mn_ 1(aG') .::; cK n-p mn_ 1(E) ,

where c is a constant which depends only on n. Proof: Choose 8 so that 0 < 8 < pact with mn_ 1(E) covering E, with

E

and (1 + 8) 2

.::;

2. Since E is com-

> 0, there exists a finite collection of open balls Uk

t\ C D and diaf(Uk) < ~ rn-1 (

E-

8 for all k, such that

dia uk)n-1 4 .:S (l + 8) mn-1(E) •

Next for each k, let Gk be the open polyhedron of Lemma 8 corresponding to Uk and 8. Then G' = Uk Gk

satisfies the above requirements with

5. Main results. We now combine various lemmas of the last two sections to establish the main theorems of this paper. We begin with a result relating the classes Qp(K). THEOREM 1. If f,

r- 1

f

Qp(K) where p ,f. n, then f, f- 1

f

Qn(K 0 )

where K 0 depends only on K, n, and p. Proof: By symmetry, it is sufficient to show that for each spherical ring R = {x: a< lx-PI < b} with

(21)

R CD

187

LIPSCHITZ MAPPINGS

where K0 depends only on K, n, and p. Suppose that n

< p < oo, fix an integer k, and for j = 1, ... , k, let i_

j-1

Rj

= lx: a(~)!{< Jx-PJ

r · a, for a < ~((f'). There are no fixed points, and

it is easy to show that the action is properly discontinuous ([8]). There is an obvious mapping of R 0 (f') onto S(f') given by the rule r which we denote by j. We have

1->

r(f'),

W.j.HARVEY

198

j

THEOREM 2. (Macbeath [8]). The mapping r r----> r([') is a local homeomorphism if 1 is of compact Fuchsian type. Note: It follows that S ([') is homeomorphic to R 0([')/ ~{([') and is a real manifold. The group ~ also acts on R 0 ([') by conjugation. If t by t* the self mapping r x

f

t* ( r) where t*(r) : x

---->

-->

f

~ we denote

C 1 r (x) t for each

1. This mapping is without fixed points since [' is not cyclic. The quotient space R 0 ([')/~* is denoted by T (['). It can be shown to

be the union of two disjoint copies of the usual Teichmiiller space, there being two possible orientations for each Riemann surface U/r(['), r

f

R 0 ([').

One may avoid this slight discrepancy by working with the full hyperbolic group of conformal and anticonformal mappings of U on itself, but then one should consider all discrete subgroups of the hyperbolic group rather than solely the Fuchsian groups. This theory provides a unified treatment of moduli of orientable and non-orientable surfaces. Note:

We remark that R 0([') may be shown to be homeomorphic to the

Cartesian product T([') x ~. since the subspace of R 0 ([') consisting of the isomorphisms normalized appropriately with respect to the action of ~* can be shown to be homeomorphic to T ([') -see for example Bers [2]. This observation is due to Singerman [10].

§5. The Action of ~f ([') x ~· We first sketch the proof of a lemma on mappings between R 0 -spaces. Let K ~ [' be an injective homomorphism between two groups K and [' of Fuchsian type such that i (K) has finite index in 1. Then there is an induced mapping LEMMA 3.

f

i:

R 0 ([')

-->

R 0 (K) given by p 1--+ p

o

i for p

f

R 0 ([').

is a real analytic homeomorphism onto a closed subspace

of R 0 (K). Proof: If x

f

['-

i (K) then, for some integer n > 0, xn

f

i (K). Let

SPACES OF FUCHSIAN GROUPS

r t R 0 (K); in order that r equal p

o

i for some p

f

199

R 0(r), it is necessary

to solve for p (x) the equation

This has essentially one solution in

f

if r(xn) is hyperbolic or parabolic

(in which case p (x) is of the same type). But one need only deal with hyperbolic elements, since every Fuchsian group of our type may be generated by a finite number of hyperbolic elements. It follows that, since

r,

i (K) has finite index in

the correspondence

pI->

r consists in extrac-

tion of a finite number of nth roots, and consequently it is analytic and bicontinuous. The image set is clearly closed in R 0 (K). There is an induced mapping

I from T (r) to T (K) which makes a

commutative diagram with the projections

"r• "K :

A

i is likewise a real-analytic homeomorphism. We consider the transformation group (IJ{ (K) x the actions of IJ{ (K) and

f•,

R 0 (K)); noting that

f* commute, we find that if the quotient space

is denoted by ~ (K), there is a commutative diagram R 0(K) - - - - - S (K)

"K T (K) - - - - ~ (K) PK

where the maps S (K)

~ (K), T (K)

-+

~ (K) represent quotient

maps with respect to the actions induced by ~· and m:(K) respectively. The action f of t G

__!___.. C

1 Gt.

f

~ on S (K) is simply conjugation: if G

Thus

~ (K)

is the space of

f- conjugacy

f

S (K), then

classes of

200

W.

J.

HARVEY

Fuchsian groups isomorphic to K. When K is a surface group of genus

y ?: 2,

!R (K)

is Riemann's space of moduli of closed surfaces of genus y.

!R (K)

Note: We remark that if K is of compact type

of the real manifold S (K) by the action of

f..

appears as the quotient

The fibre PI( 1([ G]) over a

!R (K) is isomorphic to f.;j( (G), where r£ (G) denotes the normalizer in f. of G. It will follow from further discussion that the set point [G] t:

groups G t: S (K) with

of

r£ (G) = G is dense in S (K) except in a few special

cases.

§6. The Loci of Special Branching Let r

f

R 0 (K). We denote by Stab (r) the set of elements in

which fix r. Thus Stab (r) ~ t

E

Yl.(r(K)) and a=

r£ (r (K)), since

m: (K) x f.•

= r if and only if r- 1 o(£)- 1 or. Let L(K) = {u R 0 (K): Yl.(r(K)) #. (a, t*)(r)

r {K)I. For any element r t: L (K) there is a group r of Fuchsian type with r ~ Yl.(r(()), such that for some p t: R 0 (r) there is an injection i: K .... r with poi

= r. This defines a homeomorphism

image lies entirely in L (K). Let

S: (K)

i:

R 0 (r) .... R 0 (K) whose

denote the (finite) family of all

isomorphism classes of groups r of Fuchsian type which contain normal subgroups of finite index isomorphic to K. Then we have, from the above discussion PROPOSITION 4. L (K) is a union of real analytic subspaces of R 0(K), each one homeomorphic to R 0 (r) for some

r

E

1 (K).

More precisely, we observe that for each group, r, and each surjection ¢: r .... H with H finite and ker ¢ ~ H there is an injection ir,¢: K .... ker ¢. If one takes equivalence classes of surjections modulo automorphisms of

r

and of H and isomorphism classes of groups r, there results a corresponding finite set of injections lir ,¢ l such that every subspace of L (K) homeomorphic to R 0 (r) in the above way is a translate of one of the image spaces {Im if,,¢ I by the action of ~{(K). The action of f.* takes any such image onto itself.

SPACES OF FUCHSIAN GROUPS

201

§7. The Teichmiiller Theory We now consider the implications of these results for the space T (K). One finds that since the subgroup ~ (K) (the inner automorphisms of K) of & (K) acts trivially on T (K), it is natural to consider ~ (K) as the quotient of T (K) under the action of the factor group M(K) = & (K)/ ~ (K), known usually as the mapping-class group (or Teichmiiller modular group) of K. It is known that M(K) acts properly discontinuously on T (K). The locus

of fixed points is "K(L(K))

= A(K),

and consists therefore of the union

of a finite number of homeomorphic images of Teichmiiller spaces T (r) with

r ~: :f (K),

together with translates under the action of M(K). One

may obtain simply a description of the fixed point set of an element or subgroup of M(K). THEOREM

5. The fixed point set (assumed non-empty) of any sub-

group H C M(K) of finite order is homeomorphic to a Teichmiiller space

T (r), where

r

is a group of Fuchsian type with an exact sequence

where H' is anti-isomorphic to H. For the proof see [5]. For simplicity we produce the group denote the ~ (K)-equivalence class of y

r f

for H a cyclic group. Let [y] &(K). Suppose [r]

f.

T (K) is

= t* o r. Consequently t normalizes r (K), and the group r = < r (K), t > is

fixed by [y]. Then there is an element t

= t(y)

f

f

such that r

o

y

itself Fuchsian. The exponent of t in r (K) is the order of [y], and there is a natural homomorphism onto the cyclic group

r I r (K).

Note 1. It was shown by Kravetz [6] that when K is a surface group, every finite subgroup of M(K) has non-empty fixed point set. As yet, one does not know if this extends to general Teichmiiller spaces.

Note 2. One has also a converse to the theorem ([5]): whenever there is an exact sequence

W. ]. HARVEY

202

with K and

r

of Fuchsian type and H' a finite group, there is a subgroup

of M(K) anti-isomorphic to H' with non-empty fixed point set

i (T (r)).

Note 3. In the context of the usual formulation of Teichmiiller space (see Bers [2] or Kravetz [6]), one finds that when K is a surface group of genus y, then Teichmiiller classes {[r ]I lying in the fixed point set of a subgroup

H C M(K) correspond to surfaces IU/ r (K)

= Xr l,

each admitting a con-

formal automorphism group Hr anti-isomorphic to H and such that between two surfaces xr, xs in any two classes there is a homeomorphism ers: xr ... xs which preserves the "markings" and commutes with the ac-

= 8 -lHsO. In the case when H ~ Z 2 and

tions of the groups, that is Hr

r

is the group with signature (0; 2( 2Y+ 2 ); 0) the conformal automorphisms

are the hyperelliptic sheet-interchanges, and a fixed set of such a subgroup H is a hyperelliptic equivalence class in the sense of Ahlfors ([l] page 51). An immediate corollary to Theorem 5 and Section 6 is obtained by comparing dimensions of T (r) and T (K) for groups K, with K

cr.

r

of Fuchsian type

Except for a finite number of special cases ([3]), A(K) is an

analytic set in T (K) of positive codimension. The same statement is true for L(K) as a subspace of R 0 (K), and one may deduce PROPOSITION 6.

G

f.

If K is of compact Fuchsian type, the set of groups

S(K) with i((G) = G is dense in S(K) and its complement is an ana-

lytic set. §8. An Example We consider as illustration the situation where a cyclic group Zp of automorphisms acts on a compact surface with genus y 2: 2. The uniformizing group K corresponds to an exact sequence (*)

i

¢

1 ... K - - r - z P - 1

as in note 7.2, which imposes restrictions on the signature of

r

(see

SPACES OF FUCHSIAN GROUPS

Harvey [4]). The surjection

cp

203

fixes the geometric nature of

the action of the automorphism group in that it gives the rotation angles of of the automorphism group in that it gives the rotation angles of the group elements ataany points on the surface fixed under them. A concorresponds to (that is provides the base surfaces for) an M(K)-orbit of points in T (K) and an ~( (K) x ~·-orbit in R0 (K). To be more concrete, let X be a Riemann surface of genus 3 admitting an automorphism group Z7 • Then X is uniformized by a Fuchsian group arising from a sequence (*) with one examines surjections

cp :

r

having signature (0; 7, 7, 7; 0). When

r -+ z7'

modulo the automorphism groups of

r

there are found to be only two classes and Z7 , namely x 1 1 x 2 , x 3 t-+ l, l, 5

and xl' x2 , x3 , ..... l, 2, 4 respectively, where xl' x2 , x3 are generators for

r

as in Section 2. One finds that in each case there are further automorphisms, the Fuchsian groups in question being subgroups of larger ones. The first class is in fact hyperelliptic: there is an extension r1 of r which has signature (0; 2, 7, 14; 0) which admits a homomorphism 91 : r1 such that ker

cp 1 =

K,

cp 1 I r = cp

-+

z7 z2 (E!)

and the subgroup 91 1 (Z 2) is a hyper-

elliptic group. The other case corresponds therefore to Klein's surface with automorphism group LF(2, 7) of order 168, * which cannot be hyperelliptic since it has simple automorphism group. Here the Fuchsian group K is a normal subgroup of the (0; 2, 3, 7; 0)-group. REFERENCES

[l] Ahlfors, L. V., "The Complex Analytic Structure of the Space of Closed Riemann Surfaces," Analytic Functions, pp. 45-66. Princeton, 1960. [2]

Bers, L., "Quasiconformal Mappings and Teichmiiller's Theorem,"

Analytic Functions, pp. 89-119. Princeton, 1960.

* Note.

This fact was recalled to me by R. Accola. One can in fact recover simple

equations for these curves, as respectively w 7 = z (z- 1) and w7 = z (z-1) 2 , with automorphism group the sheet interchanges w

--> 'J w,

z -+ z,

2, A(z) is called hyperbolic. It has two distinct fixed points pA and qA on the real axis R. PA =

lim D->oo

An(z) for all z

f

U, and is called the attracting fixed point of A.

ON FRICKE MODULI

209

=

lim A-n(z) for all z < U, and is called the repelling fixed point n-+oo of A. The circle hA orthogonal to R through pA and qA is called the qA

axis of A. It is a geodesic in the Poincare metric inf

o(z, A(z))

o(z 1 , z 2 )

is achieved if and only if z lies on hA.

on U.

dA

=

dA is called the

ZtU

translation length of A and is related to Itr AI by the formula (1)

ltr AI = 2 cosh dA .

See Caratheodory [6] for a derivation of this formula. The Fuchsian groups representing surfaces of signature (g ; m) contain only hyperbolic elements and are the only ones we consider here. A proof of this fact can be found in Ford [9].

III. Construction of a group of signature (0; 3)

A Fuchsian group of signature (0; 3) is a free group on two generators. Given three real numbers, the problem is to write down generators C 1 , C 2 for such a Fuchsian group. These numbers are to be the lengths of the geodesics in the free homotopy classes determined by

cl, c2

and

c2cl

on the surface corresponding to the group. If the group can be constructed from tr

cl,

tr

c2

and tr

c2cl'

by (1) the problem will be solved.

Since a Fuchsian group with a standard set of generators mined only up to conjugation, one of signature (0; 3) with

S

is deter-

S= 1Cl'C 2 ,C 2 C 1 1

can be normalized so that the attracting and repelling fixed points of C 2 C 1 are at 0 and oo respectively and so that the repelling fixed point of C 2 is at 1. When C 1 , C 2 and C 2 C 1 come from a marked surface they satisfy the following "geometric conditions." The axes of C 1 and C 2 do not intersect and the fixed points of

c 2c 1

both lie between the attracting fixed

c2

and the repelling fixed point of C 1 • This is proved in Keen [12]. This implies that the fixed points of C 1 C 2 both lie between the point of

attracting fixed point of cl and the repelling fixed point of c2 and also that the fixed points of C~ C 1 C2 1 which are C 2 (0) and C 2 (oo) lie between 0 and the attracting fixed point of C 1 • See Figure 3.

210

LINDA KEEN

1-a/b

a/c b/d -b/c

-d/c

!lcl

Pel

Figure 3. Canonical Modular polygon and the axes of some of the elements of a group of signature (0, 3). Specifically, if C 1 , C 2 and C 2 C 1 are a standard set of generators for a group normalized as above they can be written as: --b/..\ a/A )

c2 = (

a b c d)

where

The normalization conditions are: a + b = c + d,

IAI
0 , d < 0 or

II.

a< 0, b > 0, c < 0, d > 0 .

Since a/ c < - d/ c, case I implies a + d < 0 and case II implies a + d > 0. Assume a+ d > 2 and A > 0. Since - d/ c < (Ad- a/A)/(- 2cA) and - 2cA > 0, 2dA > Ad- a/A and Ad + a/A < 0. Similarly assume a+ d < - 2 and A < 0. Then - 2cA > 0 and from - d/ c < (Ad- a/A)/(- 2cA) it follows that 2Ad < Ad- a/A and Ad + a/A < 0.

q.e.d.

The geodesics on S corresponding to the generators C 1 , C 2 and C 2 C 1 go around the "holes" of S. In the following sections some of these holes will be "attached" to the holes of surfaces of signature (1; 1). The elements corresponding to the holes on each of the surfaces must agree since this "attaching" is achieved by taking the free product of the two groups and amalgamating over the cyclic subgroup generated by the element corresponding to the hole. For a surface of signature (1 ; 1) this element is the commutator of its generators; therefore the sign of its trace is independent of the signs of the traces of the generators. In fact, since the commutator is hyperbolic, formula (4) of Section VI shows that this sign is always negative. It is for this reason that the traces are always taken as negative in the construction of a (0; 3) group. Lemma 1 assures that this doesn't involve any loss of generality.

K2

Let k 1 < - 1, k 2 < - 1 and k3 < - 1 be given; let K1 = ~.::]_ and K3 = Jk'ff- 1 . Consider the matrices:

Jk/- 1,

J, L and M are to be determined. The trace of C 1 = C2 1 (C 2 C 1 ) is to be 2k 1 , hence 2k 1 = (k 3 + K3 )(k 2 + J) + (k 3 - K3 )(k 2 - J) and where

J = (k 1 -k2 k3 )/K 3 < 0. If C 2 is to have a fixed point at 1, L = M+2J. If det

c2 =

l,

k 2 - j 2 -LM =land M 2 +2JM+J 2 -K~ = 0; M = -J±K 2

and L = J ± K2 . Since the group being constructed is to satisfy (2),

LINDA KEEN

212

therefore M =

-

J + K2 , L

= J + K2

. - (K2 + J)(k3- K3) ) kl- k2K3- k3J

It is possible to check that all the conditions are fulfilled now. That is,

1. The elements Cl' C 2 and C 2C 1 constructed from k 1 , k 2 and k 3 generate a Fuchsian group of signature (0; 3). THEOREM

Proof: Since all the conditions (2) are fulfilled a canonical modular poly-

gon can be constructed for these generators and therefore the group they gen = erate is Fuchsian. These are exactly the moduli Fricke obtains. In fact, he uses the same normalization and derives essentially these transformations as generators of the group. In order to perform his construction, Fricke needs to consider complicated inequalities involving a commutator of the generators. IV. Construction of a surface of signature (0; 4)

If

S = tC 1 , C 2 , C3 , C 4 1 is a standard set of generators for a surface

of signature (0; 4) the axes of the generators are mutually disjoint and the fixed points occur on the real axis as shown in Figure 4. The axis of c 2c 1 divides u so that the axes of c3 and c4 are separated from those of cl and c2. LEMMA

erators

2. Given a group of signature (0; 4) and a standard set of gen-

S = tcl'c 2 ,C 3 ,C4 1 the axis of C3C 2 = (C 1 C 4 )- 1

separates the

axes of c2 and c3 from those of cl and c4 and so intersects the axis of c2cl. Proof: Since tel' C2 , C 3 , C 4 1 is a standard set of generators the fixed points occur as shown in Figure 4. Consequently c 2(oo), C 2(0), C 1 (oo),

ON FRICKE MODULI

213

C 1(0) , C 3(oo), C 3(0), Cioo) and C 4(0) are also in the positions shown in the figure. Let C 2(x) If x

f

X

be a fixed point of c3c2. If

(C 2(oo) ,q 2) and C 3C 2(x)

f

(-

oo, q4 ), C 2(x)

Therefore Therefore

f

f

X f

(p3, q2),

(C 3(0), Cioo)). Therefore

(C 2(oo), C 2(0)) and C 3 C 2 (x)

f

xI

(p 3 , q 2).

(C 3 (0), C 3(oo)).

xI (-oo,q4). If x f (pl'oo), C 2(x) < 0 and C 3C 2(x) < C 3(0) < p 1 . xI (pl' oo), Consequently x f (q 4 , p 3) or (q 2 , p 1) or both.

There is exactly one fixed point x in each of these intervals. Note that C 3 C 2(oo)

< C 3C 2(0) and hence that C 3 C 2 moves 0 and oo in opposite

directions. This is impossible unless only the attracting fixed point of c 3c 2 lies between them. Therefore p32

f

(q 4 , p3) and q 32

f

(q 2 , p 1).

Now since the fixed points of C 3C 2 and C 2C 1 alternate, their axes must intersect. The group G of signature (0; 4) which will be constructed will be a free product with amalgamation:

where (i) (ii) (ii)

H is the group constructed in Section III, {C 2C 1 1 is the cyclic subgroup generated by C 2C 1 and H' is generated by elements C 3 and C 4 such that C 4 C 3

(C C )- 1 2 1

and the fixed points of C 3 and C 4 occur as in Figure 4.

p

Figure 4. Surface of signature (0, 4) and a canonical modular polygon P for the group representing it.

214

LINDA KEEN

Normalize so that C 2 C 1 has attracting fixed point at 0, repelling fixed point at

00

and

c2

has repelling fixed point at 1.

To construct H' proceed as follows. Given k3 < -1, k 4 < -1, and k 34 = k 12

A~ (D, G) is a bounded (surjective) pro~ j ection of norm :; cq. For ¢ holomorphic on D, define the

Poincar~

series of ¢ by

zED, whenever the sum converges normally. THEOREM 3. (Bers [4]) @q: A~(D) .... A~(D,G) is a bounded surjective linear mapping of norm :; 1. Furthermore, for all p, l :;

1/J

E

A~(D,G), there is a¢

II¢ I q,p

f

p::;

oo,

and all

A~(D) such that 1/J =@q¢ and

::; cq II 1/J II q,p,G .

REMARK. The existence of the f3q map will show that every automorphic form is a Poincar~ series. For p

=

1, surjectivity of the@ q maps follows also

from the following two facts. The adjoint of the

@q

map is the inclusion

map. For p == oo, the inclusion map is injective with closed range.

258

IRWIN KRA

THEOREM 4. (Bers [ 4]) The Petersson scalar product establishes an anti-linear isomorphism between Ap'(D, G) and the dual space to Ap (D, G), q

where 1 ~. p

q

< oo and l/p + l/p'"' l.

Outline of proof. (1) If f is a (one-to-one) conformal map, then

is a surjective isometry. Furthermore for¢< A~(f(D), fGC 1 ) and

tf


.. completely in the interior of TT. Then, applying the residue theorem Then, applying the residue theorem to fa d/3 to the domain obtained after slitting along >.., we get 'Tlxy(W) - 'Tlxy(Z)

=

21. TTl

f

'Tlxy d'Tlzw

aTT+)..

+ - 1-

2TTi

!

>..

=

1 2 . TTl

'Tlxy d'Tlzw

1

'Tlxy d'Tlzw

aTT

'

note that the >.. under the integral really stands for both sides of the slit A. The integral over aTT is zero by equation (2.4) since 'Tlxy and 'Tlzw are normal integrals of the third kind on S.

since 'Tlxy has a jump of 2TTi from one side of >.. to the other. We thus obtain (2.8) independent of the branch of 'Tlxy chosen. We will be dealing with more than one surface in this paper. If a surface is designated by S with subscripts or superscripts, we will denote its periods, integrals, boundary of its canonical dissection, etc. with the corresponding subscript or superscript. 3. Degeneration by handle removal We summarize now the procedure used in this section. Start with a specific surface S of genus g and construct a specific surface S* of genus g*

=

g + 1. Obtain formulas relating differentials on

AARON LEBOWITZ

270

the two surfaces S and S*. Using these formulas we calculate the periods "ij on S* in terms of the "ij on S wherever possible. We get estimates for all of the "iJ, i, j = 1, ... , g*= g + 1. We now give the prescription for changing the surface S of genus g to obtain the surface S* of genus g + 1. Let 77p Q be a normal integral of third kind on S where P 0 and Q0 0 0

are not on any of the fixed cycles of the canonical homology basis, (f' ,Ll), for S. For convenience, we will index the g cycles in f' and Ll by i = 2,3, ... ,g+ 1. Consider the local parameters +77p Q 0 0 t1 = e (3.1)

on S about P 0

and (3.2) Let DP 0 and DQ 0 be parameter disks about P 0 and Q0 , respectively. These disks are chosen so as to have null intersection with the basis cycles,

(f', Ll). In the parameter disk D P , the set of points P on S, near P 0 , such 0

that Re '17p Q (P) = log f , forms a circle of radius f , I t 1 1 = f. Denote. this 0 0

circle by CP . Similarly, the set of points Q on S, near Q 0 , such that 0

Re '17p Q (Q) = log 1/f , forms a circle of radius f in the parameter disk 0 0

DQo' it 2 i = c

Denote this circle by CQ 0 . Take fixed points P' and Q'

in DP 0 and DQo' respectively, on CP 0 and CQo' respectively, with 77p Q (P') = logf+ ia' 0

and

0

'17p Q (Q') =log 1 +if3' 0 0

f

Then (3.3)

'17p Q (P') = 77p Q (Q')+2logf+i(a'-{3'). 0

0

0

0

ON THE DEGENERATION OF RIEMANN SURFACES

271

We identify the points P' and Q' and, furhter, we identify the points P and Q on CP 0 and CQ 0 , respectively, if they satisfy the following equation: T/p Q (P) = T/p Q (Q) + 2 log

(3.4)

0

0

E

+ i(a '-{3 ') .

0 0

Prescription A: Let a Jordan arc, y 1 , be drawn on S from P' to Q' which does not intersect any of the cycles in (r .~). Delete the interiors of CP and CQ and identify the boundaries of CP and CQ as described 0

0

above. We call the identified curve,

0

1. Then either there is

an anm > 0 so that z f anm(B 1), or there is a bnm < 0 so that z f bnm(B 2 ), and we assume for simplicity that the former case holds. We write anm = gn

o .. • o

g1 in normal form, and since anm > 0, we know that g 1 f G 1 - H.

It follows at once that g 1(B 1) C B 2 , and hence anm(B 1) C gn o ... o g2(B 2):

The proof is completed by observing that gn

o ... o

g2 has length n-1 and

is negative. We are now in a position to investigate S. We first observe that if ZfT 1 nB 2 thenthereisa gfG 1 -H, sothat g(z)f B 1. Henceif z f S 1 nB 2 , either z f L(G 1), or there is a gf G1 , so that g(z) f D 1 n B 2 C D. We similarly observe that if z f S1 n B 1 , either z f L(G 2 ), or there is a g f G 2 so that g(z) f D. Finally, if z f S 1 ny, then either z f L(H) C L(G 1) n L(G 2), or there is an h f H, so that h(z) f L'1 ny = D ny. We have proved PROPOSITION 6. If z f S1 , there is a g f G, so that

PROPOSITION 7. If z f S, there is a g f G, so that

Proof: Since si

c

si+ 1' for every i, there is an n, so that z f sn, z

I

sn-1'

Proposition 6 takes care of the case n = 1, and so we assume n > 1. Then

.

Zf Tn_ 1 , and so either there is a w f B 1 , and an an_ 1 m with z =

.

.

an- 1,m(w), or there is a w f B 2 , and a bn_ 1 m' with z = bn_ 1 m(w). For simplicity, we assume that the former case holds; the proof for the latter case is analogous. Now w is either a point of T 1 of of S1 . If w were in T 1, then there would be a g f G2 - H, and a point x f B 2 , so that w = g(x). Then we would have z = an- 1 ,m an- 1 ,m

o

o

g(x), where 1an- 1,m

g < 0, contradicting the fact that z

I

T n'

o

gj

=

n,

BERNARD MASKIT

306

We conclude that for every z < Sn, n > l, there is a g £ G, and a w < S 1 , so that g(z) = w. The result now follows at once from Proposition 6. We remark incidentally that Proposition 7 characterizes S. We now turn our attention to T. We first look at T n, for fixed n. We observe at once that anm(B 1) C B 2 if and only if n is odd, and bnm(B 2) C B 1 if and only if n is odd. We conclude that for fixed n, anm(B 1 ) n bnk(B 2 ) =

0.

Now consider two distinct coset representatives anm and ank, of

the same length. We write a 01;

1 o

ank = gs

o •• • o

g 1 in normal form, and

observe that s > 0, g 1 < G 1 - H, gs < G 1 - H. We conclude that a 01;;- 1 o ank(B 1 ) C B 2 and so ank(B 1 ) n anm(B 1 ) = 0. In exactly the same manner, we show that if bnm and bnk represent distinct cosets, then bnm(B2)

n bnk(B2)

=

0.

The above remarks, together with the trivial Proposition 9, below, yield PROPOSITION 8. Each connected component T nm of T n is either a set of the form l z I z

f

anm(B 1 )! or a set of the form lz I z

PROPOSITION 9. If g f G- H, then ~y)

n y

=

f

bnm(B 2)L

0.

Proof: Write g in the normal form (3), and observe that n

> 0. Assume,

for simplicity, that g > 0. Then by property (i), g/y) C B 2 • Then g 2 o gly) C B 1 and so on. We remark that D C S1 , and so each T n is bounded. We will no longer need to differentiate between positive and negative cosets. We now rewrite the coset decomposition as

where Icnml = n, and T nm is bounded by cnm(y). Now let z 0 be some point of T, then z 0 each n, there is an m(n), with z 0

£

f

T n for every n, and so for

T n,m(n) .

By Proposition 4, Koebe's theorem is applicable, and so

ON KLEIN'S COMBINATION THEOREM Ill

I. dia2 c 0 m(y) = I.

4)

n,m

dia 2 T nm

= I. I.

n,m

307

dia 2 (T nm)

< oo

•

n m

We conclude that lim I. dia 2 (T nm) n-+ m

5)

=0

As a special case of (5), we observe that 6)

lim dia (T n m(n)) n-.oo J

=0

Equation (6) asserts that for every w on y, lim

en m(n)(w)

n-+oo

= z0 ;

'

i.e., since the cn,m(n) are all distinct, z 0 is a limit point of G. We restate this as PROPOSITION

10. T C L(G).

Since S and T are complementary, Propositions 7 and 10 together with Propositions 2 and 3 assert that D is a FS for G; i.e., we have proven conclusion 3. The above remark, together with Proposition 4 yields conclusion 4. We return to equation (5) and observe that since T C T 0 , for all n, meas(T) -< meas(T n) == I.

m

meas(Tnm) -
. 0. Then one easily sees ~at

g*(B 1 ) C B 1 , and hence g* is loxodromic.

This concludes the proof of combination theorem I.

§5. PROOF OF COMBINATION THEOREM II. It should be remarked that, in combination theorem II, if H 1 and H2

are both trivial, then G is the free product of G 1 and G2 • Of course, this free product need not occur in the sense of combination theorem I ; the geometry is somewhat different. The basic idea in the proof below is to imitate, as far as possible, the elementary proof of combination theorem I given above. From here on, we assume that the hypotheses of combination theorem II hold. Since G is generated by G1 and G2 , every element g f G, can be written in the form 7)

where a 1 , ... ,an-l

.f

0, and g 2 , ... ,gn

thesis (m), we can assume that if ai-l

.fl. Using the relations in hypo< 0 and gi f H 1 , then ai S. 0; we

can also assume that if ai-l > 0 and gi f H 2 , then a i 2:. 0. An expression of the form (7) satisfying these assumptions is called a normal form for the element g. Let F be the complement of B 1 U B 2 , so that F is the interior of D 2 • We make the following observations. If a >

If a

o.

then fa(F U B 2 U y 2) C B 2

< 0, then fa(F U B 1 U y 1) C B 1

If gf G 1 , g~ l, then g(D) no=~

.

If gf G 1 -H 1 , then g(B 1 U y 1) C F -D.

ON KLEIN'S COMBINATION THEOREM III

309

If gf G 1 -H 2 , then g(B 2 U y 2) C F-D • Ifa>O and gfH 2 , then gofa(FU B 2 Uy 2) C B 2 If a < 0 and g f H 1 , then g o fa( F U B 1 U y 1) C B 1 Now let g be some non-trivial element of G, expressed in the normal form (7), and let z be some point of D. We put together the above observations as follows. If g 1 = l, then a 1 I 0, in any case, g/z) f F U y 1' a Then f 1 o g 1(z) belongs to B 1 if a 1 < 0, or B 2 U y 2 if a 1 > 0. In a either case g 2 of 1 o g 1(z) belongs to F- D, unless a 1 < 0 and g 2 f H 1 , a1 or a 1 > 0 and g 2 f H 2 . If a 1 < 0 and g2 f H 1 , then g2 of og 1(z)f B 1 , and a 2 :::; 0, so that fa 2 o g 2 o fa 1 o g 1(z) f B 1 . Similarly if a 1 > 0 and g 2 f H2 , then fa 2 o g2 o fa 1 o g 1(z) f B 2 . In any case, we can continue in this manner and observe that we have proven a PROPOSITION 12: If g = f n o ... o g 1 is in normal form, where either g1

I

l, or a 1

I

0, and if Zf D, then g(z)

I

D.

Simple modifications in the above argument, together with hypotheses (i) and (k), yield PROPOSITION 13: If gf G-H 1 , then g(y 1) ny 1 where g1 ( G1, then g(y1) ny2

0;

if g ( G, g I f 0 g1,

I 0.

By hypothesis (j), D has a non-empty interior. This statement, together with Proposition 12 trivially implies conclusions 1 and 2. It is somewhat more convenient to restate Proposition 12 as

PROPOSITION 14: If g f G, gIl, and z f D, then g(z)

I

D.

We next want to show that D C R(G). Consider the set of all the translates under G 1 of B 1 U B 2 , and call the complement of this set F 1 . Observe that for every z f F 1 , there is a g f G 1 , so that g(z) f DU y 2 U L(G 1). Hence for every z 0

f

D, either z 0 is an interior point of F 1 , or z 0 is a

point of y 1 . If z 0 is an interior point of F 1 , then z 0 trivially has a neighborhood U so that every z f U is equivalent under G 1 , to a point of D.

BERNARD MASKIT

310

If z 0 is a point of y 1 , then using hypothesis (f), there is a neighborhood U of z 0 so that for every z We restate this as

f

U, either z

f

D, or f(z)

f

F1

n R(G 1 ).

PROPOSITION 15: DC R(G).

Using hypothesis (f) again, there are neighbnrhoods V1 and V2 of y 1

y2 respectively, so that ~ 1 nv 1 C Dl' and ~ 2 nv 2 C D1 • Set V = V 1 n f- 1 (V 2 ). We define a new FS D:l for G 1 as follows. Delete f(V) n D 1 from D 1 ; for each z c F (V) n D 1 , there is an h c H 1 so that and

h o f- 1 (z)

f

~ 1 ; adjoin h o f- 1 (z) to the truncated D 1 , and in this way

we get a new FS D:l for G 1 . Similarly pick a FS Dz for G2 so that V C Dz. Thus, using Propositions 14 and 15, there is a FS D' for G, so that ~

1

nv CD'. Hence

(G,D',H 1 ,~ 1 ,y 1 )

is a conglomerate; this is conclu-

sion 4. We recall that every g f G, g 1- l, can be written, not uniquely, in the normal form 7)

where g 1 , ••• , gn gi+l

f

f

G 1 ; g 2 , ••• , gn

1-

l; a 1

, •••

,a 0 _ 1

H 1 , then ai+l :::; 0; if ai > 0 and gi+ 1

f

1-

0; if ai < 0 and

H2 , then ai+ 1 _2: 0.

Suppose now that g can also be written as s'Jme other normal form, say 8)

Proposition 12 asserts that no non-trivial normal form can be the identity. We form the expression 9)

(f

{3k

o•·· of

{31

-

og 1 )

and we know that (9) is not a normal form; i.e.,

g1

o

g 1- 1 belongs to either

H 1 or H2 , and a 1 and {3 1 have the same sign. Since the expression (9) must reduce to the identity, we easily see that as we reduce (9), we will

ON KLEIN'S COMBINATION 'TIIEOREM III

311

always get expressions of the form

Again, since (10) cannot be a normal form, we must have that

gi

o gj- 1

belongs to either H 1 or H 2 , and {3i and aj have the same sign. We conclude that

k

n .~

11)

1=1

lail =.~

lf3jl ·

]=1

We call the invariant, given by (11), the length of g, and denote it by

Igl.

If g

f

Gl' then I gl

= 0;

We trivially remark that if g 1

in particular, the identity has length zero. f

Igg 11 = Ig 1 g I

G 1 , then

=

I g I,

for every

gE G. The right hand side of (7) is called a positive normal form if g 1 and a 1

f

H1

> 0. If we also represent the same element g by the normal form

given in (8), then we again form the expression (9) and get a word which

g1

is not in normal form. It follows that

f

H 1 and {3 1 > 0; i.e., the right

hand side of (8) is also positive. It follows that we can refer to the element g as being positive. Our major interest is in elements which are non-positive; if g

f

G is non-positive, then we write g

.S 0.

In analogy with the above, the right hand side of (7) is called negative if g 1

H 2 and a 1

f

< 0.

Again, if g

f

G can be expressed by a negative

normal form, then every normal form for g is negative, and we say that g is negative. Again our major interest is in elements which are non-negative; if g f G is non-negative, then we write g 2: 0. Oneeasilyseesthatif g_$0 and hEH 1 ,then goh,$0; alsoif g;:::O and h

f

H 2 , then go h 2:.0.

We decompose G into cosets modulo H 1 • For a fixed coset gH 1 , every element has the same length; this length is also called the length of the coset. Again for fixed g, either every element of gH 1 is positive, or every element is non-positive, and we say that the coset is positive or

BERNARD MASKIT

312

non-positive respectively. For each non-positive coset of length N, we pick out some coset representative and denote it by aNM" Similarly, we decompose G into cosets modulo H2 ; define the length of the coset gH 2 as the length of g; define gH 2 to be negative if g is negative; and pick out a set of coset representatives bNM for the non-

negative cosets of length N. For N

= 0, l, 2, ... , we set

Let SN be the complement of TN; set S = UN SN' and let T be the complement of S. In analogy with the preceding theorem, our goal is to show that every point of S is a translate of some point in D U L(G 1 ) U L(G 2 ), and that T has measure zero and consists of limit points of G. For N > 0, let z 0 be some point of TN" We assume for simplicity that there is an M so that z 0

f

aNM(B 1 ); there is an obvious analogous treat-

ment for the case that z 0 f bNM(B 2 ). Let g = aNM be expressed in the normal form (7). If a 1 > 0, we set 12) Since aNM

S. 0,

g1

I

H 1 , and so g (B 1 ) C g* (B 2 ). We see at once

that I g*f = N -1, and that the right hand side of (12) is a non-negative normal form. We conclude that z 0 Similarly, if a 1

TN_ 1 •

f

< 0, then we set

13) and observe that fg*f

= N-1, g* S. 0, and g(B 1 ) C g*(B 1 ). We again

conclude that z 0 f T N- 1 • We summarize these arguments as PROPOSITION 16: For N

> 0, TN

C TN_ 1 , SN) SN-l •

ON KLEIN'S COMBINATION THEOREM Ill

313

Now let z 0 be some point of s0 . Recall that T 0 = UgfG 1g(B 1 U B 2 ), and so either z 0 f L(G 1 ) or there is a g f G 1 with g(z) f D 1 • In the latter case, g(z) either belongs to D, or lies on y 2 , in which case there is an h f H 1 so that h

o

f- 1

o

g(z) f D. We have proven

PROPOSITION 17: Let z 0 f S 0 , then there is a g f G so that g(z 0 ) f D U L(G 1 ). PROPOSITION 18 Let z 0 f S, then there is a g f G, so that g(z 0) f D U L(G 1) U L(G 2 ). Proof: Using Propositions 16 and 17, there is an N;: 0 so that

z 0 f SN+ 1 , z 0

I

SN' Since z 0 f TN, there is either an aNM with

z 0 f aNM(B 1), or there is a bNM with z 0 f bNM(B 2). We assume for simplicity that the former case occurs; the latter case is treated analogously. Let z 1 = aNM 1 (z 0) f B 1 . Either z 1 is a fixed point of f, or there is a {3 > 0 so that z 2 = f{3(z 1 ) f D 2 . In the former case,

and we are finished; we assume the latter case occurs. We want to show that z 2 in fact lies in S0 . Suppose not. Then there would be a g f G 1 so that z 3 = g(z 2) f B 1 U B 2 . Set g* = aNM

o

f -{3

o

--1

g

.

Recall that aNM .S 0 and {3 > 0, sothat N*= lg*l = lgl z 3 f B 1 , then g

I

H 1 , and so g*

+f3 >

N. If

_s 0; if z 3 f B 2 , then g( H 2 , and so

g* ?: 0. Since z 0 = g* (z 3), we conclude that z 0 f TN* contradicting the assumption that z 0 f SN + 1• Hence z 2 does lie in S 0 , and so by Proposition 17, there is a g f G, with g(z 2 ) f D U L(G 1). We remark, incidentally, that it is not clear whether or not the translates of the fixed points of f actually do occur as points of S.

BERNARD MASKIT

314

We have already observed that D C s0 , and so TN N. This remark, together with Proposition 13, yields

nD

=

0

for every

PRO POSITION 19: Every connected component T NM of TN is either a set of the form aNM(B 1 ) or a set of the form bNM(B 2 ). Each connected component T NM of TN is bounded either by aNM(y 1) or by bNM

o

f(y 1).

For each T NM we pick cNM so that T NM is bounded by cN~Y 1), and observe that the cNM all represent distinct cosets of H 1 in G. We now normalize G so that oo E D and so that y 1 and all its translates are bounded. We have already shown that there is a FS D' for G so that (G, D',H 1 , Lll'y 1 ) is a conglomerate; hence Koebe's theorem is applicable. We observe that

and so 15)

!,M dia 2 cNM(y 1 )

lim N

Now if z 0

E

T, then z 0

is an M(N) with z 0

E

E

TN for every N, and so for each N, there

TN ,M(N) . TN ,M(N)

and by (15) dia cN,M(N)(y 1 )

=0 .

-+

....

is bounded by eN ,M(N)(y 1 )

0; i.e., if z is any point of Yp then

cN,M(N)(z) .... z 0 • It follows at once that z is a limit point of G. We restate this as PROPOSITION 20: T C L(G). Putting together Propositions (18) and (20), we see that for every z

E

R(G), there is a g

E

G so that g(z)

E

D. This completes the proof

of conclusion 3; i.e., D is a FS for G. In order to prove conclusion 6, we observe that L(G 2 ) consists of two points, and L(G) zero.

n S has measure zero if and only if L(G 1 ) has measure

ON KLEIN'S COMBINATION THEOREM

III

315

Hence it suffices to prove that T has measure zero. Observe that

16)

meas(T) S meas(TN) = l

meas(TNM) S (rr/4) l

M

dia 2 TNM

M

= (rr I 4) l M

dia2 cNJY 1)

Putting together (15) and (16), we get PROPOSITION 21: T has measure zero. The only thing left is to prove that if g E G is elliptic or parabolic, then g is conjugate in G to an element of G 1 • Let g be some element of G, and let g* be a conjugate of g, where

I g* I is minimal among all conjugates I g* I > 0, then g* is loxodromic. a We write g* = f n

o .. • o

of g. It suffices to show that if

g 1 , in normal form, and we can assume that

an -/: 0. We in fact assume that an > 0; the case that an < 0 is treated in an analogous manner. With this assumption, the minimality of Ig*l implies that g* 2:. 0. Using the appropriate remarks in the proof of Proposition 12, we now observe that g* (B 2 ) C B 2 ; it follows at once that g* is loxodromic, and conclusion 5 has been established.

BERNARD MASKIT

316

REFERENCES [1] L.V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups, Proc. Nat. Acad. Sci., 55 (1966), 251-254. [2] V. Chuckrow, to appear. [3] L.R. Ford, Automorphic Functions, 2nd ed. Chelsea, New York, 1951. [ 4] F. Klein, Neue Beitrage zur Riemann'schen Functionentheorie, Math.

Ann. 21 (1883), 141-218. [5] P. Koebe, Ober die Uniformisierung der algebraischen Kurven II,

Math. Ann. 69 (1910), 1-81. [ 6]

, Uber die Uniformisierung der algebraischen Kurven III,

Math. Ann. 72 (1912), 437-516. [7] B. Maskit, On Klein's combination theorem, Trcns. Am. Math. Soc. 120 (1965), 499-509. [8]

, On Klein's combination theorem II, Trans. Am. Math. Soc. 131 (1968), 32-39.

Massachusetts Institute of Technology Cambridge, Massachusetts

ON FINSLER GEOMETRY AND APPLICATIONS TO TEICHMULLER SPACES by Brian O'Byrne* §l. Introduction.

In this paper we investigate the following situation in Finsler geometry: (X, a) and (Y, {3) are complete Finsler manifolds such that there exists a

c 1+-foliation

f: X .... Y and the Finsler structure

f3

is, in a sense to be

defined later, the infimum via f of the Finsler structure a. It is possible to define two metrics on Y which are related to a. One is the metric induced by

f3

and the other is, again in a sense to be defined later, the

infimum via f of the metric induced on X by a. Our main result is the proof of the equality of these two metrics. Earle and Eells in [2] have shown that for any Fuchsian group f' it is possible to introduce Finsler structures a([') on the space M([') of Beltrami differentials of [' and

f3 ([')

on T ([') the Teichmiiller space of f'

in such a way that (M {['), a ([')) and ( T (['),

f3 (['))

satisfy the hypotheses

of our main theorem. As a consequence of this, we are able to show that for any 11

f

M([') which is extremal in the sense of [ 4] t11 is also extremal

for 0 ,::: t ,::: l and that for any extremal 11 its norm as a linear functional on the integrable quadratic differentials of f' is equal to its supremum norm. The latter result was first proved by Hamilton in [ 4]. In §2 and §3 we describe the situation we are concerned about. In §4 we prove two preliminary lemmas, the second of which involves the lifting

*

This work was done at Cornell University as part of the author's doctoral

dissertation.

317

BRIAN O'BYRNE

318

of regular curves. In §S we prove our main theorems and in §6 we give their applications to Teichmiiller spaces. §2. Finsler Manifolds Let E be a Banach space with norm manifold modeled on E. Let rr X of X and X (x)

= rr X

1 (x)

II·IIE

and X a paracompact

c 1+-

TX -+ X denote the tangent vector bundle

:

the tangent space to X at x. Then (X, E, a) is

a Finsler manifold and a is a Finster structure on X if a: TX-+ R is a map such that for each x in X, the restriction of a to X (x) is a norm and x is centered at a coordinate chart (0, U) in which a has the following two properties: (2.1) There is a number A > 0 such that

Here (J * (x) denotes the differential of (J at x. (2.2) There is a number K > 0 such that

for all z, y in U and all v in E. These properties imply that a iS locally Lipschitz in TX and that the

I · II E

norms on X (Y) induced by a and

are equivalent uniformly for y in

u. A curve is called regular if it is continuously differentiable and a local homeomorphism onto its range. If b: [a, c] -+ X is a piecewise regular curve then its Finster length La(b) is defined by (2.3)

La(b)

=

J

ba(b'(t))dt

where b'(t)

=

b*(t)(l) c X(b(t))

a

Assuming that X is connected, for any two points, x, y c X the

Finster distance da(x, y) is defined by (2.4) da(x, y) = inf{La(b): b is a piecewise regular curve in X joining x to y} •

ON FINSLER GEOMETRY AND TEICHMULLER SPACES

319

The following lemma is well known and is fairly straightforward. LEMMA. If (X, E,a) is a Finsler manifold, then the Finsler distance

is a metric on X compatible with its topology. In the future, when speaking of a complete Finsler manifold (X, E,a) we refer to completeness with respect to Cauchy sequences in X relative to the metric da .

§3. Foliations Let X and Y be differentiable manifolds modeled on Banach spaces and f: X .... Y a differentiable surjective map. The map is called a foliation if, for each point x in X, f*(x) maps X(x) onto Y(f(x)) and the kernel Ker f*(x) of f*(x) is a direct summand of X (x), (i.e., Ker f*(x) is a closed subspace of X (x) admitting a closed supplement). Now let (X, E, a) and (Y, F, {3) be complete Finsler manifolds subject to the following two assumptions: (3.1) There is a

c 1+-foliation

f:

X ....

y.

(3.2) Given y in Y then for each x in {J(v)

r- 1(y)

and each v in Y(y)

= in£ {a (u): f*(x) u = v, u in X(x)l

For any two points y, z in Y we define (3.3)

a(y, z)

=

inf{da(x, w): x

E

f- 1(y), w

The main result of this paper is the proof that a

l

r

1 (z)l .

= d/3 and that therefore

it is a metric on Y.

§4. Preliminary Lemmas In this section and the next we assume that (X, E,a) and (Y, F, /3) are complete Finsler manifolds satisfying (3.1) and (3.2). Here we prove two lemmas that are needed for the proof of Theorem I.

BRIAN O'BYRNE

320 LEMMA 1:

Given y in Y, v in Y(y), x in f- 1 (y) and w in (f*(y)r 1 (v)

then there exists a continuous linear map S: Y(y) .... X(x) such that

The proof is quite easy and is therefore omitted. LEMMA

II: Let c: [p, q]

ple points and let

E

->

Y be any regular curve which has no multi-

> 0 be arbitrary. Then for each point

there is a regular curve bx: [p, q]

->

x in f- 1(c(p))

X such that

(4.1)

and

a (b~ (t)) :::; ,8 (c '(t)) +

(4.2)

E

for a11 t in [p, q]

Proof: First we assume that, for the moment, y in Y, x in f- 1(y) and v in Y(y) are fixed. By condition (3.2) on ,8 (v) we know there is a u in X(x) such that (4.3)

a ( u)

< ,8 (v)

+

E

/2 ,

The conditions on f imply that it is locally a trivial fibration and therefore, there are c 1+-coordinate charts ((), U) centered at

X

and

(if, V)

centered at y which trivialize f (see [ 5] Chapter II, section 2). This means that () is a c 1+ -homeomorphism of

u

onto a product of open subsets

and V1 'Jf Banach spaces E 1 and F respectively and morphism of V onto V 1 in such a way that the map (4.4)

u1 X v 1 X E1 X F

e*-1

if

u1

is a cl+-homeo-

f* if* "x 1(U)-----"y 1(V) ---v1 XF

is projection on the second and fourth factors. By Lemma I there is a continuous linear map

(4.5)

S:

Y(y)

->

X(x) such that

ON FINSLER GEOMETRY AND TEICHMULLER SPACES

321

This implies that 1/I*(y) of*(x) oe*- 1(8(x)) oe; 1 e(x) o8*(x) o S oi/I; 1 (1/J(y)) "' IF and since

e and

1/J trivialize f we have

for all z in U. Let f*TY --> X denote the vector bundle over X obtained by pulling back TY via f. Since U and V are coordinate charts, the subbundle f*rr,Y 1(V) of f*TY obtained by restricting f to U is homeomorphic to the vector bundle over U whose fibre over x is Y (f(x)). This homeomorphism is locally Lipschitz. Thus, for convenience, we may represent a point in f*rry.1 (V) as a pair (z, w) where z

E

U and w

E

Y(f(z)). Also, let L(f*rr,Y 1 (V), rr:X 1(U))

denote the set of bundle maps which are locally Lipschitz and are continuous and linear in each fibre. We define a map Sin L(f*rr,Y 1(V), rr:X 1(U)) by (4.7)

-

S(z)w

=

e; 1 (8(z))

for all (z, w)

E

~

'

1

o8*(x) oS o l/f; (1/J(y)) ol/J*(f(z))

f*rr.y 1(V) .

This map is continuous and linear in each fibre and locally Lipschitz because it is composed of maps with these properties. Clearly S (z): Y(f (z))--> X (z) and f*(z) o S(z) = IY(f (z)) for all z in U. The map P: f*rr.y 1(V) --> R defined by (4.8)

P(z,w) = a(S(z)w)-{3(w) for all (z,w) in f*rr,Y 1(V)

is locally Lipschitz for similar reasons. Therefore, if U and V are restricted sufficiently, we know that there exists a 8

> 0 such that P(z, w) < E for < 8 because S(x) =Sand

all (z,w) in f*rr.y 1 (V) such that \f3(w)-{3(v)\ a(u) ~ {3(v) + E/2.

The restriction that y, x and v are fixed will now be removed and all the maps, sets and constants from the above will be written with the sub-

322

BRIAN O'BYRNE

scripts x and v (i.e., S will now be written as Sx ,v).

If y

f

Y and y

I

c([p, q)) then we select any v in Y(y) and for each

x in f- 1(y) we construct the map S x, v as above with the additional restriction that Ux,v

nr

1(c([p,

q])) =

0.

This is possible because

r

1 (c([p,

q]))

is closed.

If y < Y and y = c (t ') for some t' in [p, q] then we set v and for each x in

r

1(y)

=

c '(t ')

we construct the map Sx ,v as above. Without loss

of generality, we can assume that Vx,v meets only a connected arc of c. Since {3 (c '(t)) is a continuous function of t, we may also assume that Vx,v has been restricted sufficiently so that lf3(c'(t))-{3(c'(t'))l

< o for

all tin [p,q] suchthat c(t) is in vxv· , The sets lux ,v! which were constructed in the above two paragraphs form an open cover of X. Since X is paracompact this open cover has a neighborhood finite subcover !Ur!. If

lpr!

is a locally Lipschitz partition

of unity subordinate to it, then we define a map S (4. 9)

S(x) w =

f

L(f*TY, TX) by

L p/x) Sr (x) w for all (x, w) in f*TY . T

This sum involves only a finite number of terms because the open cover is neighborhood finite. It is clear that the map is in L (f*TY, TX). Since f* is linear in each fibre and (4.10)

f/x)

o

Lr Pr(x)

S (x) ==

=

l for all x in X, we have

IY(f(x))

for all x in X .

Given a map S in L (f*TY, TX) satisfying (4.10) Earle and Eells have shown [3] that for x

f

r

curve bx: [p, r]

-->

(4.11)

b~(t) =

1 (c(p))

there exists a number r in (p, q] and a

X satisfying (4.1) and S(bx(t))c'(t)

forall tin [p,q].

Further ([3], Lemma 3B), we can take r == q provided that the numbers a

(S(x) c '(t)) are bounded uniformly for t in [p, q] and x in f- 1 (c (t )). But

323

ON FINSLER GEOMETRY AND TEICHMULLER SPACES

a(S(x)c'(t))

= a(I

Pr(x)Sr(x)c'(t))

r

< I Pr(x)a(Sr(x) c'(t)) r

< {3 (c '(t)) +

E

by construction. Therefore bx: [p, q) .... X is clearly regular, satisfies (4.1) and (4.11) and a (b~(t ))

= a (S(bit )) c '(t )) ::; {3 (c '(t )) +

E

by the above calculation.

Q.E.D. COROLLARY

let

E

1: Let c: [p, q]

-+

y be any piecewise regular curve and

> 0 be arbitrary. Then for each point

wise regular curve bx: [p, q]

-+

X

in

r

1(c(p))

there is a piece-

X satisfying (4.1) and (4.2).

Proof: Since c is piecewise regular it has only a finite number of zeros and points of discontinuity. Thus, there is a partition p

= c I [ti-l,

of [p, q] such that ci

ti] is a regular curve which has no multi-

ple points. By Theorem I, for each x in bx 1 : [t 0 , t 1 ]

= t 0 < ... < tn = q

r

1(c (p))

there is a regular curve

X satisfying (4.1) and (4.2) for c 1 • Similarly, for each i

-+

'

there is a regular curve bx,i: [ti-l, ti]

-+

X starting at bx,i-l (ti-l) satis-

fying (4.1) and (4.2) for ci. Therefore, the curve bx: [p, q) .... X defined by bx I [ti-l' ti]

= bx i

is a piecewise regular curve which satisfies (4.1)

'

· and (4.2) for c. §5. Main Theorems THEOREM

1: df3

Proof: Let

E

= a.

> 0 be given and let y 1 and y 2 be two points in Y. By

the definition of a there exist xl

f

f- 1(y 1) and x2

f

f- 1(y 2) such that

da(x 1 , x2 ) :S a(y 1 , y 2 ) + E/2. Let b: [0, l] .... X be a piecewise regular curve joining x 1 to x 2 for which La(b) ::; da(x 1 , x 2 ) + E/2. Then, since a (b '(t )) ::; {3 ((f

a

b) '(t )) for all t in [0, l] we have

324

BRIAN O'BYRNE

< da(x 1 , x2 ) + t::/2




R by

The map {3 ([') satisfies (3.2) and is a Finster structure on T (r). We note here that our definition of f)([') is not the same as that found in [2]. Howevery, the proofs that it depends only on (/1) and that it is a Finster structure are the same. By Theorem I we have THEOREM

III: For any Fuchsian group

r

acting on U r([') = df)([')·

Theorem II has applications to the study of extremal quasiconformal maps. We call 11 in M([') extremal if the corresponding quasi conformal map w : U --> U is extremal (see [4]). The extremal 11 are exactly those 11 /1

which are extremal with respect to the origin in the sense of §5. Thus we obtain THEOREM

IV: Let

r

be any Fuchsian group acting on U. If 11 in

M(r) is extremal, then so is t11 for ail t in [0, l].

ON FINSLER GEOMETRY AND TEICHMULLER SPACES

327

Proof: The curve y (t) = tiL, 0 ,:S t ,:S l, is an extremal curve joining 0 to IL· If L 1 ([')

= lcfJE

L 1 (u/r,C): (cpoA)(A') 2

= cpforall A in rl then

we can define a pairing ( , ) between L 1 (['), and L ""([') by

(¢, Jl)

(6.6)

=

¢ (z) 1L (z) dx dy

(

lu;r for all¢ in L 1 (['),/L in L""(f'). Under this pairing L 00 ( [ ' ) is the dual space of L 1 ([') (i.e., every complex valued linear functional 1 on L 1 ([') is equal to ( , Jl) for some Jl in L ""([') ). The space of integrable quadratic differentials of

r,

A(['), is the sub-

space of L 1([') consisting of holomorphic differentials. THEOREM V:

r

Let

be the Fuchsian group acting on U. If 1L in M(f')

S \IlL+,\ \\ 0 for ail t/J

is extremal then \\Ji.\\ 00 {,\ E

L (r): (t/J, ,\) = 00

for ail ,\ in A([')L. where A(f')J. =

00

E

A([')}.

Proof: By Theorem II we have

But f3(r)(O, *(0)11) = infla(r)(O,v): iO)v

=

*(O)Jl, v ( L 00 (r)l

infla(r)(O,/L + ,\): *(0),\"" 0, ,\ E L 00(f')l infla (r)(O, IL + ,\): ,\ E A(f')J.l because Ker *(0) Since L oo(r)

=

A([')

J.

by Bers [l].

is the dual of L 1(['), Theorem V is just a reformulation of

a theorem by Hamilton [ 4] which says that if IL in M(f') is extremal then its norm

\IlL I A(r)\1

as a linear functional on A([') equals its norm IIIL lloo

as a linear functional on L 1(['). Clearly IIIli A([') II -< IIIL II 00 • By the Hahn~ Banach theorem there is a linear functional k on L 1 (r) with norm equal to 11111 A(r)\1 such that k(¢) Therefore k

=

(¢,Jl) for all ¢ < A([').

= ( · , v) for some v in

L oo(f') and \lv \\ 00

= \\111 A(r)J\. But

328

BRIAN O' BYRNE

of course

(¢, v)

= (¢, ll) for all ¢ in A([') .

A f A(l).L. Thus lllliA(l)ll = llvlloo = llll+AIIoo > 111111 lllliA(f')ll = 111111

So v = 11+>.. with

00

by Theorem V. Therefore

00 •

REFERENCES [1]

L. Bers, On moduli of Riemann surfaces, lecture notes Eidgenossische

Technische Hochsule, Ziirich, 1%4. [2]

C. J. Earle and J. Eells, Jr., On the differential geometry of Teichmiiller spaces, Journal D'Analyse Mathematique (1%7), 35-52.

[3]

_ _ _ , Foliations and fibrations, Journal of Differential Geometry

(1967), 33-41. [4]

R. S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Transactions of the American Mathematical Society

(1%9), 399-406.

[5]

S. Lang, Introduction to Differentiable Manifolds, lnterscience, New York, 1962.

REPRODUCING FORMULAS FOR POINCARE SERIES OF DIMENSION -2 AND APPLICATIONS K. V. Rajeswara Rao * §1. Introduction: Let r denote a group of conformal self-maps of the open unit disc U of the complex plane, acting discontinuously and freely on U. Thus r is to contain no elliptic transformations. Our concern is with the Hilbert space A(r) of square integrable automorphic forms of dimension -2 belonging to r, or, equivalently, with holomorphic differentials on the Riemann surface U/r which are square integrable in the sense of Ahlfors and Sario ([1], Ch. V). We try to construct elements of A(r) as Poincare series of dimension

-2 :

I.r (r

f (Tz) . T '(z). For the existence of such Poincare series, the

natural condition to impose on r is that IT f r IT '(z)l < ""· We thus assume, throughout, that r is of convergence type, i.e., for one, and hence all, z in U (1.1)

I (1-ITzi 2) T(r

in D(R): if (3i = aidx+bidy

(i= 1,2), = Y2 ((R(a 1a2 +b 1h2 )dxdy.

, POINCARE SERIES OF DIMENSION -2

Let 11/311

= 1h.

331

If, as usual, we identify differentials which "differ"

only on a set of measure zero, it is readily verified that D(R), with the above inner product, is a Hilbert space. Note that the correspondence F .... /3 = F(z) dz is a linear isometry of A(r) into D(R). This allows us to regard A(r) as a closed subspace of D(R). We shall use this identification. The standard proof of the following lemma is omitted. LEMMA: Let an= an(z)dx + bn(z)dy < D(R) (n = 1, 2, ... oo) and a _ a(z) dx + b(z)dy be a differential on R. Assume that an converges weakly in D(R) to ii and that, in any closed parametric disc, an(z) .... a(z) and bn(z) .... b(z) uniformly as n .... oo. Then ii

= a.

§3. Poincare series: For a function f defined on U, we say that

e(f, r)

exists if every arrangement of the series ~ T and II

I

being

gn(p, q) .... g (p, q) uniformly on compact

subsets of R x R, it follows readily that, on every closed parametric disc of R, a2gn

azaw Hence sup llanll n

an < D(R) and


4 are the direct structural generalizations of it first conjectured rather imprecisely by Schottky and Jung, [12]. For convenience I call such period relations relations of Schottky type. Very succinctly, given a Riemann surface S of genus g 2: 4 and on it a canonical homology basis

y1 , ... ,y g; o1 , ... ,o g

(thus, in effect, a Torelli surface,

(7], over S), in order to write down a relation of Schottky type among the periods "ij first kind on to use

= J8. d (i , where d ( is the vector of normal differentials of

s

J

determined by oij =

fy. d(i'

i, j = 1, ... , g, one has only

J

Prescription A. Take any identity (actually only certain identities are useful-see below) among the (g- 1) -theta constants on the generalized upper half -plane @5g_ 1 , of (g- 1) x (g- 1) symmetric complex matrices with positive-definite imaginary part and replace each such constant by the square root of the product of the two g-theta constants whose characteristics consist of the (g- 1) -characteristic of that constant bordered on the left by the columns ( ~] and ( ~] , respectively, and whose period matrix arguments

* Research

partially sponsored by the Air Force Office of Scientific Research,

Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. AFOSR 1873-70.

341

342

HARRY E. RAUCH

are precisely rr

=

(rrij). (In [9] and [ 4], to which I also refer for definitions,

these are called Riemann theta constants on S with respect to the given homology basis.) In [ 4] this strikingly simple prescription is used to derive (one form of) Schottky's relation for g = 4 and a typical relation of Schottky type for g =

5. The results of which I have spoken are the outgrowth of a several year

period of intensive collaboration between Professor Farkas and me, and the final outcome is joint work. During the course of our work, Farkas, on his own, had a brilliant insight that provided us with a fundamental breakthrough and which constitutes a beautiful theorem in its own right. All of this is recorded in three PNAS notes [9], [3], [ 4] of which the first and last are joint and the middle is by Farkas. Farkas will speak about his work in his lecture. Since the note [ 4] has quite a readable outline and a fair amount of detail it seems pointless to me to rehash it here in view of time limitations. I prefer here, instead, to illustrate by a concrete example the application of Prescription A. This will serve the additional purpose of introducing the listener and reader to the marvelously fascinating calculus-perhaps I should say algebra-of the theta constants and their characteristics. This has a game-like quality to it, particularly the manipulation of the zeros and ones in the characteristics, and, indeed, many of the rules of the game, although quite recondite in origin and derivation, could probably be taught to grammar-school children. Since the key to note [ 4] is the profound study of hyperelliptic surfaces I choose to illustrate a phenomenon on such surfaces for g = 4. To explain the origin of the problem let me backtrack a little and say that the significance of Prescription A and the relations of Schottky type lies in the attempt to make quantitatively explicit the qualitative fact, observed heuristically by Riemann in 1857 and proved in sharper form by me in 1955 (cf. [7]), that the g(g+ 1)/2 periods rrij depend holomorphically on 3g- 3 complex parameters, "the" moduli, which parametrize the set of

PERIOD RELATIONS ON RIEMANN SURF ACES

343

(conformal equivalence classes of) Riemann surfaces of genus g;:::. 2 and hence are subject to g (g+ 1)/2- (3g- 3)

= (g- 2)(g- 3)/2

holomorphic

relations. The relations of Schottky type are such relations-one may view them alternatively as necessary conditions for a matrix in e>g to come from a Riemann surface as

7T

does. There is, at present, the question whether

the totality of relations of Schottky type imply all the required relations for given g, so to say whether they are ample. But it is clear even when they do as for g = 4 that they are superabundant in the sense that one derives algebraically a great many more relations than can possibly be functionally independent. An analogous but considerably simpler situation has long existed for the set of hyperelliptic surfaces. It was recognized certainly by Rosenhain in 1850, if not earlier, that the hyperelliptic surfaces of genus g depend on 2g- 1 moduli, the 2g + 2- 3 unnormalized branch points of a two sheeted representation. Again this was proved by me in sharper form in 1955 (cf. [7]). On the other hand Riemann and Weierstrass both recognized by 1865 at the latest that a necessary condition for a surface to be hyperelliptic or for a period matrix to come from such a surface is the vanishing of an algorithmically determined large set of theta constants and partial derivatives of theta functions for zero values of the argument (cf. [5], Chapter X and [7]). Thus for g = 4 one has 10 theta constants vanishing as compared with g(g+1)/2- (2g-1) one has 4.3/2

=

=

(g-1)(g-2)/2

=

3 period relations, while for g= 5

6 period relations while 66 theta constants and the first

partials of an odd theta function at the origin vanish (cf. [5], Chapter X). Max Noether accounted (cf. [5], Chapter X) for the discrepancy in these two cases by showing that for g = 4 the 10 vanishing conditions are "equivalent" to the vanishing of a subset of 3 theta constants, i.e., the vanishing of the remaining 7 imposes no new conditions and, similarly, the residual 61 vanishing conditions are "dependent" on the vanishing of a certain set of 6 theta constants for g = 5. Here, "dependent" means the existence of a linear relation among the theta constants and derivatives in

344

HARRY E. RAUCH

question whose coefficients are monomials in theta constants none of which vanishes for the particular values of the periods in question. "Equivalent" then has the obvious meaning: mutually dependent. Here, however, the reverse dependence usually means logical inclusion. But in view of the paragraph before the last the additional question arises: if one already knows a minimal functionally independent set of (g- 2)(g- 3)/2 necessary period relations (that cut down g (g+1)/2 to (3g- 3)) on a surface of genus g, is it true that the necessary conditions for hyperellipticity are "dependent" on the preceding general necessary period relations and a subset of g- 2 = 3g- 3- (2g- 1) members of the hyperelliptic conditions? I call attention to the careful formulation of the question. I do not ask for sufficient conditions that a surface be hyperelliptic-like the "g-2 conjecture" (cf. [2] and [l]). Nor do I demand that the vanishing of certain theta constants or period relations imply the vanishing of others-that would require an excursion into the mysteries of theta algebra more extensive than I am now prepared to make. I do not even wish to discuss here the question of precise sense of the functional independence of the minimal set of hyperelliptic necessary conditions since there are also mysteries there (cf. [8]). With that caveat I turn to showing that the answer to the question posed above is affirmative when g

= 4.

More precisely I show that on a hyper-

elliptic surface with g = 4 one can choose a canonical homology basis and a form of Schottky's relation such that the vanishing of a minimal set of three (out of the 10 necessary hyperelliptic conditions) is "equivalent" to Schottky's relation and the vanishing of two of them. Here, "dependence" and "equivalence" must be generalized to include the case that there is a polynomial (not necessarily linear) relation with nonvanishing coefficients in the preceding sense and with the "dependent" theta constant occurring in at least one monomial which does not contain the "independent" theta constants. It is important to observe that Schottky's relation although written in irrational form is really a polynomial relation obtained by multi-

PERIOD RELATIONS ON RIEMANN SURF ACES

345

plication of all conjugates (plus and minus combinations). This comes about by exhibiting in each term of the three term Schottky relation in the special form chosen exactly one theta constant from the special list of 10 which vanish. It seems clear to me, although I have not yet checked the eighth roots of unity which occur, that linear transformation of the theta constants leads to the conclusion that this is always true, i.e., for any choice of canonical homology basis on a hypereilliptic surface there is a form of Schottky's relation such that exactly one of the corresponding list of 10 vanishing constants occurs in each term. What I am unable to foresee now is whether or not any three of the 10 occur in some form of Schottky's relation. This contrasts with Noether's reduction (for g = 4) in which any three can serve as a minimal set. To finish off I shall exhibit Noether's reduction for the particular three constants occurring in Schottky's relation. Thus I shall show that 10 is really 3 is really 2, i.e., Schottky plus 2. 2. Hyperelliptic surfaces of genus four.

Let S be a hyperelliptic surface of genus g = 4. One may assume S represented as a two-sheeted surface over the z-plane with branch points at 0, 1, .\ 1 , ... ,.\7 , "", where no \

lies on the real axis between

-oo

and 1

and a simple (rectilinear) polygon joins the branch points in the order indicated. With these conventions I may represent S without loss of topological generality as shown by the conventional diagram in Figure 1, and I may draw a canonical homology basis as shown there. I now compute the table of 9 half-periods obtained by integrating the vector of differentials of first kind from the branch point 0 to the 9 other branch points. (See Table 1 on the following page.) The expressions on the left are defined by those in the middle. The sum of the half-periods of Table 1 with odd (or even) symbols is 0 1 1 1

1/31 , then :l

Ia + ,81 - Ia I - Re a{J < lal-

l ( Im a{J ) 2(lai-1,8J) lal

We also have for all a and ,8. la+.BI - lal - Re

afJ


( -E, E)

a differentiable (C 00 ) real-valued function. We call the corresponding classical surface Sh and we denote by 17h the mapping X .... X + hN. Deformations of a polyhedron S0 are defined analogously, except that h shall be a piecewise linear map on S0 which is zero in a neighborhood of the edges. §3. Embedding theorems Let S 0 be a fixed Riemann surface. A pair (R, f) is called a topologically marked Riemann surface, if f is a topological, orientationpreserving mapping of S0 onto R. Two marked surfaces (R, f) and (R*, f*) are conformally equivalent if f*

o

f- 1 is homotopic to a conformal mapping.

The equivalence classes are the elements of Teichmiiller space. THEOREM

A. Let S0 be any Riemann surface obtained from a compact

classical surface by deleting a finite number of points and let

E

be any

positive number. Then every topologically marked Riemann surface (R, f: S0

....

R) is conformally equivalent to a suitable

(Sh, 17h) of S0

E -deformation

•

Remarks. 1. Note that R and S 0 may be quasiconformally inequivalent. 2. If

E

is sufficiently small, all the Sh are classical surfaces without self-

intersections. 3. In the proof of Theorem A we only make use of the fact that S0 is

DEFORMATIONS OF EMBEDDED RIEMANN SURFACES

387

immersed rather than embedded in 3- space. Thus Theorem A remains valid even if S0 is merely immersed in 3- space, except that then, of course, Remark 2 is no longer applicable. 4. As a corollary of the theorem and its proof we can obtain the following result: THEOREM B. Let S0 be a compact polyhedron without self-intersec-

tions and

E

a positive number. Then every topologicaliy marked Riemann

surface (R, f: tion

n .... 4'>n· If z 0 is the origin of 4'>n corresponding to

i 0 = (0 ,z 0 ) is the origin of We x 4'>n corresponding to K . A e-valued function 71(g,z-) (g f G, if We x 4'>n) is called a (holo-

K, then

morphic) automorphy factor, if it satisfies the following conditions: (la)

71 is C00 in g and holomorphic in

(lb)

71(g1 g2 ,

z)

=

i ,

71(g1 , g 2 (i)). 71(g 2 ,

It follows from the condition (lb) that 71

z)

for all g 1 d~ 2 (

(k , ; 0 )

d,

is a character of

z(We x4'>n' K.

In the

following we will exclusively be interested in the case where this character is given by k

= (t, k) 1--> t, i.e., 71((t, 0, k), (O,z 0 ))

(lc)

= t for all t f T, k f K.

Two such automorphy factors 71, 71' are said to be equivalent if there exists a holomorphic function ¢ on We x 4'>n such that one has

LEMMA

1. There exists an automorphy factor 71 on (G, We x 4'>n) sat-

isfying the conditions (la) - (1 c). Moreover such an automorphy factor is unique up to the equivalence. In fact, one can obtain a "canonical" automorphy factor

by extending from G to G the notions of Harish-Chandra imbedding and Cayley transformation and then imitating the process of defining the canonical automorphy factor J : G x 4'>n .... GL (We) (see [6]) . The sought-for automorphy factor Tf is then obtained by taking the ex -component of

j.

The explicit form of Tf is as follows (2) Tf(g, z) = t · E{Y:!A(u, ug(z))+ A(u, J(g, z)w) + 'f 0 for

(8) is absolutely convergent; more precisely, one

has

fwe I ( -, -')

(8 ')

K Z

, Z

-)I K n), which is possible since .~) n

is simply connected. In particular, one has E(1; z ', z) = 1, which means that U z, z gives a unitary equivalence of the Fock representations

('J z'

T~,) u

and OJz ,,

T~,) (u u

f

V).

Thus all Fock representations are

mutually equivalent; this fact is, of course, a special case of the uniqueness theorem of Stone and von Neumann. For

g=

(*, *, g)

f

G , we put

. z T g = Uz,g(z)

(12)

o

z Tg

Then, by (11), one sees that the assignment g "'- T~ is a continuous map g from G into the group of unitary operators of i:s: z satisfying the relation (13)

where az (g 1 , g 2) = s(g 1 ; g 2(z), z). az is a continuous 2 -cocycle of G and, if one puts E(g) = s(arg(det(J(g, z)))), one has

Thus,

-

T~ (g G) f

g

gives rise to a (continuous) projective representation of

G with a factor set az of order 2, or equivalently, to an ordinary unitary representation of a covering group of G of order 2. The existence of such a representation (in a more general setting) has been proved by Shale [7] and Weil [8.2] using another (equivalent) representation-space L 2(W). *

*

A unitary equivalence from L 2(W) onto

iJz

is given by the

correspondence L2(W)

J

ct> .... ¢(w) = 2n/4det(z/i)-'h

det(y)~

1w

e-rriz-l[w-u]ct>(u)du.

402

ICHIRO SATAKE

3. Theta-functions (cf. [8]). Suppose there is given a lattice L in V such

n W+ L n W* and that A(f, f ') n W, L 2 = L n W* and denote by

Z for all f,

e' t

that L = L

t

L. We put

L1 = L

L/ (resp. L2) the dual lattice

of L 1 (resp. L 2 ) in W* (resp. W) with respect to the inner product = - A(w, w*) (w

f

W, w*

plies that L 1 C L2 , L 2 C L!

t

W*). Then the above condition im-

and one has L2 /L 1

:::!

L! /L 2 •

A semi -character ¢ of L is by definition a map ¢ : L 1/J (f) · E (Yz < e 1 , f 2 >) (f

f1

t

-+

T such that

L) is a character of L, where one writes f =

+e 2

with fit Li. For every r tV, r= r 1 + r2 , r 1 one can define a semi -character ¢ r by

t

W, r2

t

W*,

which depends only on r (mod. L! + L2 ). It is clear that all semi -character of L can be written in this form. When r

f

V0 = L

®

Q, ¢ r is called

rational. Now, for r

f

V, one defines a theta-function ()r as follows:

()r(w, z) =I E(~z[r 2 +f 2 ]+) f2t L2

(14)

It is well -known that this series is normally convergent and expresses a

holomorphic function on We x ~ n . Moreover, it is immediate from the definitions that one has

for all

et

L, where

ez = e1 ()rz

zf 2 • We put further

(w) = det (y)lj.. · (Jr(w, z)

and, for a given semi -character ¢,denote by

®$

the linear space formed

of all holomorphic functions () on We such that one has for all

e

f

L, where one writes

I

= (l

,e 'l) ( E G>.

Tf ()

= o/(E)- 10

Then, it is also well-

known (and immediate) that dime E>$ = [L! : L 2 ] and that, if ¢ = ¢ rO

FOCK REPRESENTATIONS AND THETA-FUNCTIONS

with r 0

E

403

V and if R denotes a complete set of representatives of L{

modulo L 2 , then

lezo r

+r

(r

E

R) l forms a basis of 8~1. over C. 'f'

For a given semi -character tj;, let f'tj; denote the subgroup of G "' Sp(V, A) formed of ally

E

G such that y(L)"' L

and tj; oy = tf;. [When

tf; is rational, f'tj; is an arithmetic subgroup of G with respect to its Qstructure defined by L.] Since we have

for

e ( 8$ '

y

E

f' tj; '

e ( L,

one has

(15)

for y

E

f' tj;

On the other hand, although 8~ is not contained in i} z itself, the operator Uz,z , can also be extended to 8~'f'1• in a natural sense. Namely, if we denote by (~z )00 the G~rding space of \5z, i.e., the subspace of i}z formed of all analytic vectors, and by ( ~ z)- 00 the conjugate dual of ( 3 z)oo , . a natural sense. * The then one has ~ z C ( 3 z)_ 00 , 8 ztj; C ( \5 z)_ 00 1n isomorphism Uz,z , can therefore be extended canonically to an isomorphism of (\5z)_oo onto (3z')_ 00 • Ifoneputs f/z)= s('liz[r 2 ]+), an easy computation shows that

f.we

K(z',z) • K(Z, z)-l f/z)

It follows that one has

* The isomorphism

uz ', z

L 2 (W)

~

~ZW

= det(J/z'-z))'li 'det(y)-'li • f/z')

e~ = e~

and hence

3 z given in the second footnote can naturally

be extended to an isomorphism from L 2 (W)_ 00 onto ( 'iY )_ 00

,

where L 2 (W)_ 00

can be interpreted as the space of tempered distributions on W. Under this isomorphism, the theta-function e~ corresponds to the tempered distribu.

tlon S(W)

E

tj;

... 2

-n/4

Ie 2 E !_- 2

-

tj; rl (r 2 + f 2 ), where S(W) = L 2 (W)00 is

the Schwartz space of W and tf;r denotes the Fourier transform of 1

404

ICHIRO SATAKE

(16)

uz,z ' ®3, "" ®~,: 'f' 'f' Combining the relations (15) and (16), one obtains the relation

(17)

for

This means that, for y = (

~ ~)

f

['

y

f

['

r/1 •

*

r/1 , one has a transformation formula

of theta-functions of the following form:

e0 r

+

(t(cz + d)- 1 w, y(z)) r

det(cz+d)'n·'T/(y,i) r

I Az, (y)0 0 ,(w,z) ' f R r ,r r +r

for all r f R. In case r/1 "" r/1 0 is rational, it is known furthermore that the r

matrix (Az, (y)) is independent of z and, when the branch of det(cz +d)

~

2

r ,r

is suitably chosen, gives a representation of ['r/1 , of which the kernel is of finite index in ['r/1.

University of California, Berkeley

*

I am indebted to J .-I. Hano for giving me a chance to read a

manuscript of his recent paper [9] before its publication, in which he gives a proof of (17) by a somewhat different method.

FOCK REPRESENTATIONS AND TIIETA-FUNCTIONS

405

REFERENCES

[l] Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform, I, Comm. Pure and Appl. Math., 14 (1961), 187214; II, A family of related function spaces. Application to distribution theory, ibid., 20 (1967), 1-101. [2] Cartier, P., (1) Quantum mechanical commutation relations and theta functions, Symposium on algebraic groups and discontinuous subgroups,

Proc. Symp. Pure Math., IV, American Math Soc. 1965, 361-383. (2) Theorie des groupes, fonctions theta et modules des varietes abeliennes, Sem. N. Bourbaki, 20e annee, 1967/68, Exp. 338. [3] Godement, R., Fonctions holomorphes de cam~ sommable dans le demiplan de Siegel, Sem. H. Cartan, E. N. S., JOe annee, 1957/58, Exp. 6. [4] Harish-Chandra, (1) Representations of semisimple Lie groups, IV,

Amer. ]. Math., 77 (1955), 743-777; V, ibid., 78 (1956), 1-41; VI, ibid., 564-628. (2) Discrete series for semisimple Lie groups, I, Acta Math., 115 (1965), 241-318; II, ibid., 116 (1966), 1-111. [5] Murakami, S., (1) Cohomology of vector-valued forms on symmetric spaces, Lecture-notes at Univ. of Chicago, Summer 1966. (2) Facteurs d'automorphie associes a un espace hermitien symetrique, Geometry of Homogeneous Bounded Domains, Centro Int. Mat. Estivo, 0

3 Ciclo, Urbino, 1967. [6] Satake, I., On unitary representations of a certain group extension (Japanese), Sugaku, Math. Soc., Japan, 21 (1969), 241-253. [7] Shale, D., Linear symmetries of fre•:! boson fields, Trans. Amer. Math.

Soc., 103 (1962), 149-167. [8] Weil, A., (1) Varietes kahleliennes, Hermann, Paris, 1958. (2) Sur certains groups d'operateurs unitaires, Acta Math., 111 (1964), 143-211. [9] Hano,

J. -1., On theta functions and Weil's generalized Poisson summa-

tion formula, Trans. Amer. Soc., 141 (1969), 195 -210.

'UNIFORMIZATIONS' OF INFINITELY CONNECTED DOMAINS

R.

J.

Sibner

1. Let ~ be a family of Riemann surfaces and

S a subfamily. In the theory of

conformal mappings, what might be called the "universal uniformization problem," is to show the existence of an element S

f

S

in each isomorphism

[i.e. conformal equivalence] class of ~For example, if ~ is the family of Riemann surfaces of genus zero and

S the subfamily of plane domains,

the above is Koebe's "general uniform-

ization principle." For ~ the family of simply connected surfaces and

S

the subfamily consisting of the disk, the finite plane and the extended plane, it is the Riemann mapping theorem. Other examples are; for ~ the family of Riemann surfaces of finite type and

S

either the subfamily of surfaces

obtained as ramified coverings of the unit disk or else as quotient spaces of the unit disk by finitely generated Fuchsian groups or for ~ the family

S as above except that the group . is taken to be a symmetric Fuchsian group.

of symmetric surfaces of finite type and

If ~ is taken to be the family of plane surfaces (i.e. plane domains),

S is

usually described by geometrical properties of the boundary components.

Some examples of families which have been investigated are those whose boundary components are:

*

(1)

parallel slits, radial slits, slits along concentric circles etc.

(2)

convex curves

(3)

circles

Research partially supported by NSF grant GP •.8556.

407

408

R. ] . SIBNER

(4)

homethetic images of given convex curves

(5)

analytic Jordan curves

The present state of knowledge on the above problem for these cases is as follows: (1) Solved for

:D

the family of all plane domains and

S

as described

(as well as many other types of slits). See [9, 14, 16, 22] for references. (2) If p is the isomorphism of a given domain D onto a horizontal slit domain obtained by variation techniques (on the coefficient of p) and q is the corresponding vertical slit mapping then p + iq maps D onto a domain bounded by convex curves, some of which can reduce to slits unless D is finitely connected. Thus the two problems have been solved: of all domains or of finitely connected domains and

S the

:D

the family

respective sub-

families of domains bounded by convex or by strictly convex curves. (3) The problem in this case is the content of the Koebe conjectureevery plane domain is conformally equivalent to a circle domain [10]. The conjecture was shown to be true for finitely connected domains and for certain symmetric domains by Koebe [ ll, 12] , for domains quasiconformally isomorphic to a circle domain by the author [18] and for domains satisfying various extremal length conditions on the "limit boundary components" by Denneberg [6], Grotzsch [7], and Strebel [20 ,21]. Other cases have been considered by Sario [15] and Meschkowski [13]. As an application of the methods developed in the present paper we will (in §5) obtain some results on domains admitting symmetries, including a new proof for the domains considered by Koebe. (4) Courant, Manel and Shiffman [5] treat the case of

:D

the family of

finitely connected domains, and Strebel [21] has obtained a generalization of domains satisfying conditions on the limit boundary components (as in (3)). (5) For

:D

the family of finitely connected domains, repeated use of the

Riemann mapping theorem (or the p + iq maps of case (2) provides a solution. (Of course a solution for any

:D

for which a solution is obtained to (3)

gives automatically a solution of (5). One of the main results of this paper

UNIFORMIZATIONS OF INFINITELY CONNECTED DOMAINS

409

(§3) is a solution of (5) in the case that ~ is the family of infinitely con-

nected domains. This result has been announced in [19]. 2. THE ORDINAL SEQUENCE. With variational techniques, used extensively for slit domains, one does not usually have to look closely at the limit points of the boundary components. For the technique which follows, it is necessary to begin by examining the structure of the limit points. Let

r 0 = r 0(D)

denote the collection of boundary components of the

plane domain D. We form the sequence PI

ro J r1

(1)

J ... J

r-cu

J

=

PI(D) (cf. [8], also [20])

rw+ 1 J

... J

ra

J ...

where w denotes the ordinal type of the natural numbers and a is an arbi trary ordinal number. If a has a predecessor a- l then

ra

is defined as

ra - 1 of components which contain points which are limit points of the components in ra- 1 so that if r 0 is given a metric space

the subset of

structure (see [20]) then ordinal, we define

ra

=

ra

is the derived set of

r a- 1•

If a is a limit

n{3

y .::; o .:S

D0 for

h (a ,f)

o

-->

Dy for all y .:S a, and isomor-

a such that (for

y .::; o .::;

E .::;

a)

h (y ,o) , h (y, d

h (y' o) is the restriction of an isomorphism h (y' o) of the domain bounded by the elements of rY(Dy).

(iii)

f(o) , h (y, a)

(iv)

The elements of AY(Dy) = r 0 (Dy) - rY(Dy) are analytic

o

f(y)

Jordan curves. S(O) is trivially true for f(O) = h(O,O) = identity. Suppose S(/3) is true for all {3 < a. We show the truth of S(a). (Case 1) If a is not a limit ordinal and hence has a predecessor

a~

l,

D 1 such that Aa- 1(D 1 ) aaconsists of analytic Jordan curves. Let Da_ 1 ;} Da_ 1 be the domain bounded by the elements of ra- 1(Da_ 1). Then ra- 1(Da_ 1)-ra(Da_ 1)

then there exists an isomorphism f (a -1): D

-->

are the isolated boundary components of Da- 1 • By Lemma 1 there exists an isomorphism

h(a- l ,a)

of

Da- 1

mapping these boundary components onto analytic Jordan curves. The elements of Aa- 1(D 1) are analytic a-

D, f(y)

' , f(a)

'' f(a-1)

"'

'"

"'

D - - - - - - - Da-1-----~ Da Y h(y,a-1) h(a-1,a)

n

1\

n

n

~

--,_;;-----+ 0 a-1--=----+

h(y, a-1)

h(a-1,a)

R. ]. SIBNER

412 Jordan curves contained-in

Da-l: and hence_are mapped by h onto analy-

tic Jordan curves. Let h(y ,a) = h(a -1, a) o h(y, a-1) and h (y, a) be the restriction of h(y, a) to Dy. Let f (a) = h (y, a) o f (y) which is well defined, independent of y, by conditions (i) and (iii), and let Da be the image of D under f(a). Then conditions (i) -(iv) for a can be immediately verified. (Case 2). If a is a limit ordinal, let I,Bkl, ,Bk

< a,

be the sequence of the

remark of §2. Then by the induction hypotheses S (,Bk) holds. For each i,

-

-

the maps h (,8 i' ,B j) :

-

D,B . .... D,B. form a normal family (if they are normalized ) be mapped onto by requiring, for example, 1 that a J fixed element of r,B·< 1 D,a. 1

the unit circle.) By the familiar diagonalization procedure we obtain a subsequence jn. such that, for every i, h(,Bi, ,Bj ) .... h(,Bi, a) uniformly on -

n

-

compact subsets of D,ai· Let h(,Bi' a) be the restriction of h(,Bi, a) to D,ai· Clearly, for k > i, by condition (i)

Since f (,8. ) = h (,8., ,B. ) o f (,8.) for every i, the f (,8. ) converge to a 1 1 Jn Jn Jn to a map f (a) : D .... Dy and (3)

f(a) = h(,Bi' a)

o

f(,Bi)

for every i.

Using the remark cf. §2 we can define for arbitrary y h(y ~) = h(,Be,a)

(4)

o

= {IRe (z- a) I < hi,

l\ =

hi and U the upper half plane. Then there exists a quasicon-

formal map of E

nU

onto (E -l\)

nU

{z : IRe (z- a) I = h and Im z > 0 I.

which is the identity on

415

UNIFORMIZATIONS OF INFINITELY CONNECTED DOMAINS

COROLLARY. If D is a domain whose boundary components are slits

along the real axis, then there exists a homeomorphism f of D onto a domain bounded by circles with centers on the real axis, such that f is quasiconformal in the exterior (with respect to D) of any neighborhood of the limit boundary points of D. (Such points occur, of course, as end points of the limit boundary slits.) LEMMA

4. Let a be a continua and fJ a circle contained in the interior

of an analytic Jordan curve y. Then there exists a quasiconformal map of the ring domain bounded by a and y onto the ring domain bounded by fJ and y which leaves y pointwise fixed.

The proof of Lemma 3 is by a trivial construction. A proof of Lemma 4 may be found in [18]. We now prove Lemma 1 of §3 ; that every domain D is isomorphic to a

domain whose isolated boundary components are analytic Jordan curves. Proof of Lemma 1. By quasiconformal deformations (Lemma 4) in the neighborhood of each isolated boundary component of D, there exists a domain K whose isolated boundary components are circles lckl, and a borneo~ f: K ... D which is quasiconformal in the exterior (with respect to K) of any neighborhood of the limit boundary components of K. Moreover, for each ck, there exists an exterior domain Nk of ck such that 11 By Lemma 2 applied to all the ck there exists a 11*

t

t

B(Nk ):

B (K* ), where K*

contains K as well as each ck, such that if g is a 11*- conformal map of K*, g(ck) is an analytic Jordan curve. Since g and f are both wconformal in K, h domain.

=

go r 1 is conformal in D. Then h(D) ( = g(K)) is the required

R. J. SIBNER

416

Remark. The stronger result that D is isomorphic to a domain whose isolated boundary components are circles is given in [19]. The proof there is similar to the one above but makes use of the full group of reflections in the isolated (circular) boundary components of K. This result is also implicit in [18, p. 294], [20, p. 20] and [13, p. 395]. See also Note end §5. 5. DOMAINS ADMITTING SYMMETRIES. As a further example of the induction technique introduced in §3 we consider now the "Koebe problem" for symmetric domains. A domain D is said to be symmetric (of degree n) if it admits anti-holomorphic automorphisms a 1 , ... , an. Using the terminology of §1 we will show that for various families g) of symmetric domains the subfamily

S

of circle domains contains an element in each isomorphism

class of g), THEOREM 2. (Koebe) Let D be a domain with countably many boundary components which admits a symmetry a with a 2 =identity and D/a

simply connected. Then D is isomorphic to a circle domain. Proof: By the Riemann mapping theorem and the reflection principle we can assume that the boundary components of D are slits on the real axis. By the Corollary to Lemma 3 there exists a circle domain E and a homeomorphism f : E -+ D which is quasiconformal on the complement E ' of any neighborhood of the limit boundary points of E. Then 11. (z) = fz /fz

f

B(E ')

and as is [18] (by a modification of the proof of Theorem 5 in Bers [3]) one obtains a locally quasiconformal homeomorphism

vP: E-+ D0 and hence an

isomorphism D .... D0 • A priori, the boundary components of D0 are as follows: Each component is symmetric with respect to reflection in the real axis. The isolated components are circles and the limit components consist of two symmetrically situated arcs y 1 and y 2 of some circle, together with the "singular" continua a and f3 as shown . The singular continua correspond, under the isomorphism D .... D0 to the end points of a limit boundary

UNIFORMIZATIONS OF INFINITELY CONNECTED DOMAINS

417

f3

a

y2 slit of D, and thus consist entirely of limit points of other boundary components of D 0 • To complete the proof we show that all singular continua are, in fact, points. This will be done by the induction method of §3. Here, however, the situation is much simpler. Consider the following statement: S(a):

The elements of Aa = Aa(D 0 ) are circles.

S (0) is vacously true. Suppose S(/3) holds for f3

< a.

Case 1. (a not a limit ordinal) The singular continua of the elements of

Aa consists entirely of limit points of the elements of Aa-l. But, by the induction hypothesis, these are circles (or points) which are symmetric with respect to reflection in the real axis and hence can accumulate only at points on the axis. Since the singular continua of Aa are therefore points, the Aa are actually circles.

Aa then, since ra(D 0 ) = nB