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Table of contents :
CONTENTS
PREFACE
A GEOMETRIC PROPERTY OF BERS’ EMBEDDING OF THE TEICHMÜLLER SPACE
PLANE MODELS FOR RIEMANN SURFACES ADMITTING CERTAIN HALFCANONICAL LINEAR SERIES, PART I
NONTRIVIALITY OF TEICHMÜLLER SPACE FOR KLEINIAN GROUP IN SPACE
THE ACTION OF THE MODULAR GROUP ON THE COMPLEX BOUNDARY
SOME REMARKS ON BOUNDED COHOMOLOGY
THE DYNAMICS OF 2GENERATOR SUBGROUPS OF PSL (2, C)
MINIMAL GEODESICS ON FRICKE’S TORUSCOVERING
ON VARIATION OF PROJECTIVE STRUCTURES
VANISHING THETA CONSTANTS
ANALYTIC TORSION AND PRYM DIFFERENTIALS
ON A NOTION OF QUASICONFORMAL RIGIDITY FOR RIEMANN SURFACES
SPIRALS AND THE UNIVERSAL TEICHMÜLLER SPACE
INTERSECTION MATRICES FOR BASES ADAPTED TO AUTOMORPHISMS OF A COMPACT RIEMANN SURFACE
HOMOMORPHISMS OF TRIANGLE GROUPS INTO PSL (2, C)
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
AUTOMORPHISMS OF COMPACT RIEMANN SURFACES AND WEIERSTRASS POINTS
AFFINE AND PROJECTIVE STRUCTURES ON RIEMANN SURFACES
BOUNDARY STRUCTURE OF THE MODULAR GROUP
A REALIZATION PROBLEM IN THE THEORY OF ANALYTIC CURVES
THE MONODROMY OF PROJECTIVE STRUCTURES
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES AND TEICHMÜLLER SPACES
COMMUTATORS IN SL(2, C)
TWO EXAMPLES OF COVERING SURFACES
DEFORMATIONS OF SYMMETRIC PRODUCTS
REMARKS ON PROJECTIVE STRUCTURES
SOME REMARKS ON KLEINIAN GROUPS
REMARKS ON WEB GROUPS
REMARKS ON FUCHSIAN GROUPS ASSOCIATED WITH OPEN RIEMANN SURFACES
ON GENERALIZED WEIERSTRASS POINTS AND RINGS WITH NO PRIME ELEMENTS
THE TOPOLOGY OF ANALYTIC SURFACES: DECOMPOSITIONS OF ELLIPTIC SURFACES
DENSE GEODESICS IN MODULI SPACE
AUTOMORPHISMEN EBENER DISKONTINUIERLICHER GRUPPEN
REMARKS ON THE GEOMETRY OF THE SIEGEL MODULAR GROUP
ON THE ERGODIC THEORY AT INFINITY OF AN ARBITRARY DISCRETE GROUP OF HYPERBOLIC MOTIONS
ON INFINITE NIELSEN KERNELS
HYPERBOLIC 3MANIFOLDS WHICH SHARE A FUNDAMENTAL POLYHEDRON
THE LENGTH SPECTRUM AS MODULI FOR COMPACT RIEMANN SURFACES
Annals of Mathematics Studies Number 97
RIEMANN SURF ACES AND RELATED TOPICS: PROCEEDINGS OF THE 1978 STONY BROOK CONFERENCE EDITED BY
IRWIN KRA AND
BERNARD MASKIT
PRINCETON UNIVERSITY PRESS AND
UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1981
Copyright © 1981 by Princeton University Press All articles in this collection are © 1981 ALL RIGHTS RESERVED
Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press
Printed in the United States of America by Princeton University Press, Princeton, New Jersey
Library of Congress Cataloging in Publication data will be found on the last printed page of this book
CONTENTS PREFACE
~
A GEOMETRIC PROPERTY OF BERS' EMBEDDING OF THE TEICHMULLER SPACE by William Abikoff
3
PLANE MODELS FOR RIEMANN SURF ACES ADMITTING CERTAIN HALFCANONICAL LINEAR SERIES, PART I by Robert D. M. Accola
7
NONTRIVIALITY OF TEICHM'OLLER SPACE FOR KLEINIAN GROUP IN SPACE by B. N. Apanasov
21
THE ACTION OF THE MODULAR GROUP ON THE COMPLEX BOUNDARY by Lipman Bers
33
SOME REMARKS ON BOUNDED COHOMOLOGY by Robert Brooks THE DYNAMICS OF 2GENERATOR SUBGROUPS OF PSL(2,C) by Robert Brooks and J. Peter Matelski
53
65
MINIMAL GEODESICS ON FRICKE'S TORUSCOVERING by Harvey Cohn
73
ON VARIATION OF PROJECTIVE STRUCTURES by Clifford J. Earle
87
VANISHING THETA CONSTANTS by Hershel M. Farkas
101
ANALYTIC TORSION AND PRYM DIFFERENTIALS by John Fay
107
ON A NOTION OF QUASICONFORMAL RIGIDITY FOR RIEMANN SURF ACES by Frederick P. Gardiner SPIRALS AND THE UNIVERSAL TEICHMULLER SPACE by F. W. Gehring INTERSECTION MATRICES FOR BASES ADAPTED TO AUTOMORPHISMS OF A COMPACT RIEMANN SURFACE by Jane Gilman and David Patterson v
123
145
149
vi
CONTENTS
HOMOMORPHISMS OF TRIANGLE GROUPS INTO PSL (2, C) by Leon Greenberg HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS by M. Gromov
167 183
AUTOMORPHISMS OF COMPACT RIEMANN SURFACES AND WEIERSTRASS POINTS by Ignacio Guerrero
215
AFFINE AND PROJECTIVE STRUCTURES ON RIEMANN SURF ACES by R. C. Gunning
225
BOUNDARY STRUCTURE OF THE MODULAR GROUP by W. J. Harvey A REALIZATION PROBLEM IN THE THEORY OF ANALYTIC CURVES by Maurice Heins THE MONODROMY OF PROJECTIVE STRUCTURES by John H. Hubbard HOLOMORPHIC FAMILIES OF RIEMANN SURFACES AND TEICHMOLLER SPACES by Yoichi lmayoshi
245
253 257
277
COMMUTATORS IN SL(2, C) by Troels J0rgensen
301
TWO EXAMPLES OF COVERING SURF ACES by T. J0rgensen, A. Marden and Ch. Pommerenke
305
DEFORMATIONS OF SYMMETRIC PRODUCTS by George R. Kempf
319
REMARKS ON PROJECTIVE STRUCTURES by Irwin Kra and Bernard Maskit
343
SOME REMARKS ON KLEINIAN GROUPS by S. L. Krushkal
361
REMARKS ON WEB GROUPS by Tadashi Kuroda, Seiki Mori and Hidenori Takahashi
367
REMARKS ON FUCHSIAN GROUPS ASSOCIATED WITH OPEN RIEMANN SURF ACES by Yukio Kusunoki and Masahiko Taniguchi
377
ON GENERALIZED WEIERSTRASS POINTS AND RINGS WITH NO PRIME ELEMENTS by Henry Laufer
391
THE TOPOLOGY OF ANALYTIC SURF ACES: DECOMPOSITIONS OF ELLIPTIC SURF ACES by Richard Mandelbaum
403
CONTENTS
DENSE GEODESICS IN MODULI SPACE by Howard Masur
vii
417
AUTOMORPHISMEN EBENER DISKONTINUIERLICHER GRUPPEN by Gerhard Rosenberger
439
REMARKS ON THE GEOMETRY OF THE SIEGEL MODULAR GROUP by Robert J. Sibner
457
ON THE ERGODIC THEORY AT INFINITY OF AN ARBITRARY DISCRETE GROUP OF HYPERBOLIC MOTIONS by Dennis Sullivan
465
ON INFINITE NIELSEN KERNELS by Judith C. Wason
497
HYPERBOLIC 3MANIFOLDS WHICH SHARE A FUNDAMENTAL POLYHEDRON by Norbert J. Wielenberg
505
THE LENGTH SPECTRUM AS MODULI FOR COMPACT RIEMANN SURF ACES by Scott Wolpert
515
PREFACE This volume contains papers and abstracts by participants of the Conference on Riemann Surfaces and Related Topics, which was held at the State University of New York at Stony Brook, June S9, 1978. This was the fourth in a series of conferences on more or less the same subject (Tulane 1965, Stony Brook 1969, Maryland 1973). We invited papers from all the Conference participants, with acceptance for publication subject to refereeing. All the manuscripts were indeed refereed by participants, and not all were accepted. As usual, thanks are due to the National Science Foundation for financial support, the State University of New York at Stony Brook for its hospitality, and Princeton University Press for providing a series where these Proceedings could be published (volumes 66 and 79 contain the Proceedings of the previous two Conferences). Most of all we thank the participants in the Conference who wrote these papers and who refereed them, who gave invited lectures and seminar talks, and who talked mathematics and created the atmosphere of excitement that made our publishing effort worthwhile. We were particularly pleased by the appearance (both at the Conference and in these Proceedings) of many new (both young and old) faces. Mathematicians from many diverse fields are now interested in Riemann Surfaces and Kleinian groups. We are delighted that the old classical theory of functions of one complex variable still shows so many signs of vitality. I. Kra
B. Maskit
MAY 1979
ix
Riemann Surfaces and Related Topics
A GEOMETRIC PROPERTY. OF BERS' EMBEDDING OF THE TEICHMULLER SPACE William Abikoff In this short note we prove a geometric property of the Bers embedding of the Teichmi.iller space. To fix the notation, let G be a finitely generated Fuchsian group of the first kind acting in the unit disc L'l . The Bers embedding of T(G)
=
T(l'l/G) represents T(G) in the space B of bound
ed quadratic differentials ch for G in the exterior E of L'l. In the usual way we associate to each ch
f
B, the normalized solution DO
of the Schwarz ian differential equation
l frh, z!
=
ch. It is important to
note that frh, hence bn(rh) is a holomorphic function on B. Set Grh frh Gfrh
1
and let i: T(G) .... B be the Bers embedding. If ch ET
=
=
i(T(G))
then frh is schlicht and Grh is a bgroup. Let A(Grh) denote the limit set of Grh and m(c/J) be the area of A(Grh). We prove the following THEOREM. If c/J
E
ai(T(G))
and m(c/J)
=
0, then c/J
E
a Ext i(T(G)).
Proof. The tripartite classification of bg;roups shows that if ch (a i(T(G))
then Grh is either totally degenerate or has accidental parabolic transformations. The two cases must be handled separately.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedin~s of the 1978 Stony Brook Conference 0691082642/80/00000303$00.50/1 (cloth) 0691082677/80/00000303$00.50/1 (paperback) For copying information, see copyright page 3
4
WILLIAM ABIKOFF
If Grh has accidental parabolic transformations, then there is some y rh
f
Grh so that r( rh) = tr 2 y rh = 4 , but r( rh) is a non constant holomor
phic function on B . Thus near rh, r takes on all values sufficiently close to 4 . It follows that there are groups Gt/J arbitrarily close to Grh which have elliptic elements of infinite order. Such groups are not Kleinian and 1/J
I
ai(T(G)).
We proceed to the case where Grh is totally degenerate. Set
A : B .... [oo, 11]
Since bn(t/J) is holomorphic on B, A is plurisuperharmonic. Further, Gronwall's Area Theorem says that if ft/J is schlicht then A(t/J) is the area of C \ft/J(E). Assume Grh
f
satisfies A(fb) ;::: (2rr) 1
Int i(T(G)). Then any holomorphic map
J
A(h(eiO))dO 2: 0.
a/1 But we may choose a holomorphic disc h(/1) with center rh and intersecting i(T(G)) along a nontrivial boundary arc and such that h(/1) C i(T(G)). It follows from the above inequality that A(fb)
> 0. But for totally degen
erate groups, A(fb) = m(ci>) and we have assumed m(ci>) = 0. We have the desired contradiction. SOME REMARKS
1) The theorem is a finite dimensional version of Gehring's theorem that the universal Teichmilller space is the interior of the Schwarzians of schlicht functions.
5
BERS EMBEDDING OF THE TEICHMULLER SPACE
2) At this conference, Thurston announced a proof that m()
=
0, for all
boundary groups of the Teichmiiller space. This result eliminates the need for our main hypothesis. 3) The second part of the proof may be repeated verbatim to prove the following statement. Given any holomorphic mapping of the punctured disc into Teichmiiller space (or equivalently, a holomorphic family of finite Riemann surfaces over the unit disc), then the puncture (or central fiber) cannot be filled in by a totally degenerate group whose limit set has zero area.
UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
PLANE MODELS FOR RIEMANN SURF ACES ADMITTING CERTAIN HALFCANONICAL LINEAR SERIES, PART I* Robert D. M. Accola**
1. Introduction Let Wp be a Riemann surface of genus p . In considering vanishing properties of the thetafunction at half periods of the Jacobian associated with Wp, one is led naturally, via Riemann's vanishing theorem, to halfcanonical linear series on Wp, that is, to linear series whose doubles are canonical. A theorem of Castelnuovo assures us that a halfcanonical grp_ 1
must be composite if p < 3r, and this leads directly to the exis
tence of automorphism of period two on Wp [2, Part III]. In this paper we are concerned with surfaces where p = 3r and Wp admits a simple grp_ 1 (which must necessarily be halfcanonical). We show that such surfaces exist for all r and we investigate the consequences. By another theorem of Castelnuovo it follows that, except for r
=
5, the existence of a simple
on W3 r insures the existence of a g 1 4 without fixed points. This in turn implies that such a Riemann surface has a plane model where the
grp_ 1
halfcanonical gr 3 r_ 1 is easily seen. From these models one easjly calculates the dimension of such Riemann surfaces in Teichmtiller space. The methods developed here also allow us to characterize, for such surfaces, when the divisors of the g 1 4 are the orbits of an automorphism group which is noncyclic of order four. *The author wishes to express his thanks to Dr. Joseph Harris for valuable discussions concerning the material of this paper. ** Research supported by the National Science Foundation
©
1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00000714$00.70/1 (cloth) 0691082677/80/00000714$00.70/1 (paperback) For copying information, see copyright page
7
8
ROBERT D. M. ACCOLA
It turns out that these methods also apply to w3r+2 's admitting two
simple halfcanonical gr 3r+l 's and to w3r+3's admitting four simple halfcanonical gr 3r+ 2 's whose sum is bicanonical. We shall consider these cases in Part II of this paper.
2.
Notation, definitions, and preliminary results
A compact Riemann surface of genus p will be denoted WP. A linear series on WP of dimension r and degree n will be denoted grn. Such a series may have fixed points, may be simple or composite, and may be complete or incomplete. For x < WP, grn  x will denote the linear series of degree n 1 of divisors of grn passing through x, not counting x. If x is not a fixed point of grn, then grn x
=
gr 1n_ 1 .
For the convenience of the reader we include the following definitions
[6, p. 257]. A linear series g~* will be defined to be simple if for a general choice of x, grn  x is without fixed points. In this situation it is known that for a general choice of x, grn  x will also be simple. A linear series g~* will be defined to be composite if for any choice of x, grn 
X
has fixed points. In this latter situation wp is a tsheeted cover
ing of a surface of genus q, Wq, and a divisor of nonfixed points of grn is a union of the fibers of the map rb: WP . Wq. In such a case Wq admits a g\nf)/t where f is the degree of the divisor of fixed points of grn, and for x not fixed for grn , grn  x has t 1 addi tiona! fixed points, the other points in the fiber of rb containing x . If grn is complete on wp' then so is gr(nf);t on wq.
If grn is a linear series, a second series gsm is said to impose (linear) conditions on grn if there is a linear series gr~m so that g rn
=
s
g m+ g
rt nm ·
This means that if D is any divisor of gsm of m distinct points, then there are t points of D, x 1 , x 2 , ···, xt so that grn  (x 1 + x 2 + · · · + xt) *Without fixed points.
grtnm + D  (x 1 + · · · + x t )
PLANE MODELS FOR RIEMANN SURFACES
9
has D  (x 1 + ··· + xt) among its fixed points. Also x 1 , x 2 , · · ·, xt impose independent conditions; that is, for each k there is a divisor in grn containing all the xi, i=1,2,···,k1, k+1,···,t, butnotcontaining xk. If g 1m imposes one condition on g~, then grn = rg 1m + Dnrm where Dnrm is the divisor of fixed points of the composite grn; for whenever a divisor of grn contains a point x, it must contain all of the unique divisor of g 1m containing x. We will use the classical fact that since a g 1n (n :S p) without fixed points imposes n1 conditions on the canonical series, it imposes at most n1 conditions on any special linear series. The extension of this is that a simple special gSm without fixed points will impose at most m s conditions on any other special linear series whose dimension is at least ms. If grn is simple (r~2) and without fixed points, then wp can be realized as a curve in pr and the hyperplane sections cut out the divisors of grn. In such cases we will say that grn has a kfold singularity if the curve in pr does. In case g~ is simple and without fixed points, WP admits a plane model of degree n. If d represents the number of double points suitably counted, then p
(n1) (n2) _ d
2
.
To compute the dimension R of all plane curves of degree n with s given ordinary singularities of multiplicities k 1 , k 2 , · ··, ks, we use the formula
R >
Often, this formula is precise. A singularity of multiplicity k will be called a kfold point of the curve or linear series. A surface will be called qhyperelliptic (q;:::O) if it is a twosheeted cover of a surface of genus q' :S q. Thus rational and elliptic surfaces are qhyperelliptic for all q.
10
ROBERT D. M. ACCOLA A divisor Dp_ 1 of degree p1 on Wp will be called halfcanonical
if 2DP_ 1 is canonical. We now discuss the preliminary results upon which the later sections of this paper depend. THEOREM 2.1 (Castelnuovo's Theorem [5) or [6, p. 295)). Let Wp admit
a simple g~ . Then p
[m/2]. Then mt
wq and a gs(mf)/t on wq which lifts to the nonfixed points of
gsm. Also g\ is lifted from Wq and we see that t = 2 or 4. If t = 4 then q = 0 and g\ imposes one condition on gsm. Consequently gsm=sg\+Df and m=4s+f. acontradiction. Suppose t=2. If
13
PLANE MODELS FOR RIEMANN SURFACES gs(mf)/ 2 is special on Wq then by Clifford's theorem we have (mf )/2 ~ 2s , a contradiction. If gs(mf)/ 2 is not special then (mf)/2 s ?:: q . But by Lemma 2. 6 q > r and by hypothesis r ~ s , so we arrive at the final contradiction (mf)/2s?:: s. q.e.d.
LEMMA 2.8. Suppose g\ and h 04 are complete linear series and g 14 is without fixed points. Suppose that 2g\ = g 28 is also complete. Sup
pose finally that 2h 04 = 2g\. Then there are disjoint integral divisors ofdegreetwo, P and Q, sothat h 04 =P+Q and I2PI=I2QI=g\. Proof. Let h 04 =x 1 +x 2 +x 3 +x 4 and 2(x 1 +x 2 +x 3 +x 4 )=0 1 +0 2 , where I0 1 1 = 10 2 1 = g\ and 0 1 and 0 2 are disjoint. Now the result follows by examining the various possibilities for 0 1 and 0 2 . q.e.d. We conclude this section with some results on W3r's admitting r
g3r1 ' s · LEMMA 2. 9. Suppose w3r admits a simple gr3r1 . Then gr3r1
IS
com
plete, without fixed points, and halfcanonical. Proof. That gr3rl is complete and without fixed points follows from Castelnuovo's inequality since the inequality is now an equality. Also 2gr3 rl = gR2 p_ 2 where R ?:: 3r 1 = p 1 . By the RiemannRoch theorem R = p 1 and so 2gr3rl is canonical. q.e.d. LEMMA 2.10. On W3r a simple gr3rl is unique.
Proof. If hr3 rl is a second halfcanonical series then gr3 rl imposes at most r conditions on hr3 rl by Theorem 2.5. Consequently each divisor of gr3rl is contained in a divisor of hr3rl. It follows that gr3r1
=
hr3r1 · q.e.d.
LEMMA 2.11. If w3r (r?::2) admits a unique complete gr3rl must be simple.
then gr3rl
Proof. Suppose gr3rl is composite. The only way that this can happen is for W3 r to be a twosheeted cover of a surface of genus q and on Wq
14
ROBERT D. M. ACCOLA
there is a complete gr( 3rlf)/ 2 where f is the degree of the divisor of fixed points of gr3rl. If gr( 3 rlf)/ 2 is special then by Clifford's theorem we have the contradiction (3r1f )/2 2r > 0. This series is not special and so q
=
(r1f)/2. By [2, p. 51] it follows that the
qhyperelliptic surface w3r admits at least 4q composite halfcanonical gr3rl 's. If q is zero the result also follows. q.e.d. THEOREM 2.12. W3r admits a simple gr3rl if and only if gr3rl IS unique. Consequently, W3r admits a simple gr3rl if and only if the
thetafunction for W3 r vanishes at precisely one halfperiod to order r + 1. The second statement of the theorem is just a translation of the first statement into the language of vanishing properties of the thetafunction via Riemann's vanishing theorem.
3.
The plane models
LEMMA 3.1. Suppose W3r (r2:4) admits a simple gr3rl and a g 1m
which imposes two linear conditions on gr3rl . Then m Proof. If
X f
w3r' then gr3r1 
X
=
4.
has a singularity of multiplicity m 1 '
namely the other points in the divisor of g 1m containing x; for if y is one of these m 1 points then (gr3r_ 1x) y has the remaining m 2 points as fixed points. Now choose a divisor Dr_ 2 = x 1 +x 2 + .. ·+xr_ 2 of r2 points in r 2 distinct divisors of g 1m. Also assume that gr3rl Dr_ 2 (= g 22 r+l) is simple and without fixed points. g 22 r+l has r 2 singularities of multiplicity m 1 corresponding to the r 2 divisors of g 1m determined by Dr_ 2 . Also g 1m imposes two conditions on g 22 r+l, so g 22 r+l has a singularity of multiplicity 2r + 1m. If Dr_ 2 is chosen in a general manner all these singularities are disjoint and so contribute (r2)(m1) (m2)/2 + (2r+lm)(2rm)/2 to the double points of the plane curve determined by g 22 r+l. Consequently p
=
3r
=
(2r)(2rl)/2 (r2)(ml)(m2)/2(2r+lm)(2rm)/2 d'
PLANE MODELS FOR RIEMANN SURF ACES
15
where d' is the contributions of the other double points of the curve. Thus 2d'= m 2(r1) + m(7r5) 12r + 4 = (m4)((r1)m(3r1)). If m ;:: 5 we see that the righthand side of this equation is negative which is a contradiction. Thus m = 4 and d' = 0. q.e.d. THEOREM 3.2. Let W3r be a Riemann surface of genus 3r(r;::4, r;t5)
admitting a simple gr3rl . Then W3r admits a plane model as a curve of degree 2r + 1 with r  2 singularities of multiplicity 3 and one singularity of multiplicity 2r  3. Denote the plane curve of the theorem t' 2r+l . Now suppose that the r 1 singularities of t' 2r+ 1 are in general
position; that is' no ( e+ 1)( e+ 2)/2 of them lie on a curve of degree
e.
(This situation is to be expected, in general, but there are examples where it is not the case.) We can simplify the model by successive quadratic transformations. First transform with the (2r3)fold point and two triple points as fundamental points. Such a transformation transforms t' 2Hl into a curve of degree 2r1, t' 2 rl, with r4 triple points and one (2r5)fold point. Second transform t' 2rl with the (2r5)fold point and two trip~e points as fundamental points to obtain a e2r3 with a (2r7)fold point and r6 triple points. Continue.
If r is even, (r2)/2 such transformations yield a curve of degree r + 3 , t'r+ 3 , with a single singularity of degree r 1 . The gr3rl is cut out by curves of degree r/2 with a ((r2)/2)fold singularity at the (r1)fold singularity of t'r+ 3 .
If r is odd, (r3)/2 such transformations yield a er+4 with a 3fold singularity and a rfold singularity. gr3rl is cut out by curves of degree (r+1)/2 with a ((r1)/2)fold singularity at the rfold singularity of t'H 4 and also passing through the triple point. It is worth remarking that in this case of r odd, these curves are the examples of Riemann surfaces for which equality is attained in the following classical inequality: p
fgf 1
=
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00002111$00.55/1 (cloth) 0691082677/80/00002111 $00.55/1 (paperback) For copying information, see copyright page
21
g', where
22
B.N.APANASOV
f: Bn .... Bn is a quasiconformal automorphism of the ball, can be represented in the form g r. AgA  1 = g', where A { 9Hn , A(Bn) = Bn. In the planer case (n=2) the described situation does not occur. Moreover, it is well known that the set of all quasiconformal automorphisms of the disk B 2 compatible with a Fuchsian group G (identifying the conformally equivalent ones) forms the Teichmiiller space of the group G and has complex dimension 3p 3 + m , where p is the genus of the surface B 2 /G , and m is the number of punctures of B 2 /G. As it may seem, after Mostow's result, that for n 2: 3 the Teichmiiller space of the Fuchs ian group G, V(Bn /G) < oo, degenerates into a point. However, our main result can be formulated in the following form. THEOREM A.
For any Fuchsian group G in Rn, n 2: 2 with compact
fundamental polyhedron Bn/G such that one side is orthogonal to all other sides that it meets there exists a homeomorphism
of the open mdimensional cube, m 2: 1 , into Teichmiiller space for the group G.
We also enlist here some applications of this result connected with spatial quasiconformal mappings and Mostow's rigidity theorem to prove the following statement. THEOREM B.
The boundary of Teichmiiller space T(G) for the Fuchsian
group G C 9.}1 3 (where G is the same as in Theorem A), contains the Kleinian groups
r
with the following properties:
1) f(r) is connected;
2)
ncn = no u n1 '
where
no
and
n1
are the connectivity
components; 3)
4)
reno)= no' rcn1) I= n1; no is a quasiconformal image of the ball;
5) the fundamental group rr 1(n 1 ) is a free infinitelygenerated group.
23
NONTRIVIALITY OF TEICHMULLER SPACE
This theorem shows once more the essential difference between the spatial case and the planar one (cf. [1]).
2. Notation and terminology For x
f
Rn we use the representation x
=
l xiei
=
(x 1 , .. ·, xn) where
e 1 , ... , en is an orthogonal basis. By Bn(x, r) we denote the open ball with center B 3 (x,
X (
r), S(x, r)
Rn and radius r > 0' snl(x, r) = aBn(x, r)' B(x, r) = =
S 2 (x, r), Hn
=
lx ( Rn: xn > O!. The hyperbolic measure
of a set M in Bn or in Hn is denoted by V(M). The closure M, the boundary aM of sets M C Rn will always be in Rn . A Mobius group G C 9Rn is said to be discrete if no sequence of distinct elements of G converges (pointwise) to 1 , or, equivalently, to any A f 9Rn . The limit set of the g~oup G C 9Rn is denoted by ~(G) and consists of the points of accumulation of the sets G(y) Its complement 11(G)
=
=
{g(y): g(Gl, y ( Rn.
(2.1)
Rn\~(G) is called the discontinuity set. The
group G is Kleinian if 11(G) /. (2). In this case one can define the socalled fundamental domain F(G) which contains a representative from each orbit G(y) (2.1). The Kleinian group G C 9Rn is called quasifuchs ian if the Jordan surface ~(G) breaks 11(G) into two connected and simply connected components. A quasiconformal automorphism f of the space Rn is compatlble with the Kleinian group G ( f is a quasiconformal deformation of the. group G), if
(2.2) Two quasiconformal deformations f and f 1 of the group G are equivalent if there exists A f 9Rn such that the equalities:
hold. We factor the set of quasiconformal deformations of G with respect
24
B. N.APANASOV
to this equivalence relation to obtain the space of quasiconformal deformations of the Kleinian group G. The space T(G) of all the quasiconformal images of the group G (with the analogous equivalence relation) is naturally associated with it. The convergence of the group can be defined via the coefficients of their matrix representations in the Lorentz group [3]. The topology thus introduced is equivalent to the one determined by the usual TeichmUller metric [8, 12] the dilatation logarithm of the quasiconformal automorphism f.
3.
Preliminary results The following statement is obtained from the wellknown results [2] on
the groups generated by reflections in hyperbolic space. LEMMA 3.1. Let the group G C Wln be generated by a finite number of
involutions Ti = :fi o()i, where :fi is inversion with respect to the sphere sn 1 (xi, ri), x~ = 0, and ()i is reflection with respect to the hyperplane Li which passes through the point xi, is orthogonal to the subspace Rn 1 and is the plane of symmetry of the polyhedron P(G)CHn bounded by the spheres sn 1 (xi, ri) and, perhaps, by the plane lx: xn=Ol = Rn 1 .
Then G is discrete iff all the dihedral angles of the polyhedron P(G) are integral parts of rr. 3.2. Under the extension of the group G C Wln into the halfspace Hn+l we understand the following. Each element g ( G is the superposition of reflections with respect to lowerdimensional planes Li C Rn and, perhaps, inversion with respect to some sphere snl(xg, rg) [4]. Consider nplanes Li the mappings
c
Rn+l' Li
g ( Wln+l
Rn' Lin Rn = Li' the spheres sn(xg, rg) and which are superpositions of reflections with re
spect to Li and, perhaps, of inversions with respect to Sn(xg, rg). We obtain the group
GC Wln+l
which leaves the halfspace Hn+l invariant
and the restriction of which to the subspace Rn coincides with the group G.
25
NONTRIVIALITY OF TEICHMULLER SPACE

If now the group G is discrete, then its extension G acts discontinuously in Hn+ 1 and the interior of the intersection int (P n Rn) = F(G) where PC Hn+l is some hyperbolically convex fundamental polyhedron
G is
of the group
(in case F(G) f Q)) the fundamental domain of the
group G. LEMMA 3.3.
Let G C 9J(n be a Kleinian group and F0 be the component
of its fundamental domain which contains a point x 0 . If for every generator A of the group G, the points x 0 and A(x 0 ) can be connected by a curve lying in rl(G), then the component 0 0
")
F0 of the discontinuity
set Q(G) is invariant. The proof of this lemma is simple and can be done in analogy with the planar case [1, 15].
4.
Proofs of Theorems A and B Though our method is independent of the space dimension we, for
simplicity, show it in the case n
=
3 , and by the example of a definite
Fuchsian group. It will be clear from the proof how to modify it in the general case. 4.1. Let Q
=
lx fR 3 : lxil C ~l be the unit cube. Circumscribe the
spheres Si , 1
:S i :S 8, of radius v3/3 from its vertices. They intersect
at the angles rr/3, and are orthogonal to the sphere S 0 Construct six more spheres Si, 9 ::; i
where r = 5/6,
p
=
S(O, v15/6).
:S 14:
= Jl0/6.
Denote as Li, 1
:=:;
i
:=:;
8, the planes passing through the points 0,
e 3 and the center of the sphere Si. Further, put
26
B. N. APANASOV
Consider the group G C 9R3 generated by the involutions Ai,
1 ::; i ::; 14, each of them being the superposition of inversion with respect to the sphere Si and of reflection with respect to the plane Li. The group G leaves the ball B(O, v15/6) invariant. It follows from Lemma
3.1 that G is a Fuchsian group of the first kind with the compact fundamental domain. Consider also the system of spheres lajCt)! 1 ::;i94 , which is obtained from the system lsi! by the substitution of the spheres sl3 and S 14 (with the centers ±~e 3 ) for the spheres a 1 it) and a 14 (t) with the centers ± t · e 3 and orthogonal to the corresponding spheres ajCt)
=
Si , 1 ::; i ::; 12 (and, consequently, having radii Jt 2  t + 5/12 ).
The group G(t) C 9R 3 , t 0
=
5/12 < t < t 1
=
5(1 + ,/5/6) generated by
the involutions A1 which correspond to the spheres ai(t), 1 ::; i ::; 14, is a Kleinian one. It directly follows from consideration of the extended (see
3.2) group G(t) and from an application of Lemma 3.1. Hence, we obtain that the fundamental domain of the group G(t) has the form: Ft
n
=
(4.4)
ext ai(t)
l:S:i:S:l4
and decomposes into two componentsthe bounded one F~ and the unbounded one F{. The set of discontinuity components Q 0 and
n1 ,
n=
D(G(t)) decomposes into two invariant
both homeomorphic to a ball. This follows
from Lemma 3.3 and properties of Ft (see [6]).
4.5. Let F 0 and F 1 denote the polyhedra F~ and F{ with t
=
5/6
(that corresponds to Fuchs ian G). Using the uniform expansions along two families of circles, orthogonal to the spheres a 13 (t) and a 1 it) correspondingly, one can construct q(t)quasiconformal mappings
(4.6) that map the sides of Ft onto the corresponding sides of F;, i
=
0,1.
27
NONTRIVIALITY OF TEICHMULLER SPACE
Here, q(t) is continuous and lim
t> 5/6
q(t)
1 .
Using wellknown facts concerning quasiconformal mappings ([9], [13], [14]), we can extend ff and
f:
periodically until we obtain the
mappings
(4.8) which, being compatible with G, will be equal to a restriction of a q(t)quasiconformal deformation t of the Fuchsian group G. It follows from the properties of s::'(G) and s::'(G(t)) that for each t', t", t 0 < t' < t"' < t 1 , deformations t' and t" are not conjugate in the group
wen.
< E we can conjugate them with
Nevertheless, when lt't"l
the help of a a(e)quasiconformal deformation which we can construct in the same way as above. Here we also have lim a(E)
E>0
=
1 .
Thus, the mapping
is a homeomorphic imbedding of the interval (t 0 , t 1 ) into the space T(G). Theorem A is proved. 4. 9. Our proof of Theorem B will be reduced to the clearing up of the structure of the groups G(t) (constructed in Section 4.4) for the parameter values t = t 0 = 5/12 and t = t 1 = 5(1 + ,/5/6). We consider the case of the group G(t 1 ). In case t
=
t 0 the situation is analogous and even
slightly simpler. First we note that the group G(t 1 ) is Kleinian. Its fundamental polyt
t
hedron has two componentsthe bounded F 0 1 and the unbounded F 1 1 . t
The polyhedron F 0 1 , like all FJ, t 0 < t < t 1 , is connected and simply
28
B.N.APANASOV
connected. At the same time the fundamental group
t
17 1 (F1 1 )
is a free
group on eight generators. It becomes clear if we note that the spheres a 1 it 1 ) and a 1 it 1 ) touch the corresponding spheres ai(t 1 ) = Si, i
=
9, .. ·, 12, where the points of contact pi, qi, 9
the spheres Sj, 1
:S i :S 12 lie outside
:S j :S 8, are the unique fixed points of some mappings
of parabolic type 1 and form five cycles. This gives us the direct calculation. The application of Lemma 3.3 gives us the discontinuity set of the tl tl group G(t 1 ) falling into two components D0 (t 1 ) :) F0 and D 1 (t 1 ) J F 1 which are invariant. At the same time, using the density of the fixed points of different classes of transformations from the group G(t 1 ) in its limit set [5] we obtain that ~(G(t 1 )) is connected. The quasiconformal equivalence of the domain D 0 (t 1 ) to the ball is proved, as in Section 4.5, by constructing the quasiconformal mapping
compatible with the group G and the last statement of the theorem on the fundamental group
1T 1
cnl (tl )) directly follows from the previous one and
from the density of parabolic points in ~(G(t 1 )) (see [5]). It is also obvious that the group G(t 1 ) cannot be obtained from G with the help of a quasiconformal automorphism of the space Rn, since the component nl (tl) is not simplyconnected. Thus, G(tl)
I T(G), and, consequently,
lies on the boundary of Teichmiiller space of the group G . Theorem B is proved.
5. Some corollaries and remarks REMARK 5.1. From the proofs of Theorems A and B it is clear that the simultaneous deformation of the Fuchs ian group G by three parameters: by generating mappings, corresponding to the spheres a 13 (t) and a 1 it), a 11 (r) and a 1 /r), a 9 (s) and a 10(s), the variations corresponding to the parameters r and s, being built over t (see 4.1), is possible. In particular, if we consider the variations corresponding to s
=
t, t t >
NONTRIVIALITY OF TEICHMULLER SPACE
29
and r < ~ , and consider the boundary group G (~, t 1 , r 0) , then we obtain the following statement. COROLLARY 5.2. On the boundary of Teichmiiller space T(G) of the
Fuchsian group G C Wl 3 from Section 4.1 there exist Kleinian groups 1. with the following properties: 1) f>(G) is connected;
2) n = no u ni where no and n1 are the connectivity components; 3) rcni) = ni' i = 0,1' i.e. the components no and n1 are
invariant; 4) The fundamental groups
77 1(n 0) and 77 1(n 1) of the invariant components no and n1 are free infinitely generated groups.
COROLLARY 5.3. The Mostow rigidity theorem [10] doesn't take place in
the case of discrete groups of isometries of the ndimensional hyperbolic space Bn, n? 2, inducing discontinuous groups on asn = sn 1 . In other words, for any n ? 2 there exist isomorphisms of Fuchsian groups of the second kind induced by quasiconformal homeomorphisms of Rn which cannot be continued up to the inner automorphism of the group
Wln. We recall that the hyperbolic volume for such groups is V(Bn /G)= oo. This corollary trivially follows from Theorem A (and in case n = 2, 3 is known even earlier). The abovementioned phenomenon of "nonrigidity" takes place also in the case of discrete groups of isometries of hyperbolic space Bn, n:: 3, whose limit set is asn. One can find the details in [7]. REMARK 5.4. Due to the fact that the fixed points of loxodromic elements are dense in ~(G(t)), t 0 < t < t 1 , we obtain that our quasiconformal automorphism t of the space Rn maps the ball Bn onto a domain n 0 (t) which has no tangent plane at any of its boundary points (see [11]). REMARK 5.5. The described method (after some modification) can be applied in the case when all the faces of the fundamental polyhedron
30
B. N. APANASOV
Bn/G of the Fuchsian group G c9J?n of the first kind contain limit vertices. For an infinitelygenerated group a result of such type was obtained by A. V. Tetenov, a student of mine (see [6]). INSTITUTE OF MATHEMATICS SIBERIAN BRANCH OF THE USSR ACADEMY OF SCIENCES NOVOSIBIRSK90, USSR 630090
REFERENCES [ 1]
.!ccola, Jl.D.M., Invariant domains for Kleinian groups, Amer.
J. of Math.§§ (1966), pp. 329336.
[2]
0 aanonHeHHH
AneKcaa~poB, A.~.
npocTpaac~Ba
uaororpaHHHKaMH,
BeCTHHK JleHHHI'pa,n;cKoro roc.YHHB., Cep. llaT.!Plrl3.XHII., 1954. II!> 2, CTp. 3343.
[3]
AnaaacoB E.H. B~
rpynn waoroMepaoro
CCCP, [4]
06 o~HOII
~
(1975),
AnaHacoB E. H.
~I,
~
AH
R." • ~oKna,n;H
3, CTp. 5095IO.
~
5, CTp. 89I898.
AnaaacoB E.H., TereaoB A.B., 0 cymecTBOBaHHH ae~pHBHanDH~ ~oKna,n;~
AH CCCP,
~
KnetiHOBHX rpynn B npOCTpaHCTBe,
{I978),
~I,
crp. I417.
AnaaacoB E.H., K reopeMe xec~Koc~• Moc~oBa, ~o&n~Y AH CCCP, 243, {1978),
[8]
~oKna,n;~
CTp. 1I14.
KBa3HKOHilJOpMH~ ~eiPOpMaUHti
[7]
npocTpaHCTBa,
AnaaacoB E.H. 0 KnetiHoB~ rpynnax B npocTpaacrBe, CH6.MaTeM. K., 1§, 1975,
[6]
aBKne~oBa
o~HOII Macae KnetiHOB~X rpynn B
06
AH CCCP, ill (1974),
[5]
aHanHTHqecKou MeTo~e B TeopHH Kne~ao
Bers
~
4,
c~p.
1029!032.
L., QU.asiconformal mappings and Teicl::llll.iiller's
theorem,
Anal.;rtic functions, Pr:l..nceton, 1960, 89120.
[9]
Deny
J.,
loions T.L., Les espaces dl1 type de Beppo Le"rt, .ADD..
Inst • .Jourier, • 5(1955), PP• 305370·
[10]
Mostow G. D., QUasicollfo:t'll81 aappings in
n apaoe
and the
rigidity of hn>erbolic space forms, l'llbl.llath. de 1 'Inatitut• des Hautes Etudes Sc1ent1f1ques, • .)4, 1968.
31
NONTRIVIALITY OF TEICHMULLER SPACE
[11]
Ko~o3
A.n.
HoBe~eHHe npocrpaHcrBeHuoro KBaBBKOB~PIHOro
0!06pazeKIH Ha DXOCKHX ceqeHHHX o6naC!B
onpe~exeBBH. lOKX~H
!H CCCP, 1§1, (!966), lt!4, crp. 743746.
[12]
Kpymxax& C.!. KBaBHKOH~PKHHe oro6pazeRBR • pBMa&OBH HOC!B, Hayxa, HOBOCH6HpCK, 1975.
noBe~
[13] Pe•e!HRK m.r. !oKanbHaR crpyxrypa 0!06pazeHBI c orpa..qeHHHK acxazeuBeM, Cz6.uareK.a., 1Q (1969), It! 6, crp. 13!9134!.
[14] PemerHRK m.r. UpocrpaHcrBeHHHe oro6pazeHRR c orpaa.qeHHHK KCK8ZeHK6M, CB6.MaTeM.K., 1 (1967), It! 3, C!p. 629658. [15]
topX !.P. AB!OKO~HHe ~HKnHH, OHTH HKTn CCCP, M.!., !936.
THE ACTION OF THE MODULAR GROUP ON THE COMPLEX BOUNDARY* Lipman Bers To ProfessorS. E. Warschawski, on his 75th birthday.
Introduction The Teichmi.iller space T p, n of Riemann surfaces of genus p , compact except for n punctures, has a canonical topology, a canonical metric and a canonical complex structure, as well as a canonical group of automorphisms, the modular group Modp,n. It has, however, several "natural" compactifications and hence several boundaries. The present note, which develops the method of [6], deals with the complex boundary obtained by identifying T p,n with the Teichmiiller space T(G) of an appropriately (but not canonically) chosen Fuchsian group G. The space T(G) is canonically embedded into the universal Teichmi.iller space T(l) and T(l) is canonically identified with a bounded domain in a complex Banach space B. The modular group Modp,n becomes identified with Mod(G) which appears as a subgroup of the universal modular group Mod(l). The main results of this note are stated and proved in §4. They include the following statements.
There is a large class of points on aT(l) on which Mod(l) acts continuously, and there is a large subgrcup of Mod(l) which acts continuously
*Work
partially supported by the NSF.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00003320$01.00/1 (cloth) 0691082677/80/00003320 $01.00/1 (paperback) For copying information, see copyright page 33
34
LIPMAN BERS
on the whole boundary aT(l). If dim T(G)
< "", then Mod(G) acts con
tinuously on almost all points of aT(G), including all socalled totally degenerate points. (The latter result uses essentially recent theorems by Sullivan [13) and by Thurston [not published).) The essential work is done in §§2, 3; some results there may be of independent interest.
§1. Background and notations We begin by introducing some (mostly standard) notations and by recalling some known facts (see [2), [5), [10) and the references given there). The main properties of quasiconformal mappings are assumed (cf. [2), [11)). We denote by Q the group of all quasiconformal selfmappings of the upper halfplane U ; for a tension of w to R
=
w
f
Q, [w) = w IR denotes the continuous ex
R U I"" l. A w
f
Q is called normalized if it (or
~
rather wiR) leaves 0, 1, "" fixed; the normalized w
f
Q form the sub
group Qn which contains the subgroup Q 0 of those w which leave every x < R fixed. Every w < Q can be written uniquely as w where
w
f
=
a
ow
Qn and a is a real Mobius transformation, i.e. an element of
the group Qc of conformal selfmappings of U. Note that Qc = PSL(2,R). By a Fuchsian group G we mean, in this paper, a discrete subgroup of Qc. In particular, 1
=
Q(G) consists of those w
I id l f
is the trivial Fuchs ian group. The set
Q for which wGw 1 is again a Fuchsian
group; we write Qn(G) = Q(G) n Qn. Let L 00 (U) 1 denote the open unit ball in the (complex) Banach space L 00 (U), and let L 00 (U ,G) 1 consist of those fJ.(g(z)) g'(z)/g'(z)
=
Every w < Q has a Beltrami coefficient
fJ.(z), /1.
/1. f
L 00 (U) 1 for which g G into < , >wGw1 . The map w*: T(G)
+
T(wGw  1) is
called an allowable map. If w normalizes G, then w* is a selfmapping of T(G). Such w* form the modular group Mod(G) of G. (In [4] the definition of Mod(G) is somewhat different, for a few groups G; this difference is of no importance for what follows.) Now let B denote the complex Banach space of holomorphic functions c/J(z), z
f
L, with
For every Fuchsian group G, let B(G) be the closed linear subspace of B consisting of those ch
f
B for which
ch(g(z)) g'(z) 2 = ch(z),
g
f
G.
The map (1.1) is a homeomorphism of T(1) onto a bounded domain in B
=
B(1); the
restriction of this map to each T(G) is a homeomorphism onto a bounded domain in B(G). By abuse of language we shall identify each T(G) with its image under the mapping (1.1 ). Thus each Teichmliller space T(G) has a complex structure and also a boundary aT(G) C B(G) , and all allowable maps are holomorphic. With every ch z
f
f
B we shall associate a meromorphic function Wch(z),
L, defined as follows:
37
MODULAR GROUP ON COMPLEX BOUNDARY
where n 1 and n 2 are solution of the ordinary differential equation 2n"(z) + c/J(z) n(z)
=
0
subject to the initial conditions n 1 (i) = n;(i)
=
1,
11~ (i)
= n/i) = 0 .
An equivalent definition reads: W = Wc/J(z) has c/J as its Schwarz ian derivative: W(z)
=
(z + i) 1 + o(1), z .... i .
The dependence of Wc/J on c/J is holomorphic. We denote by
S
the set of those c/J
B for which Wc/J is schlicht
f
(i.e., injective); by the KrausNehari inequality c/J
f
S
S
is bounded. For
it will be convenient to consider instead of Wc/J the schlicht
function
which has the Schwarz ian derivative c/J and takes (i, 2i, 3i) into (0, 1, oo). The dependence of ncb on c/J
f
S
is holomorphic.
The Teichmiiller space T(1) consists of those cb
f
S
for which Wcb
(and hence also ncb) has a quasiconformal extension from L to C. This is equivalent to the requirement that nc/J(L) be bounded by a quasicircle, an image of a circle under a quasiconformal mapping. If cjJ then cP
cjJ/1 for some 11
=
f
L 00 (U) 1 and w 11(2i)w 11(3i) w 11(2i) w 11(i)
If cb
f
T(G), then the group
is a quasiFuchsian group with fixed curve a!Jc/J(L).
f
T(1),
38
LIPMAN BERS
It is known, and easy to verify, that T(1) U aT(1) C S. Gehring [8],
[9] proved that T(1) is the interior of T(1) U aT(1) in
S
S,
and that the complement of
is nonempty.
The Banach space B(G) and the Teichmiiller space T(G) are finite dimensional if and only if G is finitely generated and of the first kind, i.e., if and only if the Poincare area of U/G is finite, i.e., if and only if PSL{2, R)/G has finite volume. If so, and if G is torsion free, the quotient U/G is a Riemann surface of some finite genus p, compact except for n punctures, and 3p 3 + n
~
0. In this case T(G) can be
identified with the Teichmtiller space T p ,n and Mod(G) with the Teichmtiller modular group Modp,n operating on T p,n. §2. Tame convergence A sequence !¢j! C B is said to converge weakly to a ¢ f B if ll¢jll
=
0(1) and lim ¢j(z)
=
¢(z) for every z fL. Every bounded se
quence in B contains a weakly convergent subsequence. If lim ¢j
=
¢
weakly' then n¢. converges to n¢ normally (uniformly on compact sub)
sets of L ). The set dense in
S
S
is closed under weak convergence and T{1) is
with respect to weak convergence. The proofs of the above
statements are obvious.
S
A sequence !¢j! in
if lim ¢j
¢ weakly and if for almost every z / fJ¢(L) there is a
=
such that z quence in
will be said to converge tamely to a ¢ f
S
I O¢.(L) for
j
J
> J. Note that a convergent (in norm) se
is always weakly convergent, but may fail to converge
tamely. PROPOSITION
2.1. If ¢
€
S
and
(2.1) then there is a sequence !¢j! C T(1) such that ¢ (Here and hereafter
a
S, J
=
lim ¢j tamely.
denotes the settheoretical boundary.)
MODULAR GROUP ON COMPLEX BOUNDARY
Proof. Let 6.n, n
39
1, 2, ... , denote the disc
=
and fn(z) the function which maps L conformally onto 6.n and satisfies the conditions fn(3i) Clearly, lim fn(z)
=
3i,
=
f(z)
=
f(z).
z normally. Set ~\b of/z)
Qc/J ofi(i)
Qc/J ofi(2i) Qc/J ofjCi) Then Vi is meromorphic and schlicht in L and takes (i, 2i, 3i) into Q¢·, where ¢i ( S is the Schwarz ian derivative J of Vi. Since Vi converges to Q¢ normally, cPj converges to ¢
(0, 1, oo). Hence Vi
=
weakly. Since ancjJ.(L) is obtained from the real analytic Jordan curve J Q¢(6j) by a Mobius transformation, Q¢· admits a quasiconformal extension to
C.
Hence
rPj
J
E
T(1).
Now let z 0 be a point exterior to Q¢(L). If i is large enough, the Mobius transformation
is arbitrarily close to the identity, so that
and, a fortiori,
Since the boundary of Q¢(L) is assumed to have measure zero, the convergence of
rPj
to ¢
is tame.
40
LIPMAN BERS
QUERY:
Is the conclusion of Proposition 1 valid without the assumption
(2 .1)? The result just proved does not, of course, imply that a
¢ < aT(G) is
a tame limit of elements in T(G), even if mes aW¢(L) = 0. A special result in this direction is, however, valid. Let G be finitely generated and of the first kind. Then the boundary points of T(G) are classified as follows (cf. [3], [12]). If ¢ < aT(G) and W¢(L) is dense in
C,
¢ and the group G¢ are called totally degenerate.
Almost all boundary points are of this nature. If ¢
is not totally degen
erate, the region of discontinuity R(G¢) of G¢ is disconnected and R(GcP)jGcP is a disjoint union of Riemann surfaces. The group G¢ and
the boundary point ¢
are called either regular or partially degenerate
according to whether the Poincare area of R(GcP)jGcP is equal on less than twice the Poincare area of U/G. If dim T(G) > 0, aT(G) contains regular points and if dim T(G) > 2, aT(G) contains partially degenerate points. PROPOSITION 2.2.
Let G he finitely generated and of the first kind.
Every regular boundary point in aT(G) is the tame limit of a sequence in T(G). The proof is a modification of an argument due to Abikoff (see [1], pp. 224231). The details are somewhat lengthy and will be presented elsewhere. §3. Formal translations The group Qn/Q 0 operates on T(1) as the group Modt of (right) translations. We will associate with every w < Qn, which has a continuous Beltrami coefficient in U, a map
S > S,
called a formal translation,
which restricts on T(1) C S to the element of Modt induced by w. Throughout this section we consider a fixed continuous () < L 00 (U) 1 and set
MODULAR GROUP ON COMPLEX BOUNDARY
We extend the definitions of w and w(z)
=
e
w(z),
41
to L by setting O(z)
=
O(z).
cPj will denote elements of S. For every 0 and c/> set
The letters c/> and
(3.1)
(3.2)
(where
I lloo
denotes the L 00 norm). Hence there is a unique quasi
conformal selfmapping AO,c/> of
C which satisfies
the Beltrami equation
and the normalization conditions
AO,c/> one/> ow(J 1 (i) = 0, Ae,c/> one/> ow(/(2i) = 1, Ae,cf> one/> ow(J 1 (3i)
0
=
oo.
The function V(z) = Ae,cf> one/> ow 1 (z), z ( L, is injective by construction and takes (i, 2i, 3i) into (0, 1,
oo).
Also, V(z) is meromorphic
since Vow 0 = Ae,c/>onc/> and one computes from (3.1) that in L the Beltrami coefficient of Ae,c/> one/> is 0. Thus V = nl/J where 1/J is the Schwarz ian derivative of V. This 1/J (
S;
and note that 1/J is defined by the relation
we write
42
LIPMAN BERS
(3.3)
e*
We must now verify that the restriction of
to T(l) is the element
(we\ of Modt. The following statement is a slight modification of a result by Gardiner [7, p. 475]. PROPOSITION 3.1.
Let Jl, v
f
L 00 (U) 1 be such that
and set
Then !/J = e*(¢), i.e., (3.3) holds. Note that the hypotheses of the Proposition mean that !/J = (we)*(¢).
Proof. Since ¢ and !/J are the Schwarzian derivatives of wlliL and wviL, respectively, there are Mobius transformations a , a J1
v
such that
in L. Using these relations we extend the definitions of f!¢ and f!!/J to all of
C.
Then the Beltrami coefficients of f!¢1U and f!!/JIU are 11
and v respectively. Let
be conformal bijections' chosen so that the maps qJl 0 n¢ and qv of!!/J keep O,l,oo fixed. Then q
of!,..~.= 11'+'
w , q of!, 1• = w J1
We define a homeomorphism a
V
a(z) is quasiconformal off the quasicircle n(R)' hence
everywhere. A direct calculation shows that the relation (Ja_ 0
az 
(),
(Ja
az
holds a.e. Also
so that a
takes the points n ow(/Ci), n owi(2i) and
0
n
{}*(¢) weakly.
=
00
Proof. It will suffice to show that (3.13) holds under the additional hypothesis that the sequence lt/lj I= le*(¢j)l converges weakly to some
t/1.
We are now in the situation described by Proposition 3.2. Hypothesis (3.12) implies that for almost every z
o(}
A.. ( z) = ''!' J
0
I
Q¢(L) there is a
for
Together with (3.11) this implies that lim
j
J
such that
>] .
oe ''~'] '"'. = o(} ,'/''"'
a. e. Hence the F
in the statement of Proposition 3.2 coincides with Ae,¢, so that
t/1
=
{}*(¢).
PROPOSITION
3.4. Assume that
(3.14)
Then {}*(¢>) depends only on
we\R
(and, of course, on ¢> ).
Proof. For ¢> < T(1) the conclusion holds, independently of hypothesis (3.14), in view of Proposition 3.1. Therefore, under this hypothesis, the conclusion follows from Propositions 2.1 and 3.3. QUERY:
Is the conclusion of Proposition 3.4 valid for all ¢>
f
aT(1)?
For all ¢> < S? PROPOSITION 3.5.
If
(3.15)
then (}* is continuous at ¢> with respect to weak convergence (i.e., if lim
rPj
=
¢> weakly, then lim {}/rf>j)
=
{}*(¢) weakly).
Proof. Note that, by (3.15), weak convergence to ¢> is tantamount to tame convergence and use Proposition 3.3.
46
LIPMAN BERS
PROPOSITION 3.6. If () has compact support in U (and hence also in U U L ), ()* is continuous with respect to weak convergence.
Proof. We may assume that we are in the situation described in Proposi
tion 3.2. We note that n. converges normally to U l¢(u)l  !lull so that
II a II
la(u)l !lull
is bounded below.
By an observation of Thurston (see [3]), if M is a compact manifold of negative curvature, then the bounded part of H*(M) is all of the cohomology of M of dimension
~
2. Roughly speaking, there are three
components of this observation: (i) a geodesic ksimplex in hyperbolic space (constant curvature 1) has bounded kdimensional volume, if

k > 1. (ii) In the universal covering space M of M, the complex of
SOME REMARKS ON BOUNDED COHOMOLOGY
61
geodesic simplices is (bounded) chain homotopic to the complex of all simplices, so that we may restrict our attention to cochains whose value on any simplex agrees with the value of the corresponding geodesic simplex. (iii) A standard comparisontheorem type argument allows one to conclude that if a is a deRham kform of M , then its value on any geodesic simplex is bounded in terms of the curvature of M, the volume of a geodesic simplex in hyperbolic space, and, of course, the size of a. Since M is a K(rr 1 (M), 1), there is a natural map c/J: Aut(rr 1 (M)) . AutH*(M). Ker(c/J) contains the group of inner automorphisms of 7T 1(M), but, perhaps, contains much more. THEOREM.
Let M be a compact manifold of negative curvature,
dim(M) > 2. Then, (a) Aut(rr 1 (M))/Ker(c/J) is a finite group. (b) The order of any element of Aut(rr 1 (M))/Ker(c/J) is bounded in terms of the Betti numbers of M. Proof. Let f be an element of Aut(rr 1 (M)).
Then there is a unique
homotopy class of homotopy equivalences, which we also denote by f, which realizes the given automorphism of rr 1 (M). Since both f and f 1 are normdecreasing, f* must be an isometry of the norm
II .
But f* also preserves the lattice H*(M, Z) in
H*(M, R). For k
~
2 , let e 1 , · · ·, er be a basis for the lattice of torsionfree
elements of Hk(M, Z). Then since f* is an isometry, there are only finitely many choices for f*(e 1 ), ·· ·, f*(er). Also, if f* is the identity on Hn 1 (M), by Poincare duality, it is so also for H 1 (M). This then proves (a). To show (b), we use the fact that f* is an isometry to show that all eigenvalues are of absolute value 1. Since f* preserves a lattice, it also follows that all the eigenvalues of f* are algebraic integers. By Kronecker's theorem, it follows that all the eigenvalues are roots of unity since each eigenvalue is the solution of an algebraic equation whose
62
ROBERT BROOKS
degree is bounded by the corresponding Betti number of M, there is a
number m which is bounded by the Betti numbers of M such that (fm) *
is unipotent. But (fm) * is an isometry, since f* is, and these two facts imply that (fm) * is the identity, proving (b). In the case where M is a hyperbolic manifold, the classical Mostow Rigidity Theorem asserts that Aut(rr 1 (M))/Inn(rr 1 (M)) is finite. Even in this case, I don't know much about the group Ker(¢)/Inn(rr 1 (M)). We may extend this discussion to arbitrary manifolds to some degree, in view of the results of §1. THEOREM. Let M be an arbitrary manifold, and f: M> M a map which induces an isomorphism on rr 1 (M). Then f* restricted to the bounded part of H*(M) has all eigenvalues of absolute value 1 . Proof. This follows from the fact that [[fk[[ is bounded independently of
k, by the result of (b). One is tempted to apply Kronecker's theorem in this context. The problem is that
a priori, the bounded part of
H*(M), need not be a ration
al subspace of M. Thus, in particular, it may happen that one eigenvalue of f* may occur in the bounded part of H*(M), and thus have absolute value 1, while its conjugate with respect to some automorphism of Q over
Q may fail to have absolute value 1.
One can show that this never happens, for instance, when rr 1 (M) has a JordanHolder decomposition in which each factor is either an amenable group or the fundamental group of a manifold of negative curvature. Intuitively, one feels that bounded cohomology detects only the "hyperbolic" structure of a group, so that there is at least some hope of the general statement being true. To sum up, we ask: QuESTION: Is it true, for all groups G, that the bounded part of H*(G) is a rational subspace of H*(G)?
63
SOME REMARKS ON BOUNDED COHOMOLOGY
An affirmative answer to the above question would sharpen the above theorem to read: f* is of finite order on the bounded part of H*(M), with the order bounded in terms of the dimension of the bounded part of H*(M). A negative answer would, of course, yield interesting examples of groups with nontrivial bounded cohomology. REFERENCES [1] Brooks, R., and Trauber, P., "The van Est Theorem for Groups of Diffeomorphisms," The Hadronic journal (1978). [2] Hochschild and Mostow, "Cohomology of Lie Groups," Ill. (1972). [3] Gromov, M., "Volume and Bounded Cohomology," preprint.
J. of Math.
THE DYNAMICS OF 2GENERA TOR SUBGROUPS OF PSL(2, C) Robert Brooks* and J. Peter Matelski A classical result of Shimizu and Leutbecher (see, for instance [6], p. 59) asserts that if
(M)
(~~) generate a discrete subgroup of
and
PSL(2, C), then either c = 0 or
lei 2:
1. This has been strengthened by
T. J¢rgensen [4] as follows: j¢RGENSEN'S INEQUALITY. If X and Y generate a discrete, nonelementary subgroup of PSL(2, C), then
In this paper, we will show the existence of a sequence of inequalities, generalizing J¢rgensen's inequality, which X and Y must satisfy in order for , the group generated by X and Y, to be discrete. These conditions are mutually independent in the sense that, for given X and Y, at most one can fail to hold. These conditions arise from the ShimizuLeutbecher process defined below. For convenience, consider the upper half space model of hyperbolic 3space. We denote a directed geodesic E by the ordered pair of its endpoints; so E =(a, b)' a, b
*Partially
£
c'
at
b. The complex distance
supported by NSF Grant #MCS 7802679.
©
1980
Princeton University Press
Riemann Surfaces and Related Topics Proceedin~s of the 1978 Stony Brook Conference
0691082642/80/00006507$00.50/1 (cloth) 0691082677/80/00006507$00.50/1 (paperback)
For copying information, see copyright page
65
T
= o(£1, £2)
66
ROBERT BROOKS AND]. PETER MATELSKI
between two directed geodesics E1 = (a 1 , b 1 ) and E2 = (a 2 , b 2 ) is defined as follows: r
C ; Re(r) ;::: 0 is the hyperbolic distance between the
l
geodesics; lm(r) is the angle made by the geodesics along their common perpendicular and is determined modulo 2rr unless Re(r) = 0, in which case ± Im(r) is determined modulo 2rr. One may compute the complex distance by the formula:
where (z 1 ,z 2 ,z 3 ,z 4 ) is the usual cross ratio, as can be checked if El
=
(1, 1) and
e2
=
(er, eT).
Let X be a loxodromic element of PSL(2, C) and axis(X) the directed geodesic in hyperbolic space joining the fixed points of X . If E is a perpendicular to axis(X), then the complex distance r between E and X(E) is called the complex translation length of X. In fact X translates Re(r) units along axis(X) and rotates hyperbolic space by lm(r) about axis(X). We have
which makes sense even if X is not loxodromic. Given X loxodromic with complex translation length r, and Y in PSL(2, C), one may check the formula: tr((YXY 1 )X 1 )2
=
(1cosh(r))(1cosh(/3)),
for {3 the complex distance from axis(X) to axis(YXY 1 ); this follows by normalizing X
=
(cosh(r/2) sinh(r/2)
vxv 1 = ( cosh(r /2)
ef3sinh(r/2)
sinh (r/2)) cosh(r/2) ef3sinh(r/2)). cosh(r/2)
2GENERATOR SUBGROUPS OF PSL(2,
C)
67
Given X and Y elements of PSL(2, C) with X loxodromic, we define the ShimizuLeutbecher sequence inductively by:
Let r be the complex translation length of X, and let f3i be the complex distance between axis(X) and axis(Y i). A necessary condition for the group generated by X and Y to be discrete is that the set lcosh(f3i)l should form a discrete subset of C. The following lemma allows one to compute cosh(f3) inductively: LEMMA. cosh(f3i+ 1)
=
(1cosh(r))cosh 2(f3i) + cosh(r).
This follows from the hyperbolic law of cosines: if E0 , E1 , E2 are given, the law of cosines gives a formula for w T1 =
8(Eo, e1)'
T2 =
aero, E2)
=
8(E 1 , E2 ) in terms of
and a which is the complex distance from
the perpendicular between E0 and E1 to the perpendicular between E0 and £2 . The formula is:
The lemma follows by setting
r1
= r2 =
f3 i
and
a
=
r .
One way to check
the law of cosines is to normalize so that eo= (0, oo)' e1 = (t1, t! 1)' and e2 = (et2, et; 1) where t1 = tanh(r 1/2)' t2 = tanh(r 2/2)' and e = ea; then compute cosh 2(w/2) = (t 1 , et 2 , et; 1 , t! 1). Note that E2 does indeed have complex distance Now let zi
T
e0
2 to
With (ea, ea) as COmmon perpendicular.
(1cosh(r))(cosh(f3)). We may rewrite the above induc
=
tive formula as: zi+1
=
zr +
c'
where
c
=
(1 cosh(r))(cosh r) '
and we have that if X and Y generate a discrete group, then lzi I forms a discrete subset of C . The dynamical behavior of C under a quadratic polynomial is well understood from the work of FatouJulia ([1], [5]; see also [2]). Let
68
ROBERT BROOKS AND
J. PETER MATELSKI
fi(z) = fof ··· of(z), where f(z) = z 2 + C; a solution p of the polynomii times al equation fi(z) = z will be called a stable periodic point of period i if Jd~ fi(p)J < 1 . Then fi is contracting on any disk Be= lz: lzp\ < el on which Jd~ fiJ < 1. The theorem of FatouJulia ensures that, for any choice of C , there is at most one stable periodic orbit. Further results of FatouJulia allow one to draw by computer the region E of C defined by E = lz: fi(z) converges to the stable periodic orbit l (see Fig. 1), and the region of C defined by IC: z 2 +C has a stable periodic orbit} (see Fig. 2). To obtain the abovementioned inequalities, let p be a stable periodic point of f of period n; we may assume that fn(z)
=
I
IPI < 1/2.
Expanding
2n
ai(zp)i as a Taylor series about p, we have i=O
I
2n
lfn(z)pl
=
lzpl
ai(zp)il
:S lzp\(2n1)m [max(1, lzp\ 2 nl]
i=l
where m = max(\ai\). Setting K < 
1J~fn(p)J ~z , we see that on the disk (2 1)m
J!!. fn(p)J , then lzp\ < min(K, 1), fn is a contracting map. If also K < dz n m · (2 1) fn(z)p has no roots other than p in the disk lzp\ < K. In the case n = 1, the fixed points of f(z) = z 2 + (1cosh(r))(cosh(r)) are cosh(r), 1cosh(r). If \1cosh(r)\ < ~' we may set p
:S 1cosh(r),
and ~~ (p) = 2(1cosh(r)), m = 1. We thus have the inequality: If 0 < 1(1cosh(r))(cosh(/3)1)\ < min(12icosh(r)1\, 2icosh(r)1\), then is not discrete. In view of our expressions for cosh(r) and cosh(/3) given above, this becomes Jr&rgensen's inequality. In the case n are
=
2, the periodic points of order 2 of f(z)
1+ y1 4(C+1) d 2 , and dz f (p) .2
=
=
z2 + C
4(C+1). Using the estimates
2GENERA TOR SUBGROUPS OF PSL(2, C)
t,
i,
69
1
we find m :S 4, so that 0 < lzpj < 1 min(14IC+1i, 4IC+11) implies is not discrete. IPI
0 define (see [9]) (3.1)
A
~
(
a {32/a
({32+:)/a)'
8=({3
~{3
~a2 /{3 ) (a 2+1)/{3
then for y = (a2+{32)( 1 +a2+{32)/a2{32, (3.2)
[A~1' 8~1]
=
A~1g~1AB
=C~ ~~}
The matrices A, B generate a free group on two elements. The above parametrization is of the most general type described by (3.3) 1 = C SL 2(R), trace [A,B] = ~2, [trace A[ > 2, \trace B[ > 2 at least to within equivalence (S~ 1 rs, S < SLiR)), with a possible sign change of A or B. We abbreviate C = AB and write
79
GEODESICS ON FRICKE'S TORUSCOVERING
(3.4)
a = trace A , b = trace B , c = trace C .
Then these relations hold:
Thus 1
is parametrized by a and
f3,
i.e., a= b/c,
f3 = a/c,
(but
not uniformly so, see [12]). The fundamental domain for
1 acting on }{, the upper halfplane, is
drawn in usual fashion as :DA,B in Figure 1, showing the action of A and B on geodesics with vertices at 1, 0, a 2 / {3 2
,
oo. For our pur
poses we need to construct :D A the special domain for A. Let ~w denote the geodetic arc (w 1 , w 2 ), the axis of W, a hyperbolic element of 1
with fixed points w 1 and w 2
~A= arc(a 1 ,a 2 ) the center a 0
=
.
We assert that for
(a 1 +a 2 )/2 satisfies
1 < a 0 < 0
(3.7)
if we use some automorphism of 1
to obtain (compare [17])
00
1
Boo
0
ABoo
=
BAoo
a2
Fig. 1. Formation of the special :!)A from the fundamental domain :DA,B in the upper half zplane :JC.
80
HARVEY COHN
(3.8)
Then the vertical arc (a 0 , oo) has a portion arc (z 0 , oo) which is normal to
~A at z*. Likewise its image arc (Az 0 , Aoo) is normal to ~A at Az*. These arcs, together with arc(O, Az 0) are the new boundary of :DA as shown in Figure 1. We next introduce an abelian integral of the first kind for :D A, B (or for :D A), namely u, to transform the z halfplane to cover a lattice in the uplane. Obviously, the covering is multivalued to the extent that each z has the same image as Xz (for X f
r' ).
In the classical modular
case (2.13), du = dJ/(J1) 112 ] 213 for J(z) the (Klein) modular invariant, (see [1] and [8]). The lattice of the covering is determined by the doublyperiodic images of z
= oo.
g) A
lA
2
z*
I •
I I
I
I I
Fig. 2.
Lattice covering of torus by :DA in the uplane. Labels refer to the
zplane (see Figure 1), and images of (z =) oo are heavy dots denoting lattice points. The complete transversals are partly dotted but all lines are geodesics. True right angles occur only along SA. Here w0 = [A 1 , B 1 ]Al and w1 = AB 1 AB are nonprimitive, (see §4).
81
GEODESICS ON FRICKE'S TORUSCOVERING
The geodesic arcs normal to ~A (but lying in
transversal field uniquely covering the image of
:D A)
:D A.
emerge as a
In Figure 2, we
label the uplane by the preimages in the zplane when multivaluedness does not interfere. We see that these normals extend outside
:D A
and
there they lose their uniqueness as they wrap themselves about the lattice points (images of z
=
oo ). It is, however, the nature of noneuclidean
distance that two nonintersecting, nonparallel geodesics have one common perpendicular at the mutually closest point of approach. Thus the distance in
TA
between arc (z 0 , oo) and arc (Az 0 , a 2 ! (3 2 ) is measured along arc
(z*, Az*). (The metric ds
=
ldzl/Im z is of course transferred to the
uplane.) 1
(U
'
Consider next all geodesics Hw for W = AX or A X, (X< 1 ). Under the action of W, ~w has a period which is measured from arc (z 0 , oo) to arc (Az 0 , Aoo), using suitable lattice translations in the uplane (see [3]). Nevertheless unless W =
s 1 AS
or
s 1 A 1 S,
then
~w cannot be wholly interior to '])A , by the uniqueness of the geodesics
in the field. There are many ways a geodesic can wind around the lattice points. In Figure 2 we see ~w
0
for
w0 = [A 1 , B 1 ]A 1 = A 1 B 1 ABA 1 .
In the
uplane it must intersect itself because in the zplane it covers the full range z>zy of [A 1 ,B 1 ]tf'', (note(3.2)). Wealsoseethe geodesic ~w
1
for W1 = AB 1 AB. It merely covers two periods of ~A
but does not intersect itself. (These cases occur in (1.4c) and (1.4b) above.) §4. Trace and distance The geodetic distance is related to trace as follows: THEOREM 4.1.
Let A be hyperbolic in SL 2 (R), acting on J{, and let
§A join its fixed points. Let z be a point on §A. Then if s is the noneuclidean distance on the arc(z,AZ) and a= ltrace AI, (4.2a)
a
=
2 cosh s/2
82
HARVEY COHN
(4.2b) If z isnotapointon ~A' thecorrespondingdistance a
on arc(z,Az)
satisfies a < 2 cosh s/2, or s > 2 cosh 1 a/2. Conjecture 1.1 then becomes the following (with M
=
A):
CONJECTURE 4.3. The shortest (periodic) geodesic between arc (z 0 , oo) and arc (Anz 0 , Anoo), i.e., through n replicas of ~A as laid out horizontally in Figure 2, is the corresponding portion of ~A. We do not prove this, but what is apparent from Figure 2 is that only a finite number of homotopy classes of the punctured covering surface can have geodesics of length below a given bound. (One must incidentally check that repeated circuits about oo do increase the length unboundedly .) From this, Theorem 1.3 follows. (Note that when b
=
c there are hori
zontal and vertical reflectional symmetries in Figure 2, so that if W is in the same r ;r~coset as An, ~w can be reflected into
:D A
for compari
son with ~A.) THEOREM 4.4 (Triangular inequality). Let (4.5) for a, b, c positive, then (4.6) For proof we construct the corresponding r let z 1 be the intersection of ~A and elegant construction). Then if z 0
=
§8
=
as in §3, and
(see A. Schmidt [18] for an
B 1 z 1 , consider the triangle formed
by arc (z 0 , Bz 0 ) on ~B, arc (Bz 0 , ABz 0 ) on ~A, and arc (z 0 , ABz 0 ) which is not on ~c, (C
=
AB). Nevertheless, the inequalities are just
right for using Theorem 4.1.
GEODESICS ON FRICKE'S TORUSCOVERING
83
§5. Markoff numbers Actually, despite the appearance of Table I, the geodesic for A uBv is not much longer than its equivalent for (A, B)u,v even when lui+ lvl is large. This is like saying that the semiperimeter of a rectangle is of the same order as its diagonal, and indeed the logarithms are asymptotic! This idea tells us the order of magnitude of Markoff numbers.
If we return to (2.14), we write the matrices (see [3]) (5.1)
M = (V
1•
V )u,v 2
'
M' = (V
1•
V )u',v' 2
'
M" = (V
1•
V )u+u',v+v' 2
where u, u', v, v' ~ 0 and uv' vu' = 1. Then for the Markoff triple (5.2)
m =(trace M)/3, m' =(trace M')/3, m" =(trace M")/3
The triangular inequality (4.6) yields 3m'+ (9m' 2 4)y,
(5.4)
2
By repeated uses (writing M" now as M = (V 1 , V2 )u, v ), we have 3m+ (9m 2 4)y, \I, v 2 < (3+5\l,)u 2. (3+8 ) '
(5.5)
if we note for V 1 we have m = 1 and for V 2 we have m = 2. In logarithmic terms, for m large, (5.6)
log 3m+ a(1)::; u log(3+5 71 )/2 + v log(3+8 71 ).
If we let mN be the Nth Markoff number, an easy consequence of (5.6) is a count of relatively prime lattice points in a triangle as a method of estimating mN : (5. 7)
84
HARVEY COHN
A "lim in£" is also seen to exist by the analogy of the rectangle cited earlier in this section, but we omit details since, recently, Christopher Gurwood [10] showed a limit to exist by direct methods. The limit has not been calculated, but the constant in (5. 7), namely 2.36247 .. ·, seems close. MATHEMATICS DEPARTMENT COLLEGE OF THE CITY OF NEW YORK NEW YORK, NEW YORK 10031
REFERENCES [1]
Cohn, H., Approach to Markoff's minimal forms through modular functions, Ann. of Math. 61 (1955), pp. 112.
[2]
, Representation of Markoff's binary quadratic forms by geodesics on a perforated torus, Acta. Arith. 18 (1971), pp. 125136.
[3]
pp. 822.
, Markoff forms and primitive words, Math. Ann. 196 (1972),
[4]
, Some direct limits of primitive homotopy words and of Markoff geodesics, Discontinuous Groups and Riemann Surfaces, Princeton, 1974, pp. 8198.
[5]
, Ternary forms as invariants of Markoff forms and other SL2(Z)bundles, Linear Alg. and Appl., 21 (1978), pp. 312.
[6]
Fenchel, W., and Nielsen,
[7]
Fricke, R., Uber die Substitutionsgruppen, welche zu den aus dem Legendre'schen lntegralmodul k2(w) gezogen Wurzeln gehoren, Math. Ann., 28 (1887), pp. 99119.
[8]
, Die Congruenzgruppen der sechsten Stufe, Math. Ann., 29 (1887), pp. 97123.
[9]
, Uber die Theorie der automorphen Modulgruppen, Gott. Nach., (1896), pp. 91101.
J.,
Discontinuous Groups, (to appear).
[10] Gurwood, C., Diophantine approximation and the Markov chain, Dissertation, N.Y.U, 1976. [11] Horowitz, R. D., Characters of free groups represented in the two dimensional special linear group, Comm. Pure Appl. Math., 25 (1972), pp. 635649. [12] Keen, L., On Fricke moduli, Advances in the Theory of Riemann Surfaces, Princeton, 1971, pp. 205209. [13] Magnus, W., Karrass, A., Solitar, D., Combinatorial Group Theory, New York, 1966.
GEODESICS ON FRICKE'S TORUSCOVERING
85
[14] Markoff, A. A., Surles formes binaires indefinies, I, Math. Ann., 15(1879), pp. 381409; II, Math. Ann., 17 (1880), pp. 379400. [15] McKean, H. P., Selberg's trace formulas as applied to a compact Riemann surface, Comm. Pure Appl. Math., 25 (1972), pp. 225246. [16] Pick, G., Uber gewisse ganzzahlige Substitutionen, welche sich nicht durch algebraische Cohgruenzen erkliiren lassen, Math. Ann., 28 (1887), pp. 119124. [17] Schmidt, A. L., Minimum of quadratic forms with respect to Fuchs ian groups, I, J. reine und angew. Math., 286/287 (1976), pp. 341368. [18]
, Minimum of quadratic forms with respect to Fuchsian groups, II,]. reine und angew. Math., 292(1977), pp. 109114.
[19] Zieschang, H., Vogt, E., Coldeway, H.D., Fhichen und ebene diskontinuierliche Gruppen, Lecture Notes in Mathematics, #122, SpringerVerlag, Berlin, New York 1970. Added in proof: Dr. Gerald Myerson kindly communicated a direct proof of Theorem 4.4.
ON VARIATION OF PROJECTIVE STRUCTURES Clifford J. Earle*
1.
Introduction The purpose of this paper is to derive a new variational formula for
the monodromy map associated with (varying) projective structures on (varying) Riemann surfaces of genus p
2:
2.
The monodromy map was investigated in a series of papers by D. A. Hejhal [6, 7, 8]. Hejhal used a cuttingandpasting technique in [6] and [7] to show that the monodromy map defines a local homeomorphism from the space of projective structures to the space of monodromy groups. He obtained a variational formula in [8], but that formula is too complicated to be useful for studying the Jacobian of the monodromy map, although it has other interesting applications [9]. In contrast, our formula implies easily that the monodromy map has nonzero Jacobian and is therefore a local homeomorphism. The formula also suggests a close relationship between the tangent space to the space of projective structures and certain Eichler cohomology groups. That relationship is explored more deeply in J. H. Hubbard's paper [10]. The survey article [5] by R. C. Gunning contains much useful information and background material about projective structures. Gunning has
*This research was partly supported by a grant from the National Science Foundation. © 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00008713$00.65/1 (cloth) 0691082677/80/00008713$00.65/1 (paperback) For copying information, see copyright page 87
88
CLIFFORD J. EARLE
independently obtained a variational formula quite similar to ours and observed a relationship with Eichler cohomology (unpublished notes).
2.
Projective structures 2.1. Let X be a closed Riemann surface of genus p:;; 2, and let
rr: D
>
X be a holomorphic universal covering of X by the bounded
Jordan region D C C. We shall assume that the group of cover transformations is a quasifuchsian group 1
(with invariant domain D).
As usual, the holomorphic function ¢: D
>
C is called a quadratic
differential (for 1) if (2.1)
¢(yz)y'(z) 2
=
¢(z)
for all
y E 1, zED.
The space of all quadratic differentials is denoted by Q(i). Given ¢ E Q(l), let f be any meromorphic solution in D of the Schwarz ian differential equation lf, z!
=
¢(z). Then f is a local homeo
morphism from D into the Riemann sphere P 1 , and there is a homomorphism p from 1
into the group PL(l, C) of Mobius transformations
such that
(2.2)
f(yz)
=
p(y)f(z)
for all
y E 1, z ED .
We say that f determines a projective structure on X, and we call p the monodromy homomorphism determined by f. The merom orphic functions g on D with l g, z! the functions g
=
=
¢(z) are precisely
A of, A E PL(l, C). The effect of replacing f by A of
is to replace p by the homomorphism y ~ Ap(y)A 1 . The projective structures determined by f and A of are called equivalent; equivalent projective structures determine conjugate monodromy homomorphisms. Thus each ¢ E Q(i) determines an equivalence class of projective structures and a conjugacy class of monodromy homomorphisms p: 1
>
PL(l, C).
2.2. In order to study what happens when we change the Riemann surface X, we introduce the Teichmuller space T p of closed Riemann
ON VARIATION OF PROJECTIVE STRUCTURES
89
surfaces of genus p 2; 2. T P is a complex manifold of dimension d = 3 p3 and can be embedded in cd as a bounded contractible domain of holomorphy. The Bers fiber space (see [1]) over T P is a region Fp C T P x C with these properties:
1 acts freely and properly discontinuously on Fp as a group of
(i)
biholomorphic maps y(t, z) = (t, /(z)) (ii) every
for all
y < 1, (t, z) < FP ,
D(t)=lz PL(l, C) so that !f, z! = ¢(z) and (2.3)
f(/z) = p(y)f(z)
for all
y < 1, z < D(t).
It is the dependence of the conjugacy class of p on (t, ¢) that we wish
to study. 2.4. To study the variation of p we need local coordinates in T p. Fix t 0 < TP and put D = D(t 0 ), I'= f'(t 0 ). Let W be the vector space of functions f1: C . C such that (2.4)
(2.5)
fl(Z)=O
forall
ziD
11(z) = A(zr 2 r/>(z) for some ¢ < Q(f') and all z < D ,
where ,\(z) ldz I is the Poincare metric on D.
90
CLIFFORD
J.
EARLE
Let W0 be the set of p. f W with supllp.(z)l; ZfDI < 1. For each fJ. f W0 there is a unique quasiconformal map w
=
wP.(z) of P 1
onto itself satisfying the Beltrami equation
in C and having the behavior w(z) = z + 0(1/z) For each p. f W0
,
as
z ....
oo •
wfl.(D) is a bounded Jordan region and the group of
Mobius transformations yfl.
=
wfJ. o yo (wP.) 1 , y f [', is quasifuchsian
with invariant domain wfl.(D). There is a neighborhood W1 of zero in W0 which provides a coordinate system at t 0 in such a way that for each p. f W1 , D( p.) is the region wfl.(D) and f'( p.) is the group of Mobius transformations y fJ.. For fJ.
f W1 and
c/lf Q(f'(p.)),
(2.3) says that foyfl.
= p(y)of
in D(p.) if lf,zl = ¢(z). Therefore, in D we have p(y)ofowfl. = foyfl.owfl. = fowfl.oy. Put h (2.6)
=
fowfl.: D> P 1 . Then h is a C"" local homeomorphism, and h(yz)
=
p(y) h(z)
for all
y f f'
and
z fD .
2.5. It will be convenient to normalize so that each ¢ f Q(f'(p.)) determines a unique function g
= ~ with l g, z I = cf>(z) in D( p.) . For
this purpose we choose z 0 f D and choose W1 so small that z 0 f D( p.) for all fJ. f W1 . We define g: D(p.)> P1 to be the unique meromorphic function satisfying
91
ON VARIATION OF PROJECTIVE STRUCTURES
(2.7) g(z 0 ) = g"(z 0 ) = 0, g'(z 0 ) = 1, and {g,zl = ¢(z) for all z
f
D(/L).
The most general function f with {f, z I = cp(z) is again f =A o g for some A
f
PL(1, C), and the function h in (2.6) has the form
h =A o g owiL. The monodromy homomorphism p: [' > PL(1, C) in (2.6) is thus a function of A , IL , and ¢ . 3.
The variation of h 3.1. In order to study the dependence of h and p on A, IL, and ¢,
we assume that (A, /L, ¢) = (Ar, ILr' ¢r) depends on a complex parameter r in such a way that A
f
PL(1, C) , IL
f
W1 , ¢
f
Q(f'( IL)) for all r , and
Our first goal is to find the variation of h . Now h=AogowiL, where g: D(IL)>P1 satisfies(2.7). The theory of the Beltrami equation (see for instance [12]) gives (3.2)
wiL(z) = z+rw(z)+o(r),
where w(z) is given explicitly by
(3.3)
w(z) = }
ff it(()((z) d~d7J. 1
D
The most important properties of w are that w_ = z and only if it = 0 .
it and that w = 0 if
Since the solution of the equation {g, z I = ¢(z) is the ratio of two independent solutions of the linear equation u"(z) =  ~ ¢(z)u(z) , it is evident that (3.4)
g(z) = g 0 (z) + rg(z) + o(r) ,
92
CLIFFORD
I g0 , z I = ¢ 0 .
where of course
J. EARLE
Therefore
h(z) = h 0 (z) + rh(z) + o(r) ,
(3.5)
3.2. It will be useful to be slightly more explicit. If we write f=A 0 g, then lf,zl=r/>(z) in D(tt), f 0 =h 0 , and
(3.6)
f(z) = f 0 (z) + rf(z) + o(r) ,
with (3.7)
f
The function
is meromorphic in D , with possible poles at the poles of
f 0 , but since all poles of f are simple, the function (3.8)
is holomorphic in D . Now h(z) = f(w(z)), so (3.2) and (3.6) give h(z) = f(z) + h 0(z)w(z) .
(3.9)
3.3. The following lemma summarizes the above observations. LEMMA
1. If A,
(3.5) with
h
tL,
and ¢ satisfy (3.1), then h = A o ~ owtL satisfies
given by (3.9). In particular h*(z) = h(z)ho(z)l
(3.10)
=
f*(z) + w(z)
is a C 00 function in D.
3.4. For our main purpose the explicit form of
h
is unimportant. We
need to know merely that h* is a C 00 function with h~ z this additional fact. LEMMA
2. h*
=0
in D if and only if
A= jJ. = ¢ = 0.
=
w_ = jJ.' and z
93
ON VARIATION OF PROJECTIVE STRUCTURES
w= 0 ,
Proof. If /1. = 0 , then
h = h*
A = /1. = ~ = 0.
= 0 if
and if ~ = 0 , then
g= 0 .
Therefore
Conversely, if h* = 0, then (since f* is
holomorphic in D )
w_ z
0 = h*
z
so
/1 = w= 0.
Therefore, by (3.8) and (3.10)
f = f*
= 0, so c/J(z) =
¢ = 0.
g = 0,
and (3. 7) gives
lf, z! = !f 0 , z! + o(r), and
A=
IJ.,
Therefore
0 . The lemma is proved. 3.5. For future reference we wish to give the precise relationship be
tween ¢. and f * .
¢=
LEMMA 3.
(f*)"' + 2c/J(f*)' + ¢ 0f*.
The proof is merely a computation. One method is to put a a 0 +ra+ o(r), with a 0
Since
f=
=
0 0.
f' /f
=
f"/f' =
Then
f 0f*, the derivatives of
f
can be expressed in terms of f 0
and f*. That gives
Finally, ¢
4.
=
a' ~ a 2
,
so
¢
=
a' a 0 a.
The lemma follows.
The variation of the monodromy map 4.1. Our main result describes the variation of the monodromy homo
morphism p in terms of h*. THEOREM 1. Let A , 11, and ¢ let p: (4.1)
r. PL(1, C)
satisfy (3.1), let h
=
A o g¢ o wll, and
satisfy (2.6). Then
p(y) = p 0 (y) +r p(y)+ o(r)
for all
y
=
4> 0 +
t~ in (3.1),
4> t: Q(r) represents a variation along the fiber Q(r). Then
f* is holomorphic in D , and
(5.6) by Lemma 3. Theorem 1 and (5.2) give
(5.7)
ay 1 to the kernel of the resolvent for D on L~ [9]. Since Qs is the Laplace transform of the fundamental solution to the heat equation on H, the reasoning in [4, §4] can be applied to give:
for all s. In order to extend (5) (8) to nonunitary
x,
we again denote by L~
the Hilbert space of all functions f on H satisfying (2) (where the norm is now dependent on g)); then D is a nonself adjoint operator on a dense subspace of L~ and one has: THEOREM
1.1. With respect to the rmrking (1),
T()()
has a unique
analytic continuation from (S 1 ) 2 g to a weiidefined holomorphic function of
x
f
(C*) 2 g satisfying T(x) = T(x 1 ) = T()?)
x.
for all
On (C*) 2 g, T(x) has a divisor of zeroes V consisting of ail representations
x
for which 0 is in the (discrete) spectrum of D on L~. +1
+1
Proof. Suppose the values of lx(AjW , lx(Bj)l
and write any y (
r
are bounded by M1 ?: 1,
N
in the form y =
l
N
B/ 1 and with ni ?: 0 and
II Yi ni
with Yi one of the A/ 1 '
1
ni a minimum. Then an elementary esti1
mate of the length r(z 0 , yz 0 ) of the nonEuclidean segment joining any basepoint z 0
f
g) to y z 0 shows that r(zo, yzo)?: C! 1
C 1 > 0 depending only on
r
(see [2] or [5]). Thus
l
N
ni for some 1
111
ANALYTIC TORSION AND PRYM DIFFERENTIALS
=IT \x(yi)\ni < M1 c1r(zo,yzo) N
\x(y)\
for any
zo ( 9) ;
1
and since \Qs(z, yz')\ = O(e(Res)r(z,yz')) as r."" uniformly for Re s 2:
E
> 0, it follows that the series (7) converges uniformly for x
with entries bounded by M1 , for z' bounded away from z in for Re s > 1 +C 1EnM 1 . Now
9J
9J
and
is compact and Gs has only the singu
larity 21 rn \z 'z I in ']); so Gs (z' z ', x) is a (generally nonself"
0
adjoint) compact integral operator on L~ for s 0 > > 0 satisfying (9)
Gs(z, z', x) Gs (z, z', x)
(ss 0 )(s+s 0 1)
JJ
0
=
Gs(z, z#, x) Gs 0 (z#, z', x) ldz"l
T
Thus from the Fredholm theory [8, Ch. IV] we conclude that Gs(z,z',x), the kernel for [D s(s1)] 1 on L~, has a simultaneous meromorphic continuation through the splane and the representation space (C*) 2 g. At a pole s 0 of order v, Gs has a Laurent development
l
+oo
Gs(z, z', x)
=
Rk(z, z')(A 0 A)k,
A0
=
s 0 (s 0 1)
k=v
m
A = s(s1), where the Rk satisfy:
as integral operators on L~. Choosing any basis e 1 (z,x), ···,em(z,x) for the (necessarily finitedimensional) A0 subspace, with (DA 0 )ej ( span le 1 ,
···,
ejl! for each j 2: 1, we can write
112
JOHN FAY
for a dualbasis lt\(z, x 1)1 of the /\ 0 subspace of D in (10) and (11) then, (12)
trace R_ 1 (z, z') = m, trace Rk(z, z') = 0
for
Kind k < 1.
Now let x' be any fixed nontrivial unitary representation of
J
r,
and set
+oo
T(x) = exp T(x')
(13)
(2s1) trace[Gs(z,z',x)Gs(z,z',x')]ds
1
where the integral, taken over the positive real axis say, is convergent near
since
oo
(Re
!im[Gs(z,z',x)Qs(z,z')]=O(e
s)r
0)
as
S>+oo,
z > z
with r 0 = inf lr(z, yz)lz £~, yf_ II the "neck" of ~. Then T(x) is well
x
defined in
as any point in the discrete spectrum crosses (1, + oo)
since the integral only changes by 2rrim, m V
f
(C*) 2 g
Z as in (12). Moreover, if
f
is the analytic subvariety given by the polar divisor of
G 1(z,z',x), T(x) will by (12) tend to 0 as
x
approaches
x0
f
V and
so, by the Riemann removable singularity theorem, can be made holomorphic near V. Thus, from (5) (8), (13) gives the unique analytic continuation of torsion from (S 1 ) 2 g to (C*) 2 g; the symmetries of T(x) follow from the reflectibn principle and the fact that T(X) = T(x) = T(x 1 ) for X
f
(S1)2g.
If
x
f
(C*) 2 g V, Gs(z, z', x) is analytic at s = 1 and one can write 4rrG 1 (z,z',x)
(14)
=en P(z,z',x) P(z,z',x)
where, for fixed z, the "primeform" P(z, z', x) is analytic in z' with only a simple zero at z'= z in
~.
B(z,z',v)
"'
=_!.Len P(z,z',x) " azaz'
is the Bergman kernel for the holomorphic Prym differentials with multipliers
x,
while
113
ANALYTIC TORSION AND PRYM DIFFERENTIALS
(15)
O(z I z ', X)
=
2 ; fn P(z aaa z z
z ', X)
I
=  1(z~z)2
+ S(z' X) + O(z ~z)
is orthogonal (under principal value) to the Prym differentials. The RiemannRoch duality then takes the following form: if ai f C, z 1 , ... ,zm are distinct points on M and
x
f (C*) 2g V,
m
l
aj O(zj, z, x) is an exact differential with multipliers x and double
j=l
l
m
poles at z
=
zj if and only if
aj B(z, zj, x 1 ) = 0 for all z f
:D.
j=l
In the elliptic case M = C/Z+Zr, Im r > 0, Kronecker's second limit formula applied to the MP zeta function for D
[6]: T(x) = \17(r)i2
()~~ x~ +X
2
=
4
a2 azaz
2 gives on LX
2
(0)
' x(A)
=
e2rrix and x(B)
=
e2rrix
T
where 71(r) is Dedekind's function, A and B are the transformations z
>
z + 1 and z
(16)
>
z + r , and
e[~] (z)
=
l
explrri(n+a)r(n+a)t+2rri(z+/3)(n+a)t!
n(Zg
r
for a, /3 ( Rg, z f Cg (here g = 1). The analytic continuation of T(x) to (C*) 2 : T(x)
for (x(A), x(B)) Z +Zr or
\.,.,(r)\2 e2rrs2 Imr
=
=
()[~ (iin 1 , it is seen that V always contains the (anti) analytically trivial representations V0 of the form: (e
or
2rris·
l, e
2rri(rs)j
)
for some s ( cg. PROPOSITION
x
1.2. V V0 is the (open) subvariety of all
< (C*) 2 g V0
for which the cupproduct pairing (17)
is degenerate (here representation
x
is the flat line bundle on M determined from the
x ).
Proof. Let wi(z, x) (resp. wi(z, x)) for i H 1 • 0 ( x)
the Prym differentials
=
H 0 • 1(
(resp.
1, ···, g1 be a basis for
x )) ;
by Grauert 's Theorem,
we may assume that these bases vary localanalytically with
x
< (C*) 2g V0 . Now if
x
< V V0 , there is a singlevalued harmonic
function f(z) with multipliers x such that df
WtH 1 • 0 (~), 1'/tH 0 • 1 (~) and w, 0
=
J
f w/z, x 1 )
a~
det D(x)
w(z, x)+ T/(z, x) for
xiVo.
bothnonzerosince
JJ
w(z, x)
A
Thus
wi(z, x 1 )
~
for any differential wj(z,
(17)'
=
17
=
x 1 ) so that
~ del ([Jw;(z, x) Awpi, Xt~
in this case. Conversely, if x
0
I V , Gs (z, z ', x) is analytic at s
=
1;
hence 1:Si,j:Sg1
for suitable aij < C will be a reproducing kernel for H 1 • 0 (~) and so det D(x) cannot vanish in this case. Thus holdsthat is, the pairing (17) degenerates.
x < V V0
if and only if (17)'
115
ANALYTIC TORSION AND PRYM DiFFERENTIALS
From (17)' one can give an equation for V in terms of the Riemann (;lfunction (16) and the SchottkyKlein prime form E(z, a) used for constructing bases for H 1 ' 0 ( x) over Zariskiopen sets in (C*) 2 g; thus if b, a 1 , ... , ag_ 1 are generic points on M, V is contained in the subvariety of e
2rris 1. 2rris1. 2 ,e ) in (C*) g defined by
Here ~
=
(g1) p + kP
f
Jg 1 (M) is the divisor class for the Riemann con
stants kP for any base point p
f
M, lifted to Cg so that the above
integrand is a welldefined form in H 1 ' 1(M); observe that V0 lies on this variety since (;lr(~~)
=0
for any positive divisor ~ of degree g1
by Riemann's theorem. THEOREM 1.3.
The divisor of zeroes V C (C*) 2 g of T(x) determines
the period matrix of M for the marking (1) of M defining T(x). Proof. For s, s ( Cg corresponding to the representation (x(A), x(B)) =
(e2rris,e2rris), let ui(z,s,s) i
=
=
ui(z,x) (resp. ui(z,s,s)
=
ulz,x)) for
1, .. ·, g be a basis for the meromorphic (resp. conjugate meromorphic)
Prym differentials with multipliers some fixed point p
f
x
and with at most a simple pole at
M; by Grauert's theorem, we may assume that this
basis varies analytically over all of (s,s) ( C 2 g covering (C*) 2g. Now if
x
f
V, there is some harmonic function f on
and we can write
:D
with multipliers
x
116
JOHN FAY
g
g
~ a.s. = ~ b.s: = 0
for some constants ai, hi, C with
""
1 1
""
i=l
1 1
'
i=l
The dimension of the space of such f is thus the column nullity of the matrix
J J
J J
uj(z, x)
uj(z,x)
A.
A.
1
D(x)
1
uj(z, x)
=
uj(z, x)
B.
B.1
1
Si
(the condition on
x (V~ ¢==::::;>
Now consider
is then redundant here); and so
T(x) = 0
~
" det D(x) =0
X
0
0
sj
det D(x) = 0 or
~
x ( V0
G 1 (z,z', x') has a pole at x' =X.
near the identity with 1 x(Ai) = Ei' 1 x(Bi) = ei;
changing the basis of uj if necessary so that
the tangent cone to det D(x) = 0 at
x =I
has the equation (up to a
constant):
l
g
g
g
~~
EE TT· • 
1 J 1J
2
l
i,k,j=l
eiek "ij Re Tjk +
l
i,j,k,f=l
EkEfTTij rie"ijk
=
0
117
ANALYTIC TORSION AND PRYM DIFFERENTIALS
where "ij is the matrix of cofactors of Im r. Thus "ij is determined up to a constant from the coefficients of eiej and then Re rij is uniquely given from the ~ g(g+l) coefficients of eiej since "ij is a positive definite matrix; the ambiguity in "ij (corresponding to some positive multiple of Im rij) is eliminated by consideration of the coefficients of eker and thus the period matrix for the given marking is uniquely deter,, mined from the tangent cone to det D(x) = 0 at x = I. §2. Differential of torsion on the moduli space Suppose 11 is a Coo Beltrami differential on M0 = M ""' 1 \H and w(z)
z + e cf>/z) + o(e)
=
is a quasiconformal mapping of H onto H fixing 0, 1, oo and with dilitation EIJ.
=
~I
rz
for E near 0. From the explicit construction of
the mapping function [1, p. lOS]:
~,.
'~'11
(z)
1,.
= '~'11
(z)
1
= Ti
rr (~')
JJ
11
z(z1)
s ,(,1)(,z)
dl"dT/ r,
'
J' =
s
c_
/z) is an Eichler integral with lc/> 11(z)l = O(lzlfulzl) as z ~ oo and with real quadratic period polynomials: (18)
If MEIL is the compact Riemann surface rEIJ.\ H formed from the Fuchsian group rEIJ.
=
1
wrw ' then the metric
gives rise to the metric on M0 :
where
yldzl
on MEIJ. pulled back by w
118
JOHN FAY
a(z)
is a real automorphic function on M0
.
If
is the Laplacian for this metric, then the spectrum of D on L 2 for M0 Ell. y is the same as the spectrum of D on L 2 for M , and the eigenfuncy Ell. tions of D on M0 are the pullback under w of the eigenfunctions of Ell. D on ME/J.. So if G~ and G~ll. are the respective resolvents for a
r
representation y of
r Ell.
and
(compatible with w ):
G~(z,z',y)=G~(w(z),w(z'),y), z,z'f1l=f'\H.
(20)
To describe the perturbation of the spectrum of D we make use of a "weightedmean" of eigenvalues as in [3]: THEOREM
2.1. For fixed y
f
(C*) 2 g, let ,\111., · · ·, ,\~ be the eigen
values (not necessarily distinct) of DEll. near the eigenvalue ,\ of D with multiplicity m, and
~uppose
R_ 1 (z, z') is the reproducing kernel
(11) for the ,\subspace. Then the weighted mean of eigenvalues has an expansion:
(21)
+4
~ ~ ,\~11
ry J... L
= ,\
+
~ {A .[f
a(z)
R_
1 (z,
.1J
rll.(z)
~ (z, z') + 11(z) a2 R_! ozoz
~
dzdz
(z,
z)
f§
z)J z ,=z dxdy
}
+ O(E) .
Proof. For any C 00 function f(z) automorphic under 1 0 with multipliers yl , (19) gives (Ds(sl)+EA)
JJG~(z,z',y)f(z')(l+ea(z')) 1l
ldz'l =f(z)+o(E)
119
ANALYTIC TORSION AND PRYM DIFFERENTIALS
as
E ~
0; expanding this out using Green's theorem and (19) we find
JG JG l 4Jryr. . . rcz') JGaz~ cz.z',x) JGaz~ cz',z",x)+ JL(z') az~ cz,z',x) az.~ (z',z",x)J dxay'
:D for all z" ,£ z in
:D.
Now if R~ (z, z ') is the sum of the reproducing
kernels on M0 for the eigenvalues of DEll near A = s(s1),
for
I;=
s'(s'1) and
o,
equation with respect to
sufficiently near 0. Differentiating this
E
E,
~ ,
and taking the trace with respect to
(22) gives
i (A~11A) . 1=1
= e
lJ ,
R,es
:D
s ( s 1 )=A
(s'(s'1)A)Hs'(z,z")l " z =z
~
+ o(e)
which is the righthand term in (21 ); here the higher order terms in the Laurent expansion of Gs do not appear by virtue of the relations (10). A Poincaretype series for the variation of the resolvent can be given in terms of the period polynomials (18): thus expressing (22) as an
120
JOHN FAY
integral over H (or differentiating (7) in e), (20) gives
# GEIL(z, z', x)l = 1 m S E=O y
~
~ Yfl
x(y)1 dQs (z, yz') lm p ~
~1
(z)ei0(yz',z)
with absolute convergence for Re s > 2 + C 1 en M1 from the growth condition on c/> 11 (as in the proof of Theorem 1.1); the series has a meromorphic continuation through the splane with poles at those of Gs(z, z', x) with twice the corresponding order. From the (anti) holomorphic part of (21) for Beltrami differentials of the form 11(z) = y 2 Q, Q a holomorphic quadratic differential, one can note also the following remark. For fixed
x,
let '\ (r ), .. ·, Ap(r) be dis tinct eigenvalues of multiplicity 1 with reproducing kernels ei(z,x 1)ei(z',x), i=1, .. ·,p onRiemannsurfaces M7 for r in some open set
'U
in Teichmiiller space Tg; then a differentia
ble function c/>(.\ 1 , · · ·, Ap) is a holomorphic function of moduli on
'U
if
and only if the Beltrami differential
on each surface M7 is orthogonal to the holomorphic quadratic differentials on M7 (a "stationary differential" in the sense of [1]). For fixed
x,
the differential of torsion T(x) on the moduli space
can now be computed from Theorem 2.1, using (21) to give a variation of the heatkernel in (3) as in [7, p. 194]; alternatively, proceeding directly from (22), one has: THEOREM
2.2. For x, x' f (C*) 2 g V,
~  2 m. ~
yLn   ,
1 0 occur when g
=
1
or 0, [16]. Let L(R) be the set of all Beltrami differentials on R. By this we mean that if 11 is in L(R) and z is a local parameter for some open set V in R, then IJ.(Z) is a measurable function on V, bounded in the su,p norm and the global differential 11 behaves in such a way that ll.(z)dz/dz is invariant. Furthermore, for the function 1111 on R, we require that the quantity ll11\l.,.,, which is the essential supremum of lll(z)l over all z in R , be bounded. Let M(R)
=
!11 E L(R); ll11ll"" < 11. Given any 11 in M(R) there will be
a Riemann surface RIJ. and a quasiconformal homeomorphism w: R which satisfies the Beltrami equation (2)
See [3]. We will denote a homeomorphic solution w to (2) by w 11 •
>
R 11
QUASICONFORMAL RIGIDITY FOR RIEMANN SURFACES
125
Now suppose h is a quasiconformal homeomorphism of R onto R. Let h*(IL) be the Beltrami coefficient of wIL o h, that is, h*(J.L)
(3)
If a(z)
=
=
(w oh)....../{w oh)z . j.L z j.L
~/hz, the explicit formula for h*(IL) is a(z) + J.L(h{z))t9(z) 1 + a(Z)J.L(h(z))t9(z)
(4) where t9(z) = (hz)lhz.
Since a, IL, and h *(IL) are the Beltrami coefficients of quasiconformal mappings with domain R, one sees that they are all elements of M(R) and formula (4) shows how the group of quasiconformal homeomorphisms of R , D(R), acts on M(R). Let D 0(R) be the group of quasiconformal homeomorphisms of R which are homotopic to the identity. Explicitly, h is homotopic to the identity if there is a continuous mapping g: R xI .... R such that g(p, 1) h(p) and g(p, 0)
=
=
p for every p in the interior of R. It follows that
for such a homeomorphism h, its quasiconformal extension to the boundary of R must preserve boundary components but it may move points along each component. DEFINITION. The reduced Teichmiiiier space of R, T#(R), is the set of orbits in M(R) under the action of D 0(R). For IL in M(R), the orbit of IL under D 0 (R) is caiied the Teichmiiller class of IL and is denoted by [J.L].
A theorem of Ahlfors and Bers and extended by Earle shews that T#(R) is a manifold, [2, 5, 6]. Moreover, the natural mapping (5)
is differentiable and each fiber at the point 0 in M(R). Hence, the tangent space to T 11(R) at the origin is L(R)/N and, under the pairing (6)
(f.l, (j:J) 1+ Re Jff.l(j:Jdxdy ,
the cotangent space is Q(R).
QUASICONFORMAL RIGIDITY FOR RIEMANN SURFACES
2.
127
The variational method The importance of the following lemma was first pointed out by
Hamilton, [11]. It is a consequence of the differential structure of the mapping (5) and the implicit function theorem. LEMMA 2.1 (Hamilton). Let
which is
C1
Jl. f
N. Then there exists a function a(t,z)
with respect to t uniformly in z for sufficiently smaii t
and such that a(t, z) is in M0 (R) for each t and a(t, z) = t JJ.(Z)+ o(t) uniformly in z . Lemma 2.1 can be extended in the following way. Let S be a finite Riemann surface and R a nonempty open subset of S and M1 (R, S) be the subset of M0 (S) consisting of elements whose support is contained in R. Let N(S)=IJJ.EL(S):JJJ1.¢dxdy=0 forall ¢in Q(S)!. Here the double integral is taken over S . LEMMA 2.2. Let
Jl. f
N(S) and suppose the support of
R. Then there exists a function a(t, z) which is
c1
Jl.
is contained in
with respect to t
uniformly in z for sufficiently smaii t and such that a) a(t, z) is in M1 (R, S) for each t and b) a(t,z)=tJJ.(z)+o(t) uniform/yin z. This lemma is proved in [10]. The method is to look at the natural mapping
(7) where M(R) consists of elements of M(S) whose support is contained in
R. In showing F has surjective derivative at each point, one uses the finite dimensionality of Tit(S). Then one uses the differentiability of (7), as developed by Ahlfors and Bers, and the implicit function theorem. It is possible to prove a version of Lemma 2.1 by using the main in
equality of Reich and Strebel, [14], and this approach will be investigated in later work. Lemma 2.2 seems to defy this approach.
128
FREDERICK P. GARDINER
Now let R be connected and of finite type and notice that restrictions of elements of M(S) to R are Beltrami differentials on R . The boundary of R in S may not be analytic. Nonetheless, by the uniformization theorem, it can be realized as a subsurface with analytic boundary curves contained in a larger surface. Thus Tit(R), Q(R), and N can be defined just as in section 2. Let f : Tit(R) . R be a differentiable function. Then f o
R such that
fo in S which does not lie in the closure
of T. The domain D in Theorem 2 can be described in a very explicit manner. Namely, D = C y, where y is the arc y
and a
i
(o, 8~).
=
lz = ± ie(a+i)t: t dO,
oo)l
U
IOI
Hence it is not difficult to derive an analytic expression
for the conformal mapping g of L onto D, and cf>
=
Sg turns out to be
a rational function. The idea behind the proof of Theorem 2 is quite simple. For a let
i
(0, oo)
SPIRALS AND THE UNIVERSAL TEICHMULLER SPACE
147
Then a 1 and a 2 are logarithmi~ spirals in D which converge onto the point 0 from opposite sides of aD. Next suppose that f is any conformal mapping of D which fixes the points 1, 1 ,
oo.
As 11Sfll 0
approaches 0, f converges to the identity in D. Hence for 11Sfll 0 small, f maps a 1 , a 2 onto a pair of disjoint open arcs a 1 *, a 2 * which spiral onto f 1(0), fiO.), the points which f(z) approaches as z
>
0 from the two sides of
ao.
Now the rate at which a 1 and a 2 , and hence a 1 * and a 2 *, spiral depends on a. If a is sufficiently small, then a 1 *, a 2 * will spiral very slowly onto f 1 (0), fiO). Since a 1 *, a 2 * are disjoint, the points f 1 (0), f 2 (0) will either coincide or be separated by a distance greater than a positive constant d. Finally if we make 11Sfll 0 still smaller, we can arrange that f 1 (0), f 2 (0) lie near 0 and hence within distance d of each other. Then f 1 (0) and f 2 (0) will coincide and f(D) will not be a Jordan domain. Complete proofs for Theorem 2 and its corollary are given in [7]. UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN
REFERENCES [1] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109(1963), pp. 291301.
[2] L. Bers, On boundaries of Teichmiiller spaces and on Kleinian groups I, Ann. of Math. 91 (1970), pp. 570600. [3]
, Universal Teichmiiller space, Analytic methods in mathematical physics, Gordon and Breach (1970), pp. 6583.
[4]
, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), pp. 257300.
[5]
, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), pp. 10831100.
148
F. W. GEHRING
[6] F. W. Gehring, Univalent functions and the Schwarz ian derivative, Comm. Math. Helv. 52 (1977), pp. 561572.
[7]
, Spirals and the universal Teichmi.iller space, Acta Math. 141 (1978), pp. 99113.
[8] Z. Nehari, The Schwarz ian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), pp. 545551.
INTERSECTION MATRICES FOR BASES ADAPTED TO AUTOMORPHISMS OF A COMPACT RIEMANN SURFACE Jane Gilman and David Patterson
1.
Introduction Consider a conformal automorphism h of prime order p with t fixed
points on a compact Riemann surface of genus g (g 2: 2). In [2] ] . Gilman describes an integral homology basis (called the basis adapted to h) with respect to which the matrix of the automorphism h has a particularly nice form. For many applications we are interested not only in this homology basis, but also in the intersection matrix of this basis. Whereas the matrix representation for h depends only on the numbers p, g, and t , the intersection matrix also depends on the conjugacy class of h in the mapping class group of the surface. This conjugacy class is determined by a (p1)tuple of integers. For an automorphism with fixed points, only part of this intersection matrix was obtained in [2]. The purpose of this paper is to obtain the complete intersection matrix. In recent discussions with M. Tretkoff, the authors have discovered that he has devised a similar method for constructing a basis for surfaces that are branched coverings of the sphere and for computing intersection numbers of such curves [6].
2.
Notation We now fix some notation and terminology to be used throughout the
paper. For h, p, g and t as above, let H be the cyclic group
©
1980 Princeton University Press
Riemann Surfaces and Related Topics Proceedin{5s of the 1978 Stony Brook Conference
0691082642/80/00014918 $00.90/1 (cloth) 0691082677/80/00014918 $00.90/1 (paperback) For copying information, see copyright page
149
150
JANE GILMAN AND DAVID PATTERSON
generated by h . The quotient surface W/H will be denoted by W0 and its genus by g 0
.
We assume that t > 0. At each fixed point Pk (1 :S k:::; t)
of h, we can assign a rotation number qk (1 :S qk :S p1). The rotation number is characterized by the fact that there is a local coordinate system z near Pk such that the action of h is z
1>
exp(2rriqk/p)z. (Here we
adopt the convention that a positive rotation is one in the counterclockwise direction.) Associated with each rotation number qk, there is a complementary rotation number sk (1 :S sk Sp1) satisfying qk · sk = 1 (mod p). For 1 :S s :S p1 , let ns be the number of sk is equal to s. It is known [4], [1] that the (p1)tuple of integers (n 1 , ···,np_ 1 ) deter
mines the conjugacy class of h in the mapping class group of the surface W. We introduce further notation that will simplify many of the formulae that follow. Let
q
s
be the smallest integer s such that ns ~ 0 and let
be the integer satisfying 1
S q S p 1
and
q · s =1 (mod
p). We may
assume that the fixed points are ordered so that the complementary rotation numbers are in increasing order. In this case we have
s= s1
and
q=
q 1 . For any integer m, let [m] denote the least nonnegative residue
of
q· m
mod p. Thus the integer [m] satisfies 0 :S [m] :::; p 1 and
s · [m] = m 3.
(mod p).
The main result
Before we state the main result we shall describe the basis adapted to h. The following result was suggested by the work of
J.
Nielsen [4].
THEOREM 1 (J. Gilman [2]). The surface W has an integral homology basis consisting of the homology classes:
(1) hj(Aw), hj(Bw)
(1::; w:::; g 0 , 0::; j:::; p1)
(2) h k(Xi)
(3 :S i :S t, 0 :S k :S p2) .
The elements of the first set are permuted cyclically by h :
h(hj(Aw)) ~ hj+l(Aw) h(hp 1 (Aw)) ~ h 0 (Aw)
(1::; j:::; p2)
151
BASES ADAPTED TO AUTOMORPHISMS
(with similar equations for the classes involving the Bw's ). The elements of the second set are mapped as follows: h(hk(Xi)) ~ hk+ 1 (Xi)
(1Sk
Ka C C can be extended to an isomorphism a: L > L. It is known that 2 77i 2 rrix
the isomorphisms of L are induced by maps e t
1>
e
t
, where x is
an integer relatively prime to t. Thus, the isomorphisms a: K
>
Ka C C
are induced by maps: COS
:!I. a
t> COS 7TX
COS
:!I. b
1> COS 7TX
COS
:!I. c
1> COS 7TX
a
b
c
where (x, 2a) = (x, 2b) = (x, 2c) = 1. THEOREM
3. Let Fg denote the fundamental group of a surface of genus
g. Then there exists an injection Fg C
Proof. It is well known that F 2
:;
S0(3).
F g, and is easily seen that
F 2 C T(2, 8,8). (Consider F 2 as a Fuchs ian group, whose fundamental domain D is a regular, nonEuclidean octagon. The lines from the center to the vertices and the perpendicular bisectors of the sides divide D into 16 congruent triangles, with angles 77/2, rr/8, rr/8. The group generated by reflections in the sides of any of these triangles is R(2, 8, 8). From this we see that F 2 C R(2, 8, 8), and so F 2 C T(2, 8, 8) .) For T(2,8,8), the corresponding field is K = Q(cos [}, and t=16. If we take x = 3, we obtain an isomorphism a: K
>
Q (cos
1
77 )
,
and
176
LEON GREENBERG
1
COS
cos 377 8
cos 377 2
1
cos 377 8
cos 377 8
1
cos 377 2
Ma
377 2
Since (77/2, 377/8, 377/8) are the angles of a spherical triangle, Ma is positive definite. Therefore T(2, 8, 8) can be injected into S0(3), and so this is also true for Fg C T(2, 8, 8). THEOREM 4.
q.e.d.
The homomorphism ea,{3,y: T(a, b, c)> T(if,
duced by a field isomorphism a: K __, Ka (and consequently
€. t)
1s m
ea,tJ, a y
is an
isomorphism) if and only if the following two conditions are satisfied: (1) (a, 2a)
(/3, 2b)
=
=
=
(y, 2c)
=
1,
/3 mod 2(a, b) a { (2) /3:y mod2(b,c) y =a
mod 2(c,a),
or if conditions analogous to (1) and (2) are satisfied when the triple (a, /3, y) is replaced by (a, b/3, cy), (aa, /3, cy) or (aa, b/3, y).
/377 y77 . Proof. Let ~ be the circular triangle with angles a77 a• b' c' wh1ch determines the homomorphism ators of T (if.
f t) .
ea,FJ,Y a ,
and let
If we extend the sides of
x , y, z
6:
be the gener
to complete circles,
we obtain a configuration containing several triangles, whose vertices are fixed points of
x, y and z. (There are 8 triangles in the spherical and
hyperbolic cases, and 4 triangles in the Euclidean case. See the figures in Remark 2, following Theorem 2.) All of these triangles determine the
e
same homomorphism. ( a,fJ, Q y is really determined by the reflections in the circles, rather than the triangles.) These triangles have angles ( a 77 /3 77 y 77) a ' b ' c '
(aa77 '
(b/3) 77 (cy) 77) ((aa) 77 /3 77 (cy) 77) d an b ' c ' a ' b ' c
77 (b/3) 77 y 77 ) Thus 8 is induced by a field isomorphism, ( (aa) a ' b ' c · a,/3,y if and only if one of the following maps a extends to a field isomorphism of K:
HOMOMORPHISMS OF TRIANGLE GROUPS INTO PSL(2,
cos
11 b' cos
g:.)
( cos aa• 11 cos (311 c ' b' cos Y11)
(cos~.
cos
11 b' cos
g:.)
11 ( cos aa' 11 cos (b{3) b
(cy) 11) cos c ,
11 ~. cos b' cos
g:.)
(cos (aa)11 cos (311 b a '
(cy)11) cos c ,
a 3 (cos a4
177
~.
a 1 (cos a2
C)
( cos 11a• cos b' 11 cos c 11)
11 y 11) (cos (aa) 11 cos (b{3) b' cos c a '
0
We consider the case a= a 1 . From the previous discussion on field isomorphisms, we see that the map
a(cos~.
cos~·
cos
E)= (cos ~11 ,
cos (3b11 , cos Yc11 ) extends to a field isomorphism of K, if and only if there is an integer x, such that (x, 2a) congruences
=(x, 2b) =(x, 2c) =1,
and the
mod 2a
mod 2b mod 2c admit a solution. By the Chinese remainder theorem, these congruences have a solution if and only if
'a= (3
is an isomorphism only if it is
Ka.
is an isomorphism, then a, (3, y
cannot all be even. For consider the spherical, Euclidean or hyperbolic · 1e tnang
17 ~ Wit· h ang 1es a (3 17 a, b,
y 17 L et R C.
generated by the reflections in the sides of
A.
(aa• b, fi Cy ) b e t h e group T(~. ~· t) is the sub
group of index 2 in R( ~. ~·
t) , consisting of the orientationpreserving
transformations. Let A,
v be the angle bisectors in ~ , and
11.,
e,
m,
178
LEON GREENBERG
n the reflections in A, 11, v. If a, {3, y are even, then R(\i.
~· ~)
, and the products
bisectors A, 11,
E, m, n
E
Em, mn, nEE T(\i. ~· ~) . Since the angle
v meet at a point, the products Em, mn, nf have the
same fixed points, so they commute. But the corresponding products in T(a, b, c) do not commute, because in this case, the corresponding reflection lines lie outside the triangle, and do not meet at a common point. Thus if a , {3 , y are even, then T(\i. not satisfied in T(a, b, c), and
ea, {3 ' y
~· ~)
has a relation which is
is not an isomorphism.
Since a , b, c are pairwise relatively· prime, at most one of these integers is even. The argument now proceeds by considering various cases where a , b , c , a , {3 , y are even or odd. In each case, a suitable replacement of (a, {3, y) by (a, b{3, cy), (aa, {3, cy) or (aa, b{3, y) transforms the situation to one of the cases: (i) a, {3, y are all odd, or (ii) a, {3, y are all even. In case (i), Theorem 4 shows that ea,{3,y is induced by a field isomorphism. In case (ii), ea,{3,y is not an isomorphism. We illustrate the argument with a typical case: Suppose that a, a, {3 are even, and b, c, y odd. Replacing (a, {3, y) by (aa, {3, cy)' we obtain 3 even integers, so e isomorphism.
{3 y is not an a, '
q.e.d.
It may be true that Theorem 4 is valid more generally, and that all isomorphisms ea,{3,y are induced by field isomorphisms. There is a good deal of evidence for this. An interesting case study is the group T(6, 9, 18), which has three fieldisomorphism classes of nonEuclidean homomorphic images: T (6, 9 , 18)
Tl
~
T2
~
T
T3
~
T
c e
1 18 7) 6'9' 1 18 5) 6'9'
~
~
~
T T T
c c
5) 6' 49' 18 1) 6' 49' 18
c
2 1) 6'9' 18
~
T
~
T
~
T
c
7) ' 6' 29' 18
c
7 18 5) ' 6' 9'
c
5 18 7) . 6' 9'
HOMOMORPHISMS OF TRIANGLE GROUPS INTO PSL(2,
T
(~, ~' 1~)
and T(t•
~' { 8 )
C)
179
are spherical groups, and the others are
hyperbolic. T 1 is not isomorphic to T 2 , because there are 3 reflection lines through the vertices of the
(~· ~77 ,
fs)
triangle, which are concur
rent. I don't know if T 1 is isomorphic to T 3 .
4.
Quadratic differentials We shall now calculate some quadratic differentials F(z)dz 2 , which
are invariant under T(a, b, c) and induce the homomorphisms
ea, (3 ' y.
Let 11 be a hyperbolic triangle (in the unit disc D) with angles
Jf
~' ~,
at its vertices A, B, C. Let 11 be a circular triangle with
angles aarr,
~ 77 ,
ycrr at its vertices A,
B, C.
(We assume the triangles
have the same orientation.) There exists a conformal map f: 11 that f(A) = A , f(B) =
B
and f(C) =
>
11
such
C.
The function f(z) = fa, (3 ,y(z) can be extended by reflection to a meromorphic function in D , such that f(17z) = 8a,{3,y(17)f(z), for 17 < T(a, b, c). We shall compute F(z) = Fa,{3jz) =If, z l.
Let H denote the upper halfplane. We consider conformal maps w: 11
>
H, v: H . 11 and u: H . 11, such that w(A) = 0, w(B) = 1,
w(C) = oo, u(O) =A, u(1) = B, u(oo) = C, and v = w 1 . The map w(z) can be extended by reflection to a meromorphic function, defined in D. It then becomes an automorphic function, which generates the field of
automorphic functions of T(a, b, c). Geometrically, w(z) is a ramified covering of D over the extended plane
C.
It is the projection map from
D to the quotient D/T(a, b, c). Any quadratic differential in D, invariant under T(a, b, c), is the wlift of a quadratic differential on
C,
and
we shall describe them in this way. The conformal map f: 11
>
~ can be expressed: f = u ow= u ov 1 .
The Schwarzian !f,zl = lu,w!
(~;) 2
0=lv,w!(~;) 2 +lw,z!, or
lw,z!=lv,w!(~;) 2 .
+ lw,z!. Since z = v(w(z)), Usingthevariable
w = w(z) in H, so that ~; · ~: = 1, we now have Cayley's identity:
180
LEON GREENBERG
!f, zl
(1)
!u,w! !v,wl
(~:)2
Formulas for the Schwarz triangle functions u(w), v(w) are known, and can be found, for example, in Schwarz [2] or Caratheodory [1]. These formulas are the following
(2)
!v,wl
(3)
!u,w!
1_l _..!.. +1... a2 b2 c2 2w(1w)
When !1 is suitably normalized,
(4)
K
dv dw aw
1!.
a(lw)
1!.
bF(f,m,n;w) 2
where F(f, m, n; w) is the hypergeometric function, (5)
(6)
K
l(n) ['{1 f) ['{1m) a
['{2n) l{nf) l{nm)
'
(7)
a Y(z) Summing up these equations, Fa,f..J,
=
!f,zl satisfies:
HOMOMORPHISMS OF TRIANGLE GROUPS INTO PSL(2,
{1
(8)
F a (z) a,,._,,y
=
[
~2\
~
2a 2w2 2
x!!._w k2
+
C)
1__b~2
1:;2 + + y:;1 + ""__;:__ _....::...__ 2b2(1w) 2 2w(1w) (1{32)
2~
a(1w)
181
J
2~
bF(e,m,n;w) 4 .
The quadratic differential Fa,{3,y(z)dz 2 is the wlift of Ga,{3,/w)dw 2 , where
(9)
The homomorphism ea,{3,y: T(a, b, c). T
(~· ~· t)
is induced by the
quadratic differential w a,,._,,y a = Fa,,._,,y a (z)dz 2 . Summing up our results, we have the following. THEOREM
6. The homomorphism 8 a,,._,,y a is induced by the quadratic
differentials w a,,._,,y a , w a, ba a c y and wa_a, ba ,._,, c y , wa a,,._,, ,._,,y . These are ail obtained from the Schwarzians of conformal maps. DEPARTMENT OF MATHEMATICS UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND 20742
REFERENCES [1] C. Caratheodory, Theory of Functions of a Complex Variable, vol. 2, Chelsea Publishing Company (1960). [2] H. A. Schwarz, Ueber diejenigem Falle in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, Jour. Reine und Angew. Math. 75, 292335. [3] E. G. Van Vleck, On the Combination of Nonloxodromic Substitutions, Trans. A.M.A. 20(1919), 299312.
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS M. Gromov 0. Introduction This lecture gives an outline of basic geometric notions and ideas associated with the conception of hyperbolicity. Very little is said here about the hyperbolic space itself (the main source of knowledge is Thurston's lectures [21]), but it is shown how the phenomenon of hyperbolicity appears in Riemannian geometry, topological dynamics, combinatorial group theory and geometrical theory of mappings. Our presentation is expository, proofs are only sketched, but constructions and definitions are illustrated by examples. The comprehensive theory of hyperbolicity is yet to be built and my purpose here is to provide motivation for future research. (Questions and conjectures are italicized.)
1. Length Spaces 1.1. Riemannian spaces with singularities Our main example is the following. Take a piecewise smooth polyhedron X C RN. The space X carries the metric d induced from RN. We get another metric if we use the length function on curves in X and define R(x, y), x, y d and the equality holds iff X is a convex set.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00018331 $01.55/1 (cloth) 0691082677/80/00018331 $01. 55/1 (paperback) For copying information, see copyright page
183
184
M. GROMOV
DIGRESSION. The quantity sup (R(x,y) · d 1 (x,y)) measures the distortion of X. One can easily show that the distortion of any topological circle is at least rr/2 and consequently, Distortion (X) < rr/2 implies that X is simply connected. Probably this inequality implies that X is contractible, but I can show it only under the stronger assumption Distortion (X) < rr/2v2. * A different approach to the distortion can be found in [10]. DEFINITION. A length structure in a space X is given by a metric and a length function on curves, such that the distance between any two points is equal to the length of the shortest curve (supposed to exist) joining these points. We shall also assume all metric spaces to be complete unless stated otherwise. FURTHER EXAMPLES. Let V be a Riemannian manifold with boundary. Its length structure is more complicated than that of a manifold due to the fact that the shortest curves between interior points can touch the boundary. Take a Riemannian manifold and divide it by a group of isometries. When the action is not free then the natural length structure again has new features. DEFINITIONS. A map from one length space (space with length structure) to another is called isometric if it preserves the distance function. Such maps are always embeddings. A map preserving the lengths of the curves is called pathisometric. Isometric maps from [a, b], R+, R into X are called straight segment, ray and straight line correspondingly. A locally isometric map R . X is called a geodesic. There is a natural Raction in the set of all geodesics called the geodesic flow.
1.2. Horofunctions and the ideal boundary Let X be a complete metric space. The distance function determines an isometrical embedding x . dist (x, y) from X into the space C(X) of
*Unpublished.
185
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
the continuous functions on X . Consider the factor space C' = C(X)/(constant functions) with the topology of uniform convergence on bounded sets in X. The space X is now embedded into C' and we can define the closure Cf(X) and the boundary
ax =
Cf(X) \ X . The
space X is assumed further to be countably compact. In this case, Cf(X) and a(X) are compact. A function h ( C(X) that projects into a point b ( horofunction centered at b.
ax C C'
is called a
The sets h 1 ( oo, c) C X are called (open)
horoballs centered at c; the levels h 1 (c) C X are called horospheres and the sets h 1 (c, oo) are called horospaces. The limit set
ax 0
(or the set of the limit points) of a closed X 0 C X
is defined as the intersection of its closure in Cf(X 0 ) with
ax.
When X is a length space, every ray has exactly one limit point. (We identify rays and straight lines with their images.) The corresponding horofunction is called the ray function or the Busemann function.
1.3. lsometries With an isometry y: X .... X , one associates the displacement function oy: X .... R+, oy 0, is unbounded. Every such isometry has, obviously,
a fixed point at
ax .
(All isometries can be continuously extended to
CfX.)
1.4. The word metric Consider a group r
generated by {31' ... ' f3k and denote by
II Yll
the
186
M. GROMOV
length of the minimal word (in {3i and
lly\1
=
IIY 1 11
f3i 1 )
IIY1 Y2 II:S IIY 1 11 + l\y2 11. The invariant metric IIY1 1 Y2 11 on f'.
to the left
and
representing y. Obviously length function gives rise
Quasiisometry. A map f from a metric space X to Y is called quasiisometric if the ratio dist(x 1 ,x 2 )/dist(f(x 1 ),f(x 2 )) x 1 ,x 2 pinched between C and
c
f
X, is
1 . Sometimes one specifies C and calls
such an f a Cquasiisometric map. Obviously, two different word metrics, corresponding to different choices of generators, are quasiisometric. Coarse quasiisometries. A coarse quasiisometric map X .... Y is a map f 0 from a subset X 0
C
X to Y such that X0 is edense * in X
(i.e., its eneighborhood coincides with X) and f 0 is Cquasiisometric. We call f 0 a coarse equivalence if its image is edense * in Y . This is an equivalence relation if the constants C and e are allowed to vary. The following obvious fact plays an important role in connecting geometry of a space with its fundamental group. Let X be a compact length space and X be its universal covering. Then X (with the induced length structure) is coarse equivalent to the fundamental group rr 1(X). (We suppose here X to be topologically a polyhedron or a more general space admitting covering space theory.)
1.5. Convex sets A set A in a length space is called convex if its intersection with every geodesic segment is connected. A set is called locally convex if for any a
f
A there is an e such that the intersection of A with any
geodesic segment of the length e passing through a is connected. 2. Convex Spaces
2.1. Definition A length space X is called convex if the distance function is convex; namely, for any two geodesic segments x: [a, b] .... X, y: [c, d] .... X dist (xH 1 ), y(t 2 )) is convex on [a, b] x [c, d].
*In
this paper E is positive but not always small.
187
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
X is called locally convex if it can be covered by open Ui such that
dist(x, y) is Convex when
X,
y belong to Ui.
EXAMPLES. A Riemannian manifold is locally convex iff K sectional curvature
~
0 ). It is convex iff K
~
0 and
rr 1 =
~
0 (i.e.,
0. In the
context of length spaces one has by adapting the classical arguments, THE CARTAN HADAMARD THEOREM. A simply connected locally con
vex space is convex. Every convex space is straight (any two points can be joined by the unique geodesic segment) and hence contractible. QUESTION. Are there convex length spaces which are topological mani
folds different from Rn? 2.2. Manifolds with boundary Let V be a Riemannian manifold with smooth boundary B. The boundary is called convex when the second quadratic form is nonnegative; it is called concave when this form is nonpositive. The boundary is called kconvex if the second quadratic form looks infinitesimally as
I
n1
aixi2 , n1
=
dim B ,
1
where among ai there are at least k nonnegative numbers. When k
=
n2, we call B nexttoconvex. ( k
=
n1 corresponds to convexity.)
REMARK. Riemannian manifolds with K < 0 and with nexttoconvex
boundary B have locally convex length structure. For example, surfaces with boundary are locally convex when k
~
0.
More generally, take a 2plane r tangent to B and denote by K'r the sectional curvature of the induced metric in B. The following property is necessary and sufficient for local convexity of V (viewed as a length space): K
< 0 and for any r where the second quadratic form is negative
is nonpositive. The proofs of all these points are straightforward.
K~
188
M. GROMOV
2.3. Digression: kconvex hypersurfaces in Rn Let V C Rn be a compact domain with smooth boundary B. The classical fact stating convexity of a locally convex connected set can be formulated in the following fancy fashion: B is (n1)convex iff for any straight line ~ < Rn the homomorphism H 0{e n V) > H 0 (V) is injective. By using rudimentary Morse theory one generalizes this theorem: B is kconvex iff for any (nk)dimensional plane P C Rn the homomorphism Hnk 1(P n V) > Hnk 1 (V) is injective. In this case V has the homotopy type of an (nk)dimensional polyhedron. The most regular behavior is shown by V's with the nexttoconvex boundary: Let V have unit volume. Denote by A the (n1)dimensional volume of its boundary and by b 1 the first Betti number of V. Then V
contains a ball of radius e > F(A, b 1 , n), where for F one can take exp ( exp (A+b 1 +n).
Sketch of the proof. We actually find a cube C of the size e contained in V : moving an ecube by parallel translation, we first inscribe its 1skeleton cCl) into V such that the homomorphism H 1(cCl))> H 1(V) is trivial. Because V is nexttoconvex, the inclusion c 2 in£ oy(x) ; XtX
X£X
Y
this implies the theorem. COROLLARY.
If y is fixed point free and o~ = 0, then y Is
parabolic.
2.7. Straight subgroups and asymptotic torsion Let [' be any group and 1°
c ['
be a finitely generated subgroup.
We call 1° straight in [' if for any finitely generated ['' c 1°' the inclusion 1° c__.
r'
to the word metrics in [' 0 and
['
containing
is a quasiisometric imbedding with respect
r'.
We call a y < [' an asymptotic torsion element if for some finitely generated
r'
containing y' we have fim n> oo
it llYn II
=
0. This happens
exactly when the group generated by y is finite, or when it is infinite cyclic but not straight.
An example. Let f' be a nilpotent group without torsion. Then [' has asymptotic torsion unless it is Abelian. Combining this with the theorem from the previous section, we conclude. THEOREM.
Let f' be a group of isometries of a convex space. If ['
has no elliptic or parabolic elements, then every nilpotent subgroup in f' is Abelian.
191
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
An analogous argument shows:
If there is an
E
> 0 such that 0 y (x) >  E.
X f
X' y
r*
f
(= I'\ identity)
then any Abelian subgroup in
r
Abelian group with oyCx)::::
must be finitely generated and torsion free.
E
is straight. Observe also that an
As a corollary, we get a generalization of the GromollWolfLawson
y au theorem (see [5]):
r
Let oyCx) ::::
E
be a soluble group of isometries of a convex space X. If
> 0'
y (
r*'
X (
X' then
r
contains an Abelian subgroup of
finite index. Proof. If every Abelian subgroup in a soluble group
r
is finitely gener
ated and straight, then I' obviously contains the required Abelian subgroup. Observe that for the branched torus the subgroup
17 1 (F)
C
17 1 (V)
is
usually not straight.
2.8. The uniformization problems Let X be a finite dimensional polyhedron of K(77; 1) type. Does there exist a locally convex space (noncompact but preferably finite
dimensional) homotopy equivalent to X? The Thurston theory [21] says "yes" for a vast class of 3manifold containing all the known examples. Let X be any finite dimensional polyhedron. Does there exist a con
vex space X and a discrete group (with fixed points)
X
such that
X/I'
r
is homeomorphic to X? When dim X
of isometries of =
2 , we have
an easy "yes''. 3. Strict Convexity
3.1. Definition We shall assume below that our length space X satisfies the following property: for any x < X there is an e such that every sphere centered at x of radius
~
e contains no geodesic segments. (This is always so
for Riemannian manifolds.) When X is convex, the above property implies that large spheres contain no geodesic segments as well.
192
M. GROMOV
oconvexity. We call a convex X strictly oconvex if there is a
positive v such that for any convex set A C X the normal projection P:X
>
A ( P sends x to the nearest point from A ; this is a well
defined map due to the absence of geodesic segments in spheres) satisfies: for any curve C C X with dist (C, A)~
o,
we have length (P(C)) ::;
(1v) length (C).
Observe that (O+O 1 )convexity follows from 8convexity. X is called strictly convex if it is oconvex for a positive o.
EXAMPLE. A simplyconnected Riemannian manifold (possibly with nexttoconvex boundary) with K ::; actually 8convex for all
K,
K
> 0, is strictly convex (it is
o > 0 ).
We call X locally oconvex (strictly convex) if its universal covering is oconvex (strictly convex). This is really a "local" notion, because oconvexity follows from 8convexity of the balls of radius 0. The double construction from section 2.4 gives a strictly convex
space when the underlying manifold has K ::;
K ,
example, take a manifold X with
K 2
K 1
::; K::;
K
> 0 . As a pleasant
< 0 and with finite total
volume. In this case, one can find a locally convex A C X such that the complement X\ A is bounded and homeomorphic to X itself. The double space Y in this case has the same homotopy type as the usual double of X'= X\ Int A. When X has constant curvature, the double of X' can be
itself equipped with a metric of nonpositive curvature. This metric can be chosen real analytic and with negative Ricci curvature (The last possibility was pointed out to me by Ernst Heintze.) Observe that there are compact manifolds that have C 00 metrics with K::; 0 but have no real analytic metrics with K < 0 . The proofs of these facts are easy (see also
[11]). 3.2. Closure Let X be strictly convex. In this case every horofunction is a ray function. Moreover, there is a natural onetoone correspondence between
193
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
the points from
ax
and the rays starting from a fixed point Xo ( X .
Intersection between any two horoballs with different centers is compact, and thus we have the Vproperty: any two distinct points from
ax
can
be joined by a straight line. For any straight line f C X, the normal projection X
>
f can be
continuously extended to C f{X) and the only fixed points of the extension are the "ends" f + ' f
c ax
of this line.
All these facts are classical for Riemannian manifolds and the classical proofs work in our case with no problems. When X is the hyperbolic space (K = 1), there is another important property: Let A C ax be a closed set and C be its convex hull. Then the limit set of C is equal to A . This feature is probably shared by all strictly convex spaces, but I could prove it only when X is a Riemannian manifold with K '5 1 and A
c ax
is a finite set. (Recall that the convex hull of A is defined as
the minimal set C C C fX containing A such that An X C X is a convex set.)
3.3. Isometries Let X be strictly convex and y: X .... X be an isometry. There are only three possibilities: a) y is elliptic. In this case it has a fixed point and the topological group generated by y (in the group Is(X) of all isometries) is compact. b) y is parabolic. Then y has no fixed points in X but has a
unique fixed point
X
in
ax .
This point is called the center of y.
All horospheres centered at x are invariant under y. The group generated by y is isomorphic to Z and its action in C f(X) \{xI is discrete. c) y is hyperbolic. In this case y has two fixed points y+ and y in
ax '
it generates
z
that acts freely and discretely in
194
M. GROMOV
C f(X)\ !y+, y1. The point y+ is an attractor for y; moreover, for every compact set A, A C Cf(X)\Iy1 and any neighborhood U C C f(X) of y + there is an n such that yn(A) C U. The point y is an attractor for y 1 . This classification is well known for Riemannian manifolds, and the proof depends only on the properties stated in the previous section.
3.4. Special groups of isometries X is again supposed to be strictly convex. Consider a group ing by isometries on X.
r
r
act
is called special when one of the following
three cases occur: 1)
r
keeps fixed a point
X (
X. In this case the closure of
r
in
the group Is(X) of all isometries is compact. This is a Lie group when X is a Riemannian manifold (possibly with boundary). When X is a polyhedron, this closure is usually a profinite group; this is always so when X is onedimensional. 2) There is a straight line f C X invariant under X . In this case
r
r' . . r . . re . . 0' where r' c r is the subgroup keeping f fixed and r e is a subgroup in Is( f) . 3) r keeps fixed a point ax . The group r can be very compli
factors as follows: 0 ....
X (
cated even when X is a Riemannian manifold, but when the curvature is pinched, i.e.,
Kl ~
K(X) ~
K2
< 0' the closure of
r
is isomorphic to
an extension of a soluble group by a compact Lie group. This follows from the Margulis lemma (see [11]).
3.5. Nonspecial groups
r
Let
be a discrete nonspecial group of isometries of a strictly
convex space X. THEOREM.
a) L
There is a closed set L C C f X with the following properties:
is an infinite set without isolated points;
b) the action of
r
closed subsets;
in L is minimal, i.e., there are no invariant
195
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
c) the action of
r
in C f(X) \L is discrete.
Denote by L ( 2 ) the set of distinct pairs f 1 , f 2 f L. Such a pair is called axial if there is a hyperbolic isometry from r keeping this pair fixed.
r
d) Axial pairs are dense in L( 2 ), unless ail isometries from
are
elliptic. Denote by L( 3 ) the space of pairwise distinct triples from L.
r
e) The action of
in L ( 3 ) is discrete. If the action of
r
is
uniform in X (i.e., the translates of a compact set cover X) then L
=ax
and
r
acts uniformly in L( 3 ).
r
All these properties are well known when
is a Kleinian group.
(Many of them I have learned from Dennis Sullivan.) Classical proofs use only the properties stated in sections 3.3 and 3.4. (See [6] for more information.) There is another important property of
r
that can be established by
Klein's argument: Let y 1 , · · ·, Ym C
r
be hyperbolic elements where no two of them are
powers of a third element from k.
r.
Then there is a number k such that
powers yi 1 for ki ;:: k, i = 1, · · ·, m, generate a free group of rank m . To complete the list of basic properties of
r,
we must mention
another one, obvious but quite important: Every hyperbolic y (
r
can be uniquely written as y~ where y 0 is
not a proper power.
3.6. Remarks on not strictly convex manifolds There are some conditions weaker than strict convexity that lead to the conclusions analogous to the theorem from 3.5. Consider first the case when X is a Riemannian manifold of nonpositive curvature. We suppose that X is simply connected and the group of isometries Is(X) acts on X uniformly, for example, X is the universal covering of a compact manifold. Then X shares all global
196
M. GROMOV
properties of a strictly convex space (in particular Theorem 3.5) iff it contains no flats. (A flat is a totally geodesic submanifold isometric to
Rk, k ~ 2 .) As an example, one can take a not flat surface with K
~
0 . When
this surface has a closed geodesic such that in a neighborhood of this geodesic curvature vanishes, then the universal covering has no strict convexity (because some horospheres contain geodesic segments). The "noflats" condition is also satisfied by the universal coverings of branched tori from 2.5 when V0 intersects all 2dimensional flat subtori in V. Though V is not a Riemannian manifold, its metric can be smoothed to a Riemannian metric with K < 0 and with no flats in the universal covering. It is unclear whether in the "noflats" case V carries a Riemannian metric with K < 0. QUESTION.
Let V be a compact Riemannian manifold with K < 0 and
its universal covering contains a flat. Does it follow that
rr 1 (V)
con
tains Z + Z? When the universal covering of V has no flats, the periodic geodesics are dense in the unit tangent bundle (because Theorem 3.5, in particular, d) is applicable to the universal covering of V ), but if there is a flat, it is unclear whether V has more than one simple closed geodesic. The positive answer to the question would provide infinitely many such geodesics. Of course, there are many compact manifolds with K < 0 where the density of closed geodesic in the unit tangent bundle is established. This is the case when V is compact and locally symmetric (see [17]). Another (very easy) example is provided by compact connected manifolds having a tangent vector such that the sectional curvature is negative on all tangent 2planes containing this vector. Observe in the end that the whole discussion can be extended to a certain general class of locally convex spaces.
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
197
4. Isoperimetric Inequalities
4.1. Openness at infinity Consider a domain Q in an ndimensional Riemannian manifold V. We denote by Vol(Q) its volume and by Vol(aQ) the (n1)dimensional volume of the boundary. V is called open at infinity (see (19]) if there is a constant C such that any Q
cV
satisfies Vol(Q) 'S C Vol(aQ). When V is orientable,
openness at infinity is equivalent (see (19]) to any one of the following three conditions: a) The volume form in V is the differential of a bounded form w (i.e., llwllv.C:const, VfV). b) Any bounded nform on V is the differential of a bounded form. c) There is a bounded vector field on V such that its divergence is greater than a fixed positive number. EXAMPLES. A closed manifold is not open at infinity. Rn is not open at infinity. Strictly convex Riemannian manifolds (possibly with boundary) are open at infinity. THEOREM (Avez (3]). Let V be a simply connected manifold (without
boundary) of nonpositive curvature such that the group Is(V) acts uniformly in V. If V is not open at infinity, then it is isometric to the Euclidean space.
4.2. Amenable groups A group
r
is called amenable if any action of
r
on a compact
space has an invariant measure. Basic properties of such groups can be found in (9]. Here are a few simple facts: A finite extension of a soluble group is amenable. If a group then
r
r
contains a free subgroup of rank :::: 2,
is nonamenable. Every finitely generated nonamenable group
has exponential growth: the number N of words with N ~~ (const)R.
II I 'S
R satisfies
198
M. GROMOV
The following simple fact gives a geometric interpretation of amenability:
Let V be a compact Riemannian manifold with fundamental group 1.

This group is nonamenable iff the universal covering V is open at
infinity. We shall use this fact in a slightly more general situation: take a submanifold V0 C V. If rr 1 (V \ V0 ) is nonamenable, then the universal ~
covering V \V0 is open at infinity.
4.3. Quasiconformal maps Let V be a complete Riemannian manifold of dimension n. Take a point v 0
f
V and denote by s(r) the (n1)dimensional volume of the
radius r sphere centered at v 0
.
J
00
AHLFORS' LEMMA.
If the integral
1
(s(r)) n 1 dr diverges, then V
1
cannot be made open at infinity by any conformal change of metric. Proof. Take an arbitrary function f > 0 on V, multiply the metric by f, and show that there is a ball 0 (with respect to the old metric) such that Volnew(O)!Volnew(aO) becomes arbitrarily large. When 0 =Or is a ball centered at v 0 of radius r, we have
HYPERBOLIC MANIFOLDS, GROUPS AND ACTIONS
199
A straightforward calculation shows now that the divergence of
J oo
(s(r))
n~1
n makes the ratio
Approachable
1
Let V, W be two orientable Riemannian manifolds of dimension n, where V is complete and W is open at infinity. Let h : V
>
W be a map
with the following properties generalizing the notion of quasiconformality: a) The map h factors as
where V 1 is a Riemannian manifold, the map g is a conformal equivalence and h' is a pathquasiisometric map, i.e., for any curve const < b) Consider the balls Ur
ec v1 '
we have
e
length < consC 1 . length h'(f)
cV
as above and denote by w* the pull
back h*(w) of the volume form in W. Our second condition is the following:
as a subgroup of the Teichmtiller modular group (Earle (3], Harvey (9], EarleKra (4]). We have to study the local behavior of T and T' near the fixed points. Let 11 = exp (2rri/6). The rotation constant at the fixed point of T is either 11 or 11 5 (primitive sixth roots of 1 ), the rotation constant at the fixed points of T 2 is either 11 2 or 11 4 (primitive cubic roots of 1 ), and the rotation constant at the fixed points of T 3 is 11 3 (primitive square root of 1 ). Topological restrictions (Harvey (9] Lemma 6, Guerrero (7] Lemma 1.2) reduce the possibilities to 11 , 11 2 , 11 3 and 11 5 , 11 4 , 11 3 for the rotations at the fixed points of T , T 2 and T 3 respectively. Note that if the rotation constants for T, T 2
,
T 3 are 11, 11 2 , 11 3 , then the rotation constants
for V=T 5 , V 2 , V 3 are 11 5 , 11'4 , 11 3 andfurther =. Of course the same analysis is valid for T'. It follows that < T~ > = < T'~ > with T and T"' having the same rotation constants at corresponding fixed points (T"'= T' or (T') 5 ). We conclude that and are conjugate (see Harvey (9]). Our family
0
>
3" has a holomorphic section s : 3"
fixed point of TT: MT .... MT. Define strass point!. Locally we can write
rn ltJ
>
0,
s(r) = the
= {u3"[s(r) is an ordinary Weier= {u3"[w(r,s(r))=O!. Here,
w(r, z) is a holomorphic function determined by the representation in local coordinates of the Wronskian of a basis for the space of holomorphic differentials on Mr (see Bers [2]). It is clear now that there are three possibilities: (1) of codimens ion 1 .
rn
= Ql' (2)
rn
= j"' (3)
rn
is an analytic subset of j"
220
IGNACIO GUERRERO
Let p be the fixed point of T: M . M and p' the projection of p on M/< T >. If p' is an ordinary Weierstrass point, there is a holomorphic differential w' with ord p ,w' >  g'. The lift w of w' satisfies ordpw 2: 6g' + 5 > g, thus p is an ordinary Weierstrass point. If we start with a Riemann surface M' of genus g' and a Weierstrass point p' on it, we can construct a Riemann surface M with an automorphism T such that M/< T > ""' M' and the covering M . M' has the same branch structure as the members of our family
G.
j". Further, we require the
fixed point of T to be over p'. To construct M, first we uniformize M' by a Fuchsian group 1 of signature (6, 3, 2; g') (the points over which branching occurs can be arbitrarily prescribed) and then apply Harvey's theorem (Harvey [8], Theorem 4). The fixed point of T must be an ordinary Weierstrass point. This shows that ill We have been unable to show that l~
f 0.
I j"
(we believe that this is the
case) i.e. to establish the existence of automorphisms fixing a single point which is not an ordinary Weierstrass point.
3.
Automorphisms with two fixed points
3.1. Let T: M. M be an automorphism fixing two points p 1 and p 2 . Assume that T has prime order N and that M/< T > has genus 1 . The RiemannHurwitz relation shows immediately that M has genus g = N. Denote by w the lift to M of a nonzero holomorphic differential on M/< T > (unique up to a constant). The divisor of w is (w)
(N1)p 1 + (N1)p 2 (g1)pl + (g1)p2.
Suppose p 1 is a qfold Weierstrass point. By definition, there is a holomorphic qdifferential r whose divisor is of the form (r)
o
=
[(2q1) (g1) +a] p 1 + {3p 2 +
o.
is a divisor not containing p 1 or p 2 . Of course, a + {3 + dego = g1. We can choose r to be an eigenvector for Tq, Tqr = Ar. Here
221
AUTOMORPHISMS OF COMPACT SURFACES
This implies that the divisor
o
is invariant under T and therefore deg
is a multiple of N = g. Since deg
o< g
we must have
o= 0 .
o
Hence
(r) = [(2q1) (g1) +a] p 1 + {3p 2 , a + {3 = g1 Now, f = r/wq is a meromorphic function on M and (f)
[(q1)(g1)+a]p 1 + [{3_:_q(g1)]p 2 [(q1)(g1)+a](p 1 p 2 ).
The function fN is invariant under T , so it projects to a function h on M/
where ai is the image of Pi under the projection M .... M/< T >. Using Abel's theorem (identify M/< T > with its Jacobian) we conclude that a 1 a 2 is a rational point. Conversely, assume that a 1 a 2 is a nonzero rational point in M/. Then, there exists a meromorphic function h with (h)=k(a 1 a 2 ), k:: 2. Let f be the lift of h to M, (f)= kN(p 1  p 2 ). Consider = f(gl)WkN
T
(r) = 2kN(g1)p 1 . therefore, p 1 is a kNfold Weierstrass point. The above arguments are of course valid for the fixed point p 2 . Thus we have shown that p 1 and p 2 are qfold Weierstrass points for some q if and only if a 1  a 2 is a nonzero rational point on the torus M/< T >. Now it is clear how to construct examples of automorphisms fixing two points which are not qfold Weierstrass points for any q :: 1. Simply construct suitable coverings of a torus with branch points over a 1 and a 2 with a 1a 2 not rational. 3.2. Let M be the hyperelliptic Riemann surface defined by the equation
222
IGNACIO GUERRERO
Choose n = 2g' +1, then M has genus g = 2g'. Define T: M . M by z . z , w
>
w . T is an involution fixing the two points over 0 and
M/ is a Riemann surface of genus g'. Note that the two points over are not fixed by T, actually the RiemannHurwitz relation shows that
oo
a Riemann surface of even genus cannot have an involution with four fixed points. We will show that it is possible to choose a 1 , ···,an so that the fixed points are not Weierstrass points of any order. It is not hard to see that the space of holomorphic qdifferentials on M has a basis lzjwqdzq, zkwq+ldzq I, j k
=
0, · · ·, (q1)(g1) 2 (except for q
=
=
0, · · ·, q(g1) and
1 and q
=
2, g
=
2. Then the
differentials zjwqdzq are sufficient). The points over z
=
0 are not ordinary Weierstrass points (for any
choice of a 1 , ···,an), so assume q 2 2. By definition a point over z =0 is a qfold Weierstrass point if and only if there is a relation
Here P and Q are polynomials of degree s
=
q(g1) and t
respectively, S(z) is a power series convergent near z d
=
=
=
(q1)(g1)2
0 and
(2g1)(g1). Note that we have two expressions for w(z), corre
sponding to the two points over z
=
0. From now on assume that one of
these has been chosen and write
We have P(z) + Q(z)w(z)
=
zdS(z)wq(z).
Substituting the expression for w(z) and comparing coefficients one sees that the existence of P and Q (not both zero) is equivalent to the vanishing of
w
AUTOMORPHISMS OF COMPACT SURFACES
223
Now w(z)=(z 2 ns 1z 2 n 2 +· .. +sn_ 1 z 2 sn)'h where s 1 , .. ·,sn are the elementary symmetric functions in a
f, ···,a~ .
The coefficients of the
power series for w(z) are of the form
for a polynomial F . We conclude that the vanishing of W is equivalent to the vanishing of a polynomial in s 1 , .. ·, sn or, equivalently, to the vanishing of a polynomial in a 1 , .. ·,an . A careful analysis of the expansion of w(z) shows that W is not identically zero as a function of a 1 ,
... ,an.
In fact, the leading coeffi
cient in sn_ 1 will be a determinant involving binomial coefficients. This determinant can be shown to be nonzero. It will be crucial the fact that g is even. We have shown that the points over z = 0 will be qfold Weierstrass points when a pair of polynomials P~(a 1 , .. ·,an) vanish. Choosing a 1 , ···,an off the zero set of the countable collection of polynomials
!P~\q:::: 21 we obtain the example sought. DEPARTMENT OF MATHEMATICS UNIVERSITY OF GEORGIA ATHENS, GEORGIA 30602
REFERENCES [1] [2] [3] [4] [5] (6] [7]
Accola, R. D. M. On generalized Weierstrass points on Riemann surfaces (to appear). Bers, L. Holomorphic differentials as functions of moduli, Bull. Amer. Math. Soc., 67(1961), 206210. Earle, C. J. Moduli of surfaces with symmetries, Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies 66(1971), 119130. Earle, C. J. and Kra, I. On sections of some holomorphic families of closed Riemann surfaces, Acta Math., 137(1976), 4979. Eichler, M. Introduction to the theory of algebraic numbers and functions, Academic Press, New YorkLondon, 1966. Farkas, H. and Kra, I. (Forthcoming book on Riemann surfaces.) Guerrero, I. On Eichler trace formulas (to appear).
224 [8]
[9]
IGNACIO GUERRERO
Harvey, W. J. On cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford (2), 17(1966), 8697. . On branch loci in Teichmi.iller space, Trans. Amer. Math. Soc., 153(1971), 387399.
[10] Kuribayashi, A. On analytic families of compact Riemann surfaces with nontrivial automorphisms, Nagoya Math. J., 28(1966), 119165. [11] Lewittes, J. Automorphisms of compact Riemann surfaces, Amer. J. of Math., 85(1963), 732752. [12] Olsen, B. On higher order Weierstrass points, Ann. of Math., 95(1972), 357364.
AFFINE AND PROJECTIVE STRUCTURES ON RIEMANN SURF ACES R. C. Gunning 1. Let M be a compact Riemann surface of genus g > 0, represented as

the quotient of its universal covering space M by the group of covering translations ['. A projective structure on M is described by a complex

analytic local homeomorphism f : M > P 1 with the property that for any
T
f
f', f(Tz)
(1)
for some pT
f
=
pT(f(z ))
PL(1, C); here P 1 is the onedimensional complex projec
tive space (the Riemann sphere), and PL(1, C) is the group of projective transformations (linear fractional transformations) acting on P1
.
The
mapping f can be viewed as describing a special complex analytic coordinate covering of the Riemann surface M. The coordinate transformations of this coordinate covering are not merely complex analytic mappings but actually projective mappings; so this coordinate covering determines a projective structure on M , in a manner analogous to the determination of a complex structure by a coordinate covering with complex analytic coordinate transformations. If f is any complex analytic local homeomorphism satisfying (1) and a is any element of PL(1, C) then the composition f' = a of is also a complex analytic local homeomorphism, and satisfies a condition of the form (1) with pT replaced by p 'T = a opT o a~ 1 ; th e
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00022 52 0$01.00/1 (cloth) 0691082677/80/00022520$01.00/1 (paperback) For copying information, see copyright page
225
226
R. C. GUNNING
mappings f and f' are considered as describing equivalent projective structures. It is clear from (1) that the mapping T phism from the group
r
>
pT
is a homomor
into the group PL(1, C); this homomorphism p
is called the representation of the projective structure described by the mapping f. Note that the representations of equivalent projective structures are also equivalent, in the sense of being conjugate representations. The study of projective structures on Riemann surfaces is of course closely related to the study of Fuchsian and Kleinian groups, a currently very active field of research; but it involves a slightly different point of view, and suggests a different set of problems for investigation. The aim of this paper is to survey some of these problems and to describe some of the results known about them. First, however, to indicate what a complete theory of projective structures on Riemann surfaces might look like, the corresponding theory for the much simpler case of affine structures will be sketched; things are so simple in that case that very explicit results are easily obtained. Some results about the natural extension to branched structures will also be included. 2. An affine structure is described by a complex analytic local homeomorphism f: M .... C with the property that for any T form (1) holds for some pT
f
f
r
an equation of the
A(1, C); here A(1, C) is the group of affine
transformations on C, so that pT(z) = aTz + bT for some complex constants aT, bT with aT~ 0. The notions of equivalent affine structures and of their representations are the obvious ones. If M admits an affine structure then it has a coordinate covering for which the Jacobians of the coordinate transformations are constants, hence for which the canonical bundle is flat, using the terminology of [4 ]; and that implies that g
=
1 , which both simplifies and limits the theory
of affine structures. To be more explicit, consider therefore a marked Riemann surface M of genus g
=
1 ; the fundamental group
r
of M is
a free abelian group of rank 2, and a marking is merely a choice of two free generators A, B for the group
r.
The universal covering space M
227
AFFINE AND PROJECTIVE STRUCTURES
can be identified with the complex plane C in such a manner that A(z) z + 1, B(z)
=
=
z +w, for some complex number w having positive imagi
nary part; the Teichmiiller space of all such marked Riemann surfaces can be identified with the upper half plane H , with the point w
f
H repre
senting the given marked Riemann surface. An affine structure on M is described by a complex analytic local homeomorphism f : C
>
C such that
(2) The derivative f' is nowhere zero; and it is apparent from (2) that the quotient function f"/f' is a holomorphic rinvariant function on is hence a constant 2rric . If c
=
c'
and
0 then f is itself an affine transforma
tion; and of course any affine transformation f clearly describes an
f 0 then f(z) = ae 277icz + b for some complex constants a, b with a f 0; and it is clear that any such mapping
affine structure on M. If c
also describes an affine structure on M. By passing to an equivalent affine structure, by replacing f by the composition a of for some affine transformation a, it is clear that when c
=
0 the function f can be re
duced to the form f(z)
=
z, while when c
f 0 the function f can be re
duced to the form f(z)
=
e 277icz; and no further equivalences are possible.
Thus in this manner the set of equivalence classes of affine structures on the marked Riemann surface M can be put into canonical onetoone correspondence with the complex plane C. Using this explicit form, note that the mapping f: C
>
C describing
an affine structure on M is always a covering mapping; indeed, in the normal form chosen above, f is the identity mapping for the affine structure parametrized by c image f(C) any c
=
C*
=
=
lz(C:
0 , while f is the universal covering of the
zfOl for the affine structures parametrized by
f 0. The image p(r) of the representation of the affine structure
parametrized by c is a group of affine transformations acting on the image f(C); indeed p(r)
=
r
for c
=
o,
affine transformations on C* generated by
while p(r) is the group of
228
R. C. GUNNING
(3)
for c
f
0. It is easy to verify that the group p(r) is a properly discon
tinuous group of affine transformations on f(C) if and only if c = r/(p+qw) for some integers p, q, r ; hence for general parameter values c the group p(1) is not discontinuous. If p(1) is discontinuous the quotient space f(C)/ p([') is also a compact Riemann surface of genus 1 , and f induces a covering mapping f* : M > f(C)/ p([') . The order of this covering mapping f* is easily seen to be equal to the number of distinct pairs ({vp/rl, lvq/rl) as v ranges over all the integers, where c
=
r/(p+qw)
and lx I = x [x] is the fractional part of x; thus if p, q, r are coprime the order of this covering is equal to lrl whenever r
f
0.
To describe some further properties of the representations p
w, c of the affine structures associated to the parameter values c £ C, the =
p
representations given explicitly by (3), note that the set Hom (1, A(1, C)) of all homomorphisms from 1 into the affine group A(1, C) can be given a natural complex structure. Indeed viewing 1 as the free abelian group generated by two elements A, B, any p £Hom (1, A(1, C)) is completely described by the affine transformations pA(z) a 8 z + b8 PsPA.
;
=
aAz + bA and p 8 (z)
=
and these can be any affine transformations satisfying p APB =
Thus the set Hom (1, A(1, C)) can be identified with the three
dimensional complex analytic subvariety
This subvariety has only one singular point, the point (1, 1, 0, 0) corresponding to the identity representation; the remaining points of V form a connected threedimensional complex manifold. Actually of course the principal interest is not so much in the set of all affine structures as in the set of all equivalence classes of affine structures; and hence it is more interesting to consider in place of Hom (r, A(1, C)) the set of all equivalence classes of affine representations of
r,
thequotientspace Hom(r,A(1,C))/A(1,C) where A(1,C)
AFFINE AND PROJECTIVE STRUCTURES
229
acts by conjugation on the representations in Hom (1, A(1, C)). If a(z) = az + b < A(1, C) and if p
f
Hom (1, A(1, C)) is described by the coordi
nates (a A, a 8 , b A' b 8 ) then the conjugate representation p' = ao p o a 1 is described by the coordinates
This exhibits A(1, C) as a group of analytic automorphisms of the complex analytic subvariety V, and leads to the question whether the quotient space W = VI A(1, C) can also be given a complex structure. Note first that the singular point (1, 1, 0, 0) tions a
f
f
V is left fixed by all transforma
A(1, C) as of course it must be. Note next that the orbit under
A(1, C) of any point of the form (1, 1, b A, b 8 ) is the set of points l (1, 1, ab A, ab 8 ): at C*!, hence is an analytic submanifold of V isomorphic to C*. Note finally that the orbit under A(1, C) of any point of the form (a A, a 8 , bA, bB) where a A f. 1 is the set of points l(aA, a 8 , z, (a 8 1)(aA1r 1 z):ztC!, hence is an analytic submanifold of V isomorphic to C; and similarly for any point for which a 8
I=
1 . Thus there are
three categories of points of V , corresponding to the three types of orbits under the action of the group A(1, C) on V. However upon restriction to the open subset (6)
each orbit of A(1, C) in V except for the singular point (1, 1, 0, 0) itself is a submanifold of V1 isomorphic to C*. These orbits are moreover just the fibres of the complex analytic mapping r:f;: V1 . C* x C* x P1 definedby r:f;(aA,aB,bA,bB)=(aA,aB,[bA,bB]) where [bA,b 8 ] isthe point of P 1 having homogeneous coordinates (bA, b 8 ). Therefore, excluding the singular point of V, the remainder of the quotient space W = VI A(1, C) can be identified with the twodimensional complex submanifold wl (7)
c
wl
C* c c* X pl ' where 
=
!(aA, aB, [bA, bB]tC*x C*x pl: (aA~1)bB = (aB1)bA!.
230
R. C. GUNNING
Note that the restriction to this submanifold natural projection C* XC* X p1
>
w1
c
C* XC* X p1 of the
c* XC* is a complex analytic mapping
!{l:W1 .C*x_C~ suchthat !{l 1 (aA,aB) isasinglepointwhenever (a A, a 8 ) J (1, 1) but !{1 1 (1, 1) = P 1 ; thus W1 can be described as the complex manifold arising from C* x C* by applying a quadratic transform at the point (1, 1), blowing up the point (1, 1) to P 1 . Now the representation p w, c = p w (c) described by (3) for c t 0, when considered as a point p (c), V, has coordinates (e 277ic, e 277icw, w 0, 0); and the orbit of A(1, C) through this point is the subvariety
!(e2rric, e 277icw, b(1e 277ic), b(1e 277icw)) : b f duce to the singular point (1, 1, 0, 0) pw(c)
(8)
f
Pw
w1;
f
C!. This orbit does not re
V, hence represents a point
and indeed that point is
( c)  (e2rric e2rricw [e2rric_1 e2rricw_1]) ' ' c ' c '
w c c* X C*x p 1
1
On the other hand the representation of the affine structure parametrized by c = 0, when considered as a point pw(O) < V, clearly has the coordinates (1, 1, 1, w). The orbit of A(l, C) through this point does notreduce to the singular point of V either, so represents a point pw(c) < W1 ; and indeed that point is also given by (8), when extended to the value c = 0 by analytic continuation. The mapping Pw: C . W1 thus defined is clearly an injective mapping; so distinct equivalence classes of affine structures have distinct representation classes. The mapping p w is moreover a nonsingular holomorphic mapping; thus the set of representation classes of all affine structures on M is the onedimensional complex analytic submanifold pw(C) C W1
,
isomorphic as a complex manifold to C.
Finally considering the mapping Pw as a function of the point w
f
H,
note that whenever w , w' are distinct points of H there are infinitely many pairs of parameters (c, c') such that p w (c)= pw (c'); indeed this equality holds precisely when c = (mw' + n)/(w' w) for some integers m, n, not both of which are zero. Thus although on any fixed Riemann surface distinct affine structures have distinct representation classes, none
231
R. C. GUNNING
theless for any pair of distinct marked Riemann surfaces there are affine structures having the same representations. The mapping p: H x C .... W1 taking any points w f H, c f C to the point pw(c) f W1 given by (8) is thus not an injective mapping. On the other hand it is easy to see that p is a nonsingular holomorphic mapping; thus the image p(H x C) is an open subset of W1 . Whenever c I= 0 the image pw(c) the natural projection
f
W1 is a point at which
1/1: wl > C* XC* is a local homeomorphism; and the
composite mapping 1/J o p: H x C* > C* x C* has the explicit form 1/fop(w,c)
= (e 277ic,e 277icw). It is easy to see that the image of this map
ping is precisely the complement of the subset {(x, y) Indeed whenever (x, y)
f
c* XC* write
are integers m, n such that c
=
X
l:"+ m I=
=
e 277icf' y
0 and w
=
f
C*x C*: lxl = IYI = 11.
= e 277i77; then there (7J+n)/((+ m) satis
fies Imw > 0 precisely when not both l;, 77 are purely real. On the other hand when c
=
0 the image pw(c)
f
W1 is in the subset
1/f 1 (1, 1) C W1 ,
hence can be viewed as a point in P 1 ; and as such, clearly pw(c)
=
(1' w). Thus altogether the image p(H X C) is a proper open subset of
wl'
the union of the two sets just described. This provides an explicit determination of precisely which homomorphisms p
f
Hom (1, A(1, C)) can be
the representations of some affine structures on some compact Riemann surfaces of genus 1. Note that this provides a solution to the purely topological problem of determining for which homomorphism p
f
Hom(1,A(1, C)) there exists a local homeomorphism f: C> C, not
necessarily complex analytic, satisfying (2); a more direct, purely topological solution of this problem might be interesting. 3. Turning then to the case of projective structures on compact Riemann surfaces, there is a unique such structure on P1 , and it is easy to verify that any projective structure on a surface of genus 1 is equivalent to an affine structure; hence it can be assumed that M is a marked Riemann surface of genus g > 1. The universal covering space M can then be identified with the unit disc tl in the complex plane; and 1 then becomes a group of projective transformations acting on tl with .tl/1
=
M.
232
R. C. GUNNING
The Schwarz ian differential operator O(f) = (f"/f')' 1/2(f"/f') 2 plays the role for projective structures that the differential operator f"/f' played for affine structures. The characteristic properties of the Schwarzian differential operator are that O(f) = 0 precisely when f is a projective transformation, and that O(f o g)= 0(f)(g') 2 + O(g). From these properties it is easy to see that a complex analytic local homeomorphism f satisfies (1) precisely when O(f)(Tz). (T'(z)) 2 = O(f)(z), hence precisely when f/J(z) = O(f)dz 2 represents a quadratic differential on ~/1 = M; and conversely any quadratic differential on the form
O(f )dz 2
~/['
= M can be written in
for some complex analytic local homeomorphism f
satisfying (1), the most general such function f being a of for any projective transformation a. Thus the set of equivalence classes of projective structures on M can be put into onetoone correspondence with the (3g3)dimensional space of quadratic differentials on M. The quadratic differentials f/J(z) on M cannot be considered as being very explicitly known, and solutions f of the Schwarzian differential equation O(f )dz 2 = f/J(z) are even less explicitly known; and that complicates the study of projective structures. Nonetheless some interesting general properties of the mappings f:
~ >
P1 can be established.
The mapping f is not always a covering mapping; indeed it is a covering mapping only for a set of projective structures corresponding to a compact subset of the vector space of quadratic differentials, [10], [11]. The following three conditions are equivalent: (i) f is a covering mapping; (ii) the image
is a proper subset of P 1 ; (iii) the group p(f') is a properly discontinuous group of transformations acting on the image f(~), f(~)
[5], [10]. As in the affine case, whenever p(f') is a properly discontinuous group of transformations on f(~) the quotient space f(~)/ p(f') is a compact Riemann surface, and f induces an analytic mapping f* : M > f(~)/ p(f'); and again this is not necessarily a onetoone mapping, [11]. Most of the further questions one can ask about these mappings f and f* remain open.
AFFINE AND PROJECTIVE STRUCTURES
233
Turning next to the representations of these projective structures, the set Hom(f', PL(l, C)) can be given the structure of a complex analytic variety just as was done for the corresponding set in the case of affine structures. If A 1 , ... , Ag, B 1 , ... , Bg are the standard generators of f', subject to the relation C 1 ... Cg =I where Cj = AjBjAj 1 Bj 1 , then an element p f Hom (f', PL(l, C)) is determined by the 2g elements Xj = p(Aj), Yj = p(Bj) of PL(l, C); and these can be arbitrary elements of PL(l, C), subject to the condition imposed by the defining relation. Thus Hom (f', PL(l, C) can be identified with the complex analytic subvariety
of the complex manifold PL(l, C) 2 g; this is a subvariety with singularities. The affine transformations A(l, C) can be viewed as forming a complex submanifold of PL(l, C); and the subset V0 = IZ(Xj, Yj)zl: Xj, Yj f A(l, C), Z f PL(l, C)l is an analytic subvariety of the product manifold PL(l, C) 2 g, with the property that the complement V1 = VV0 is a (6g3)dimensional complex analytic manifold. Furthermore the quotient space W1 = V1 /PL(l, C), where PL(l, C) acts as a group of analytic automorphisms of vl by conjugating all the elements xj ' y j simultaneously, has the natural structure of a (6g6)dimensional complex analytic manifold, [6], [7]. Identifying the vector space of quadratic differentials on M with C 3 g 3 , it follows readily from wellknown properties of the Schwarzian differential operator that the mapping
PM:
C 3g 3
>
V, which associates
to each quadratic differential the representation of the associated projective structure, is a complex analytic mapping. The image must lie in the open subset V1 C V, since any
p f
V0 is equivalent to an affine repre
sentation and M admits no affine structures; so this mapping actually extends to a holomorphic mapping
pM :
C 3 g 3
>
W1 . Using the properties
of the Schwarzian differential operator again, it is not difficult to see that this mapping p is a nonsingular injective holomorphic mapping, [4], [9].
234
R. C. GUNNING
Thus distinct projective structures on M have distinct representation classes; and the image p(C 3 g 3 ) is probably a complex submanifold of wl
isomorphic as a complex manifold to C 3 g 3 ' although it is not yet
known to be a closed subset of wl . The set of all marked Riemann surfaces of genus g is parametrized by the Teichmiiller space Tg, a (3g3)dimensional complex manifold; and there is a natural complex analytic vector bundle Q of rank 3g3 over Tg such that the quadratic differentials on a marked Riemann surface M can be identified with the fibre of Q over the point representing M, [2]. There is a complex analytic mapping p: Q .... W1 such that the restriction of p to the fibre over the point representing M is just the mapping pM ;
and it can be demonstrated that this mapping p is a nonsingular
complex analytic mapping, [3]. Thus the image p(Q) C W1 , the set of all those equivalence classes of homomorphisms p f Hom (r, PL(1, C)) which are representations of some projective structures on some compact Riemann surfaces of genus g, is an open subset of W1 . As in the affine case this mapping p: Q .... W1 is not injective; perhaps the most interesting illustration of this is provided by the simultaneous uniformizations described by L. Bers, [1]. The problem of determining the image p(Q)
c W1
is really the purely
topological one of determining for which homomorphisms pfHom(r,PL(1,C)) there exist local homeomorphisms f : L\ .... P1 , not necessarily complex analytic, satisfying (1 ); but analytic methods provide several partial results. First, since Q is connected the image p(Q) C W1 must be contained in a connected component of W1 ; and this is a nontrivial result, since W1 is not connected. To analyze this situation more closely, recall that every projective transformation can be represented by a 2 x 2 complex matrix of determinant 1 , leading to the exact sequence of groups
cP PL(1, C).... 0 ; 0.... Z/2Z .... SL(2, C).... and the homomorphism c/J induces a mapping
AFFINE AND PROJECTIVE STRUCTURES
~:
235
Hom (1, SL(2, C)) . Hom (r, PL(l, C)) .
The set Hom (r, SL(2, C)) can be given the structure of a complex analytic subvariety
v* c SL(2, C) 2 g,
paralleling the imposition of the com
plex structure V on the set Hom (1, PL(l, C)); and in these terms the mapping ~ :
v; =
g 1 (V1 )
v* . V
is a complex analytic mapping. The inverse image
is the set of irreducible linear representations in
Hom (1, SL(2, C)), and it has the structure of a (6g3)dimensional complex manifold. The quotient space
w; vt /SL(2, C), =
where SL(2, C)
acts by conjugation, has the structure of a (6g6)dimensional complex manifold, [6], [7]. It is not hard to see that
w;
is a connected complex
manifold; the traces of the appropriate elements give coordinates by means of which this can be demonstrated, following the approach of [8] extended naturally to the complex case. Now using yet again the properties of the Schwarz ian differential operator, it can be seen that p(Q) C ~(W;) C W1 , [4], [5]; thus each p
f
Hom {r, PL(l, C)) that is the representation of a
projective structure on M lifts to a representation p* t: Hom(1, SL(2, C)). That is the essence of the connectivity condition. The condition that each representation of a projective structure on M lifts to a linear representation provides a connection between the projective structures on M and the complex analytic vector bundles over M, since each p*
t:
Hom (1, SL(2, C)) determines a complex analytic vector
bundle of rank 2 over M; and that leads to some interesting further results, the statements of which do not require any knowledge of the properties of complex analytic vector bundles. These results principally involve an analytic invariant of p* called the divisor order which can be defined as follows. There are infinitely many pairs of meromorphic functions f 1 , f 2 on f':... such that the vectorvalued meromorphic function F =
(!~)
satisfies F(Tz) = p*T F(z) for all T
t:
f'. For each such func
tion F it is clear that there are complex analytic functions g, h on f':... having no common zeros on f':..., such that gf/h, gf 2 /h are also complex
R. C. GUNNING
236
analytic functions on L'1 having no common zeros on L'1 ; and orderMF
=
orderMh  orderMg is independent of the choice of these functions g, h, where orderMg is the total number of zeros of g, counting multiplicities, in a fundamental domain for the action of 1 on L'1 • The divisor order of p* is defined to be div p* = maxF orderMF; and it can be shown that it
is a finite integer, in the range [l;gJ < div p* < g1 , where [x] is the greatest integer ;; x, although these inequalities are not needed here. Then the representations p*
f
Hom (1, SL(2, C)) corre
sponding to projective structures on M are precisely those representations for which div p*
=
g1, [5]. The explicit determination of the
divisor order of p * is as yet generally impossible, since few relations between the divisor order and any other invariants of the representation are known. However it is easy to see that div p*;; 0 whenever p* is a unitary representation; and that implies immediately that no unitary representation can correspond to a projective structure on any Riemann surface, [6]. It should be mentioned in passing that the most spectacular result about the divisor order is the rather deep converse assertion that div p* ~ 0 implies p* is analytically equivalent to a unitary representation, [15]. It is tempting to conjecture that all equivalence classes of representa
tions p*
f
Hom (1, SL(2, C)) correspond to some projective structures on
some compact Riemann surfaces of genus g, except for those representations already shown to be excluded; the excluded representations are the reducible representations and the unitary representations, and any representations that become reducible or unitary when restricted to any subgroup of finite index in 1 . 4. The preceding discussion can be extended to cover structures with possible branching. A branched projective structure on M is described
237
AFFINE AND PROJECTIVE STRUCTURES

by a complex analytic mapping f : M .., P 1 , not necessarily a local homeomorphism, satisfying (1); and a branched affine structure on M is described correspondingly by a complex analytic mapping f: M > C satisfying an equation of the form (1) but with pT
f
A(1, C). Thus the only
difference between regular and branched structures is that in the latter case the mapping f is allowed to have branch points. The notions of equivalent structures, and of the representations of structures, are introduced just as in the case of unbranched structures. These branched structures were studied by Mandelbaum in [12], [13], [14]. (It should be pointed out though that Mandelbaum uses a slightly different terminology when considering affine structures. What are here called branched affine structures he calls regular branched affine structures; his branched affine structures are the more general ones for which f is a mapping to P 1 rather than merely to C . ) To discuss the branched affine structures an approach rather different from that used by Mandelbaum will be followed here.

If f: M > C is a complex analytic mapping such that for any T < 1 (10) for some constants aT
C*, bT
f
f
C, then the differential ¢(z) = df(z)
is a complex analytic differential form on M such that for any T
f
1
(11)
such a differential is called a Prym differential on M associated to the representation a
E
Hom (1, C*), where
a
assigns to any T
f
1 the
value aT ( C*. Conversely if is a Prym differential on M associated Hom (1, C*) then there are holomorphic functions
to the representation a
f
f on M such that df
0. The zeros of the Prym differential c/J
=
df are precisely
the branch points of f, and the total order of the differential c/J on M is just the total branching order of the restriction of f to any fundamental domain for the action of the group r 2g2 for any representation a
f
on M ; and that order is equal to
Hom{r, C*), [4]. Thus when g = 1 a
branched affine structure always reduces to an unbranched affine structure; so further consideration can be limited to Riemann surfaces of genus g > 1 . The situation is rather more complicated than that for the case g
=
1. Although the mappings f:
easy to see that the image
~
f(~)
.... C can be somewhat complicated, it is is either all of C or all of C except
for a single point; and this illustrates a recurrent dichotomy in the discussion of these structures. First, if
f(~)
is all of C except for a single
point then after replacing f by the composition of f with an affine transformation it can be assumed that f(~) serve C*, all the transformations and hence f(Tz) sentation a
f
=
aT f(z) for all T
pT f
=
C*. Since p(r) must pre
reduce to the form
pT(z) =
aTz;
r. That implies that the repre
Hom (r, C*) is analytically trivial; and moreover f(z)
=
exp w(z) where dw(z) is an abelian differential on M , viewed as a r invariant differential form on
~
. Since w(z) is readily seen to take
on all values, it follows that f(C) does equal c*, so this possibility does exist. Next, if
f(~)
omits at least two points of C then the family
of complex analytic functions !f(Tz): T f r! is a normal family in ~;
239
AFFINE AND PROJECTIVE STRUCTURES
and consequently the image group p(l) is a subgroup of A(1, C) having compact closure. It is easy to verify that the only compact subgroups of A(1, C) are finite cyclic subgroups or are conjugates of the subgroup
lz .az: Ia I= 11; hence after conjugation all the transformations pT reduce to the form pT(z) = aTz. However this reduces to the case con
c*'
sidered before, in which f(!i) =
contradicting the assumption that
f(!i) f, C*. Thus the only case in which f(!i) I= C is that in which a
f
Hom (1, C*) is analytically trivial and f is equivalent to the function
exhibiting this triviality, that is to say, f(z) = ag(z) + and g(Tz) =aT g(z) for all T
f
f3 where
a
I= 0
1; and in this exceptional case f(!i) is
the complement of a single point in C . This exceptional case is also the only one for which the branched affine structure is not uniquely determined by its representation class. For if f 1 , f 2 are two complex analytic mappings fj : 1'1 > C such that fj(Tz) = aTfj(z) + bT then the difference g = f 1 f 2 is a complex analytic function such that g(Tz) =aT g(z); thus a
f
Hom (1, C*) is analyti
cally trivial and g exhibits this triviality. Conversely whenever a
f
Hom (1, C*) is analytically trivial and g exhibits this triviality, then
the branched affine structure determined by any mapping f: 1'1 . C for which f(Tz) = aTf(z)+ bT has the same representation as the branched affine structure determined by the mapping f + g; but these are not equivalent branched affine structures. Of course it is possible to modify the notion of equivalence to avoid this exception; but the modified notion is less natural while the exceptional case is exceptional in many other ways as well. Turning next to the question whether p(1) is a properly discontinuous group of transformations, it is convenient to consider three separate cases. The general classification of properly discontinuous groups of complex analytic automorphisms of C or of C* is simple and will be assumed known; details can be found in L. R. Ford's Automorphic Functions, for example. (i) The first case is that in which f(!i) = C and p(l) is the lattice subgroup of C generated by the translations z . z + 1 , z . z + w, where Imw > 0; the mapping f : 1'1
>
C exhibits M = 1'1/1 as a branched
240
R. C. GUNNING
analytic covering of the to.rus Cl p(f'), and the function f is an abelian integral. Note that in this case the representation a (Hom (1, C*) is the identity representation. The condition that there exists such a mapping is of course readily expressed in terms of the period matrix of the abelian differentials on M. (ii) The second case is that in which
f(~) =
C* and p(f') is a purely multiplicative group, as in the case of unbranched affine structures; again f: ~ .... C* exhibits M = ~If' as a branched analytic covering of the torus C* I p(f'), and f = exp w where w is an abelian integral. Note that in this case the representation a ( Hom(f', C*) is analytically trivial. The condition that there exists such a mapping is again readily expressed in terms of the period matrix of the abelian differentials on M . (iii) The third case is that in which f(~)
= C and p(f') is the extension of a lattice subgroup of C by a
cyclic group of order v, where v = 2, 3, 4, or 6; thus p(f') consists of transformations of the form z .... ekz + m + nw for arbitrary integers k , m , n, where ev = 1 and w = e if v > 2 . The mapping f : ~ . C exhibits M =
~If'
as a branched covering of the onedimensional projec
tive space P1 = Cl p(f'). Note that in this case the representation a ( Hom (f', C*) is never analytically trivial, indeed is a homomorphism for which the image a(f') is a cyclic subgroup of C* of order v. There is a subgroup 1 0
C
f' of finite index for which the restriction p(f'0 ) is a
lattice subgroup, so the induced branched affine structure on M0 = ~110 is of the type considered in case (i); and M is the quotient of M0 by a cyclic group of order v. In all other cases, in particular whenever the representation a (Hom (1, C*) is not analytically trivial and the values aT are not vth roots of unity for v = 2, 3, 4, or 6, the group p(f') does not act discontinuously on
f(~).
Letting Aj , Bj be the canonical generators of f' corresponding to the marking of the Riemann surface M , the set Hom (f', A(l, C)) can be given the structure of a complex analytic variety as in the other cases considered before. Indeed associating to any p (Hom (1, A(l, C)) the coordinates (aj,aj,bj,bj) where PA.(z)=ajz+bj,Ps.(z)=ajz+bj, J
J
241
AFFINE AND PROJECTIVE STRUCTURES
establishes a onetoone correspondence between the set Hom(r,A(1, C)) and the complex analytic subvariety
contained in (C*)g
X
(C*)g
cg
X
X
cg; this is an irreducible analytic
subvariety of dimension 4g1 , the only singularity being at the point aj = aj = 1 , bj = bj = 0 corresponding to the identity representation. This reduces to the subvariety (4) when g by an element a
f
=
1. Conjugation of Hom(r, A(1, C))
A(1, C) of the form a(z)
=
az + {3 has the effect of
transforming (aj, aj, bj, bj) to
and this e11hibits A(1, C) as a group of complex analytic automorphisms of V. To describe a complex structure on the quotient space it is convenient to introduce the auxiliary 2 x 2g complex matrix a 1 1 I ··· 1 a g1 I a'1 1 1 ... I a'g1) (14)
M =
( bl
and to decompose
v
,· .. ,bg
'b~
into the three subsets
,···,bg
v = vou vl u v2
where
Vv = {(aj, aj, bj, bjhV: rank M= vl; note that each subset Vv is mapped to itself under the action of any automorphism a
f
A(1, C). Here V0 is
just the singular point of V, a separate orbit by itself. The discussion of the subset
vl
:xactly parallels that in the case g
=
1 ; thus the orbit
space W1 = V1 I A(1, C) can be described as the complex manifold of dimension 2g arising from (C*) 2 g by blowing th~ point aj
=
aj = 1 up
to the projective space P2 gl of dimension 2g1. On the subset V2 each orbit is a complex submanifold of V2 of the form {(aj, aj, abj {3(aj 1), abjf3(aj1): a£C*,{3ECI, hence as a complex manifold is equivalent to C*x C; the orbit through any point (aj, aj, bj, bj) is the product of the
242
R. C. GUNNING
point (aj, ap f (C*) 2 g with a twodimensional linear subspace L C C 2 g containing the complex line joining the point (ar1, aj1) ( C2 g to the origin. Note that when (aj1, aj1) = (1, 0, · · ·, 0) ( C2 g each such twodimensional linear space L can be described by a point (O,z 2 ,···,z 2 g)fL for which not all of the coefficients z 2 , ···,z 2 g are zero, and two such points (O,z 2 , ···,z 2 g) and (O,z;, ···,z;g) describe the same linear subspace precisely when (z;, ···, z;g) = (cz 2 ,
···,
cz 2 g) for some c f C*;
thus this. set of linear subspaces is parametrized by the complex projective space P2 g_ 2 of dimension 2g2. Since the points (aj1, aj1) can be reduced to the form (1, 0, · · ·, 0) by nonsingular linear changes of coordinates in C2 g, and these changes can moreover be taken to be complex analytic functions of the variables aj , W2
=
aj
locally, it follows that
V2 / Aut (1, C) can be described as an analytic subvariety of dimen
sion 4g3 contained in a complex analytic P2 g_ 2 bundle over (C*) 2 g. The identity homomorphism p f V0 cannot be the representation of any branched affine structure on M, since C/ p(r) is not compact. For a homomorphism p f V1 either pT is a pure translation for each T ( r (in case aj = aj = 1 ), or after suitable conjugation pT(z) = aTz for each T ( r . In these cases ture f: L\
>
p
is the representation of a branched affine struc
C on M only when f(z) = w(z) or f(z) = exp w(z) for some
abelian differential dw on M; thus the possible representations p can be determined explicitly from the period matrix of the abelian differentials on M . "The only homomorphisms
p
f V1 that can be the representations
of some branched affine structures on some Riemann surfaces M are those for which the quotients C/ p(r) or C* I p(r) are compact; but it is not clear that enough is yet known about the possible period matrices of Riemann surfaces to show that all these homomorphisms can be representations of some branched affine structures. Determining which homomorphisms p f V2 are representations of branched affine structures on M leads to the difficult problem of determining the period classes of Prym differentials on M. Again it is tempting to conjecture that all
p (
V2 for
which C/ p(r) is compact are representations of some branched affine structures on some Riemann surfaces.
243
AFFINE AND PROJECTIVE STRUCTURES
5. Mandelbaum's approach to these branched structures involves examining the meromorphic functions f"/f' or O(f), and leads to interesting results about the branch points of the mapping f; details can be found in his papers [12], [13], [14]. Let it suffice here, in conclusion, to report that he demonstrates that any p
f
Hom (r, PL(1, C)) is the representation of some
branched projective structure on some Riemann surface. What is much more interesting, though, is his result that on a fixed compact Riemann surface M of genus g > 1 , any irreducible homomorphism p
f
Hom(r, PL(1, C)) for which div p
=
k is the representation of a
branched projective structure with branching order 2g  2  2k ; and conversely, if p is irreducible and is the representation of a branched projective structure with branching order 2g 2  2k then div p ;:; k, and this is actually an equality if k ;:; 0, [14]. REFERENCES [1]
L. Bers. Simultaneous uniformization. Bull. American Math. Soc. 66 (1 %0), 9497.
[2]
. Fiber spaces over Teichmiiller spaces. Acta Math. 130 (1973), 89126.
[3]
C.
[4]
R. C. Gunning. Lectures on Riemann Surfaces. Princeton University Press, (Mathematical Notes 2), 1966.
[5]
. Special coordinate coverings of Riemann surfaces. Math. Annalen 170(1967), 6786.
[6]
. Lectures on Vector Bundles over Riemann Surfaces. Princeton University Press, (Mathematical Notes 6), 1967.
[7]
. Analytic structures on the space of flat vector bundles over a compact Riemann surface. Several Complex Variables II, Maryland, 1970. Springer Lecture Notes 185(1971), 4762.
[8]
H, Helling. Diskrete Untergruppen von SL 2 (R). Invent. Math. 17 (1972), 217229.
[9] [10]
J. Earle. On variation of projective structures. This volume.
I. Kra. On affine and projective structures on Riemann surfaces.
J. d'Analyse Math. 22(1%9), 285298.
. Deformations of Fuchs ian groups, I, II. Duke Math. (1969), 537546 and 38(1971), 499508.
J.
36
244
R. C. GUNNING
[11] I. Kra and B. Maskit. Remarks on projective structures. This volume.
[12] R. Mandelbaum. Branched structures on Riemann surfaces. Trans. American Math. Soc. 163(1972), 261275. [13]
. Branched structures and affine and projective bundles on Riemann surfaces. Trans. American Math. Soc. 183(1973), 3758.
[14]
. Unstable bundles and branched structures on Riemann surfaces. Math. Annalen 214(1975), 4959.
[15] M.S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact Riemann surface. Annals of Math. 82 (1965), 540567.
BOUNDARY STRUCTURE OF THE MODULAR GROUP
W. J. Harvey*
§0. Introduction In this note we introduce a simplicial structure for the collection of simple loops in a surface, with the main purpose in mind being to provide an appropriate combinatorial framework for studying the geometry of how the modular group r(S) of a surface S acts at infinity on the Teichmiiller space T(S). A model for such a study is the paper of Borel and Serre [4], which analyses the structure of arithmetic groups by adding suitable boundary components to the relevant homogeneous spaces. We content ourselves here with a description of the basic facts and some simple deductions. A detailed study of the implied algebraic structure for the modular group and a related geometric description of the compactified space of moduli will appear elsewhere. The main point which emerges is the remarkable closeness of the analogy with arithmetic groups; underneath the definite lack of homogeneity in the complex analytic character of Tg, there lies concealed a beautiful real analytic structure which is mirrored in the action of rg.
*Preliminary report presented at the Riemann surfaces conference in July 1978 at S.U.N.Y., Stony Brook. Partial support for the work was provided by NSF Grant MCS 7718723 AOl. The author is grateful for the hospitality of the Institute for Advanced Study during preparation of this manuscript. © 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedin~s of the 1978 Stony Brook Conference 06 91082642/80/00024507 $00.50/1 (cloth) 0691082677/80/00024507$00.50/1 (paperback) For copying information, see copyright page
245
246
W.j.HARVEY
I want to record here my indebtedness to J.P. Serre, both for rendering palatable to my tender stomach the rich diet of buildings and arithmetic groups and for demonstrating that one should not be put off by locally infinite phenomena. After all, think of the structure of a cusp! §2. Partitions of a surface Let S be a surface of finite type (g, n). We denote by ~(S) the set of all partitions of S . These are systems A of disjoint simple loops in S , such that (i)
no loop in A bounds either a disc in S or a single boundary component of S;
(ii) no pair of loops form the boundary of an annulus.
The group of all homeomorphisms of S acts naturally on
~(S), and
the orbit of A consists of all partitions which determine topologically equivalent ways of dissecting the surface. Normally we regard as identical two partitions that are equivalent by a homeomorphism isotopic to the identity, and our aim is to understand how the group r(S) of isotopy classes of homeomorphisms acts on the classes of ~(S) . We observe that ~(S) is partially ordered by inclusion, so that it is possible to assemble
~
as an abstract simplicial complex by taking
singleton loops as vertices with edges corresponding to pairs of loops, the edge joining the vertices representing the individual loops of the pair, and so on. For a ksimplex, which represents a partition by k+1 loops, there will be k +1 faces, each a (k1)simplex obtained by omission of one loop. The resulting simplicial complex is denoted
§{s)
or
§'.
It plays
in the theory the role of Tits building, although it lacks certain essential features of that object. PROPOSITION
1. /!J{S) is a thick chamber complex of dimension N 1
(N =3g+n3), on which f'(S) operates simplicially. The quotient is a finite complex.
BOUNDARY STRUCTURE OF THE MODULAR GROUP
247
[A chamber complex (see for instance [11]) of dimension m is a simplicial complex with the property that every simplex is a face of some msimplex. It is termed thick if each (m1)simplex abutts at least three msimplices
(chambers).] Any partition extends to a maximal one with 3g + n 3 loops, which determines a chamber. This can be done in infinitely many ways if A has fewer than N loops since some subsurface of S \A must have negative Euler characteristic and thus contains infinitely many distinct simple loops. Therefore /!!'is locally infinite. We shall abuse notation by using the same symbol for the geometric realization of of all functions ,\ from the vertex set
/!J: which is the space
~ to [0, 1] whose support is a
simplex and which are such that the sum of the ,\values is 1 . Equipped with the weak topology,
f!J{s)
is a C W complex, on which r(S) acts
by the rule
for ..\
f
/!f' and
g
f
r. This action is certainly simplicial, and a finite
connected subcomplex
Xr:; /!!'
containing a simplex from every rorbit
is easily obtained by choosing one partition for each of the finite number of ways in which S can be dissected into a collection of 3holed spheres. After passing to the barycentric subdivision of /!!'(and construct a precise fundamental domain; the quotient
(f{),
f!Jir
one can
is therefore a
triangulable finite complex. EXAMPLES. (a) In genus 1 with 1 hole /!!'is a discrete set of vertices. The quotient is a point. (b) If S has genus 2, there are two orbits of 2simplices which represent the following partitions:
W.J.HARVEY
248
In the barycentric subdivision of
:J{,
we find the shaded fundamental
region.
Notice that all the loops are fixed setwise up to homotopy by the hyperelliptic involution. We end this section by stating the following elementary yet powerful result (cf. Birman's discussion in (D.G] Chapter 6). PROPOSITION
2.
/!i(s)
is connected if it has positive dimension.
The proof consists in determining, for a given pair of loops f, ( in
s
(with intersection number m > 0 ), how to walk from
e to
e'
by a
sequence of loops each disjoint from its predecessor. If m = 1 , this is two steps when S has type (g, n) ~ (1, 1), and if m > 1, an inductive argument due to Lickorish (9] applies. Since this implies that r{S) is generated by stability groups of vertices, one obtains by induction on the topological type of S the corollary that r(S) is generated by a finite number of Dehn twists. A more careful study yields the precise form of Lickorish's result, as sharpened recently by Humphries, that 2g + 1 twists are sufficient to generate r(Sg, 0 ).
§2. The cuspidal boundary of Teichmiiller space A method used by several authors ([1], [2], (6], [7]) to construct a compacification of T(S)/r(S) has important connections with the complex
/!i(s)
of §1. It stems from the observation that to any given partition A of S
BOUNDARY STRUCTURE OF THE MODULAR GROUP
249
there is associated a way to degenerate the surface into a singular topological space, known as a surface with nodes, or stable surface, by shrinking of the loops in A to points. Formally, we make the following definition. DEFINITION.
The stable surface SA of a partition A i.s the topological
space S/A, obtained from the equivalence relation "'A determined by the rule x "'A y
¢::=:::?
x and y lie on the same loop of A . ~··
The projection of loops in A to SA determines the set of nodes in SA. Note that homotopically equivalent partitions determine isomorphic stable surfaces; again we shall abuse notation by identifying these. The partial ordering on partitions induces a partial ordering on stable surfaces, according to which there exists a morphism from SA to SA' whenever A C A' here a morphism is a continuous map whose fibers are either points or simple loops, disjoint and homotopically distinct from the nodes of SA. For our purposes we regard all boundary components of S as points and require morphisms to fix each one. Now one associates to each partition A of S a Teichmiiller space of stable surfaces T A= T(SA) which classifies marked complex structures
on SA. A natural way to parametrize T A comes from the wellknown FenchelNielsen coordinates, taken with respect to some maximal partition
A 0 containing A. These determine a homeomorphism from T(S) onto the product of N = 3g + n 3 copies of f) = R x R+ , well defined after choice of an ordering A 0 , with the R+components representing the set of lengths for the minimal geodesics in the homotopy classes of loops in A 0 , measured in the Riemannian metric on S given by the choice of complex structure Sx
f
T(S), and Rcomponents giving the twist parameters
which describe the procedure for assembling Sx from the component parts of S \A 0 in terms of angular deviation from a fixed one. For more details, the reader is referred to [D.G] Chapter 9 and further references to be found there.
250
W.J.HARVEY
The various faces of the corner R~ obtained by setting Alengths to 0 are now to be viewed as the result of shrinking S by the relation A for the various A C A 0 which amounts to identifying a given face with a parametrization of the product of Tspaces for the parts of S \A with the
Atwist parameters added. In view of the naturality of the procedures and the mutual compatibility of changing to different maximal partitions, it turns out that one can adjoin to T(S) in this fashion the various boundary components aTA for all A ( fYJ(S), with appropriate identifications arising from the partial ordering corresponding to the procedure of collapsing TA' onto TA c aTA' when A' cA. This is the cuspidal boundary
structure for T(S) which we denote aT(S). THEOREM
1. The bordified space
T
U aT is a connected Hausdorff
real analytic manifold with boundary, on which the Teichmiiller modular group
r
acts properly discontinuously.
THEOREM
2. There is a weak homotopy equivalence between aT(S) and
the complex
/!i(S) equivariant with respect to the two r(S)actions.
The boundary structure is closely related to the boundaries studied by Abikoff, by Bers and by Earle and Marden, and all have a common root in Mumford's work on stable curves and the studies of Bers and Maskit (2, 10] on Kleinian groups. The main novelty here is that one blows up one real dimension for each degeneration curve, thereby rendering the action of
res)
proper, since the stability group of a boundary component TA
X
R#(A)
contains the group of Dehn twists about the loops of A and these act as translations on the corresponding twist parameter space R#(A). §3. Comments The FenchelNielsen coordinates are in some respects more naturally defined by using the reciprocal of the lengths of Aloops as R+ variables. It is then possible to make canonical the Ashrinking process as passage
to the cusp at
oo
for the translation group of Dehn twists about Acurves,
BOUNDARY STRUCTURE OF THE MODULAR GROUP
251
and there is then a natural interpretation in terms of the action of SL 2 (R)N on ~N. As a final remark we observe that it is an easy consequence of Theorem 2 that the quotient moduli space
[J({s) =
T(S) U aT(S)/f'(S) is
a compact real analytic space with boundary. Known properties of f'(S) imply (see e.g. [8]) that there is a finite covering of
ffcs)
that is a
smooth manifold. This result does not extend immediately to the complex analytic Mumford compactification of moduli space. REFERENCES [D.G] Discrete Groups and Automorphic Functions (edited by W. J. Harvey), Acad. Press (London), 1977. [1] W. Abikoff, "Degenerating families of Riemann surfaces," Ann. of Math 105(1977), 2944. [2] L. Bers, "On boundaries of Teichmiiller spaces and on Kleinian groups I," Ann. of Math. 91 (1970), 570600. [3] , "Spaces of degenerating Riemann surfaces," in Ann. of Math Studies no. 79(1974), 4355. [4] A. Borel and J.P. Serre, "Corners and Arithmetic Groups," Comment. Math. Helv. 48(1973), 436491. [5] P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Publ. Math. I.H.E.S. 36(1969). [6] C. J. Earle and A. Marden, unpublished (but see reference [D.G], Chapter 8). [7] W. J. Harvey, "Chabauty spaces of discrete groups," in Ann. of Math Study no. 79(1974), 239246; see also Chapter 9 of [D.G]. [8] , "Geometric structure of surface mappingclass groups," in Homological Group Theory (ed. C. T. C. Wall), London Mathematical Society Lecture Notes #36, Cambridge University Press (1979), 255269. [9] W. B. R. Lickorish, "A finite set of generators for the homeotopy group of a 2manifold," Proc. Comb. Phil. Soc. 60(1964), 769778. (Also corrigendum, ibid., 62 (1966), 679681.) [10] B. Maskit, "On boundaries of Teichmuller spaces and on Kleinian groups II," Ann. of Math. 91 (1970), 607639. [11] J. Tits, Buildings of spherical type and finite B N pairs, Lecture Notes in Math 386, Springer Verlag 1974.
A REALIZATION PROBLEM IN THE THEORY OF ANALYTIC CURVES Maurice Heins
1. My talk at the Conference was based on a paper [3] that appeared in the volume of the Bulletin of the Greek Mathematical Society dedicated to the memory of Christos Papakyriakopoulos. For this reason, the present note has the restricted objective of summarizing the paper and of indicating two further realization problems that may be treated with the aid of the methods of the paper. We start with these related questions. 2. One of these is the problem of characterizing up to conformal equivalence a Weierstrass class [2] taken with its center map. The answer is simply stated. The pairs, (Weierstrass class, associated center function), are conformally equivaleni: to exactly the pairs (S, f), where S is a noncompact Riemann surface and f is a locally simple analytic function on S. By the GunningNarasimhan solution [1] of the problem of Karl Stein concerning the existence of locally simple analytic functions on noncompact Riemann surfaces, all noncompact Riemann surfaces admit locally simple analytic functions. Consequently, there exists a Weierstrass class conformally equivalent to a given noncompact Riemann surface. Further, for a giver. pair (S, f) there exists g analytic on S such that (S, f, g) is equivalent to an
© 1980 Princeton University Press Riemann Surfaces and Related Topics ProceedinJ2s of the 1978 Stony Brook Conference 0691082642/80/00025303$00.50/1 (cloth) 0691082677/80/00025303$00.50/1 (paperback) For copying information, see copyright page
253
254
MAURICE HEINS
analytic entity ( = analytische Gebilde) so that the Weierstrass class induced by (S, f, g) does not admit adjunction of poles or algebraic elements. On communicating this result to Robert Gunning at the Conference I learned from him that essentially the same question had been proposed to him by Barry Simon of Princeton University. I am indebted to Robert Gunning for this information and for also stating to me a second problem proposed by Barry Simon, namely: Given a region 0 C2
c
C, does there exist a holomorphic map of 0 into
whose first projection is the identity map on
n
and which is maximal
in the sense that it is not representable as the composition of a (univalent) holomorphic map of a Riemann surface S into C 2 and a proper injective holomorphic map of
n
into
s?
This problem has an affirmative answer as does its generalization which replaces 0 by a noncompact Riemann surface and takes the first projection as a given analytic function on S . It is planned to give a unified account of these questions in the Pro
ceedings of the projected 1979 Durham Instructional Conference (to appear in the L .M.S. Series of the Academic Press). 3. The problem treated in my talk is the following: Let F be holomorphic on C 2 , not the constant 0, but taking the value zero somewhere. Such F will be termed allowed. Let A be an analytic entity annihilating F in the sense that with c and v denoting respectively the center and value functions of A the equality F[c(p), v(p)]
=
0 holds for all p (A such that (c(p), v(p)) ( C 2 . One asks
for the conformal equivalence classes containing such A . The answer is simple: all. Given a Riemann surface S, there exist an allowed F and an analytic entity A that is conformally equivalent to S and annihilates F. The facts are classical for compact S. It suffices to refer to the results concerning the problem of Riemann and Klein, cf. [2].
THEORY OF ANALYTIC CURVES
255
To treat the case where S is not compact we proceed as follows. We construct with the aid of the theorem of Behnke and Stein and PickNevanlinna interpolation theory on Riemann surfaces analytic functions f and g on S which have the following properties: (1) C 1 (liz I< r l) is not empty and its components are relatively com
pact, 0
< r < + oo.
(2) Let
n
be a component of C 1 (l[z I< r!). Then f has a zero on
each component of
s ~ n.
(3) 0/ g(S). (4) For some a < C such that f has multiplicity one at each point of f~ 1 (lal)
it is the case that g[f~ 1 (lal) is univalent.
(5) The infinite product
(3.1)
II
[
f(p )=z
w Jn(p;f) g(p)
1~
is uniformly convergent on liz I;::; rl x llw I;::; rl, 0 < r < + oo. We see that the infinite product defines an allowed F which is annihilated by the pair (f, g). Further, it is concluded with the aid of (1), (2), and (4) that (S, f, g) is equivalent to an analytic entity taken together with its center and value functions. These facts permit us to conclude that the analytic entity induced by (S, f, g) annihilates F. We conclude that every conformal equivalence class of Riemann surfaces contains an annihilator of an allowed F . UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND 20742
REFERENCES
[1] Gunning, R. C. and Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103108 (1967). [2] Heins, M., Complex Function Theory. New York and London. Academic Press, 1968. .. , A realization problem in the theory of analytic curves. Bull. [3] Gr. Math. Soc., Papakyriakopoulos Memorial Volume 1978.
THE MONODROMY OF PROJECTIVE STRUCTURES John H. Hubbard
Introduction In this paper we shall give a new proof of the result, due to Hejhal [5], that the map associating to an isomorphism class of projective structures its conjugacy class of monodromy homomorphisms is a local homeomorphism. We shall follow the following plan: show that the domain (Prop. 1) and the range (Prop. 4) are manifolds, identify their tangent spaces (Prop. 2 and 4), and compute the derivative of the map above. It turns out that one space is an Eichler cohomology space and the other is the cohomology of a group; they are canonically isomorphic by a classical theorem of algebraic topology. The derivative is the canonical isomorphism. The idea of using differential calculus on this problem is not new: both Earle [2] and Gunning [4] have proposed similar proofs; this paper explains the appearance of Eichler cohomology in their computations. Many of the other results I establish in this paper were already known to Hejhal, Kra, Gunning, Maskit, Earle, Weiland no doubt others. The exposition is, I hope, in the spirit of Gunning's book and in fact the paper is largely a matter of putting parameters in arguments appearing there. I wish to thank Earle, Douady, Kra, and Gunning for helpful conversations, and the N .S.F. for financial support during part of the preparation of this paper.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00025719$00.95/1 (cloth) 0691082677/80/00025719$00.95/1 (paperback) For copying information, see copyright page
257
258
JOHN H. HUBBARD
NOTATION.
P 1 is the complex projective line (the Riemann Sphere)
G = PGLiC) = Aut P 1 ; A C G is the subgroup of affine maps z
1>
~
=
az +b. pgl 2 (C)
=
Space of analytic vector fields on P 1 .
The adjoint action of G on ~ corresponds to the direct image of the corresponding vector fields. We will speak of the fundamental group of a space only after a universal covering space has been chosen; the fundamental group is then the group of automorphisms of the universal covering space. All universal covering maps will be denoted u .
1.
Projective structures A projective atlas on a Riemann surface X is an open cover U i of
X and analytic maps ai : U i .... P 1 which are homeomorphisms onto their images such that aj
o
ai 1 is the restriction to ai(Ui n Uj) of an element
of G. Two projective atlases are equivalent if together they form a projective atlas; a projective structure on X is an equivalence class of projective atlases. EXAMPLES.
(i) If X is compact of genus 2' 2 and H is the upper
half plane, there is a covering map u: H .... X by the uniformization theorem. Sections of u over simplyconnected open subsets of X define a projective atlas. (ii) Other planar covering spaces of X, such as the Schottky covering space, can be used to describe projective structures. (iii) If r
cC
is a lattice and X
=
C/r, appropriate restrictions of
the canonical coordinate z of C define a projective structure on X as above; restrictions of eaz also do for any a< C{01. In these cases, the changes of coordinates are affine; a projective structure which can be defined by an affine atlas is called an affine structure.
259
THE MONODROMY OF PROJECTIVE STRUCTURES

Let a be a projective structure on a Riemann surface X, and X be

a universal covering space of X, with u : X .... X the covering map. LEMMA
1. (i) There exists an analytic map f: X > P 1 such that on any
contractible open subset U C X the composition f chart. Any other such map is of the form a
o
o
u 1
is a projective
f for some a
f
G.
(ii) To every such f there corresponds a unique homomorphism Pf:rr 1 (X)>G suchthat Pf(y)of=foy, and Paof=aopfoa 1
.
Proof. Cover X by open subset Ui on which u is injective, and such that there exist projective charts ai: u(Ui) .... P 1 ; let {3i = ai ou 1 . Then aij.= aj oai 1 is a 1cocycle on X with values in G. Since X is contractible, this cocycle is a coboundary, after refining the cover if necessary, and there exist ai
f
G such that aij = aj 1 a ai. Then on Ui
ai ai = aj aj so all the ai 0
0
0
n Uj ,
ai are restrictions of a global map
f : X .... P 1 with the appropriate properties. The second part of (i) is obvious. In any Ui there is a homomorphism Pf,i: rr 1 (X)> G such that Pf,i(y)of(x) = f(y(x)) for
X f
Ui, since both fa u 1 and fay a u 1 are
projective coordinates on X. But it is clear from analytic continuation


that Pf ,i(y) of= f oy on all of X, since X is connected and both sides Q.E.D. are analytic functions of x . Such a map f is called a developing map of X; Pf is the corresponding monodromy homomorphism. We will need the following fact: LEMMA
2. A projective structure on a compact Riemann surface is
equivalent to an affine structure if and only if the surface is of genus 1. Proof. See [3], p. 173. The result is purely topological, essentially saying that if a surface admits an affine structure, the cotangent bundle is trivial.
Q.E.D.
260
JOHN H. HUBBARD
COROLLARY. If X
is a compact Riemann surface of genus 2: 2 and a
is a projective structure on X, then the monodromy homomorphism Pf for any developing map f has noncommutative image. Proof. Any commutative subgroup of G is conjugate by an appropriate
a
f
G to a subgroup of the affine group A C G . The projective atlas
formed by maps of the form a of o ul on contractible open subsets of X
Q.E.D.
is affine.
The real interest of projective structures is the geometry of developing maps and the monodromy homomorphism. Beyond this corollary there is little to be said in general; the developing maps may fail to be covering spaces of their images, the monodromy homomorphisms may fail to be isomorphisms, and their images may fail to be discrete. In fact all of these pathologies occur for the family given in example (iii) for appropriate values of a.
2.
The Schwarzian derivative and the affine structure of P(X) Let U be a Riemann surface, x
tions on U such that f'(x)
f
U, and f, g meromorphic func
I= 0, g'(x) I= 0. There exists a unique a f G
such that f and a o g agree to order 2 at x. Then d 3 (f aog)(x) is naturally a cubic map TxU. Tfcxt 1 , and f'(xr 1 a d\faag)(x) is a cubic map TxU> TxU. But for any one dimensional complex vector space V , the cubic maps V > V correspond naturally to the quadratic maps V > C. Therefore the construction above defines a quadratic form S(f, g)(x) on T xU, and it is easy to see that S(f, g) is a meromorphic quadratic differential form on U , holomorphic at those points x where f'(x)
I= 0 and g'(x) 1/0.
If U f" f'
C
C and z is the canonical coordinate on C, then S(f, z)
! (f") /(f') 2
2,
=
as the classical definition requires.
For any Riemann surface U, let Q(U) be the space of holomorphic quadratic forms on U .
THE MONODROMY OF PROJECTIVE STRUCTURES
261
LEMMA 3. The Schwarzian derivative has the following properties : (i)
S(f, g)= S(f, aog) = S(aof, g) for all a
G.
(ii)
S(f, g) = 0 if f = ao g, and conversely f = ao g if S(f, g)= 0
f
and U is connected. (iii) S(f, g)+ S(g, f)= S(f, h), and S(f, g)= S(g, f). (iv) If U is simply connected, f is schlicht on U and q
f
Q(U),
there exists a solution g schlicht on U to the equation S(f, g)= q.
Proof. Parts i, ii, iii are obvious. Part (iv) is similar to Lemma 1. Q.E.D. Suppose the projective structures a and {3 on X are defined by
(\j, {3j).
atlases (Ui,ai) and fined in
ui n vJ.
Then the quadratic forms S(ai, {3j), de
coincide on open sets of form
ui n ui n v
are induced by a quadratic form q = a {3 on X . Conversely, if a structure {3
f
f
P(X) and q
f
1
2
so they
Q(X), there is a unique projective
P(X) such that {3 a = q , we shall denote it q + a . If
(Ui,ai) is an atlas defining a and the Ui are simply connected, then {3 may be defined by (Ui, {3i) where the {3 are solutions of S({3i, ai) = q in Ui, which exist by Lemma 3, (iv). LEMMA 4. The map Q(X) x P(X)
>
P(X) given by (q, a)
1>
q +a makes
P(X) into an affine space under Q(X).
Proof. All that is left to show is that P(X) is not empty. This follows from the uniformization theorem, as in the example (i) §1, or from RiemannRoch as in [3], p. 172.
Q.E.D.
COROLLARY. The space P(X) is canonically a complex manifold, and
for all a
3.
f
P(X), we have TaP(X) = Q(X).
Relative projective structures Let rr: X
>
S be a smooth family of compact Riemann surfaces para
metrized by a complex manifold S (i.e. a proper analytic submersion with
262
JOHN H. HUBBARD
fibers X(s)
= 77 1 (s)
of dimension 1 ). A relative projective atlas on X
isarelativeatlas (Ui,ai) wherethe Ui formanopencoverof X, and the ai : Ui .... P 1 are analytic maps where restrictions to fibers of isomorphisms onto their images, such that over each s
f
77
are
S, the pair
(Ui(s),ai(s)) is a projective atlas on X(s). As above, two relative projective atlases are equivalent if together they still form a projective atlas, and a relative projective structure on X is an equivalence class of relative projective atlases. REMARK. To say that a family of projective structures a(s) is induced by a relative structure is to say that a(s) depends analytically on s. EXAMPLES. The family of projective structures obtained by applying the uniformization theorem fiber by fiber does not define a relative projective structure: the normalization requiring the images of the universal covering spaces to be the upper half plane cannot be made analytic. The generalization of the uniformization theorem given by Bers [1] does give relative projective structures on the universal curve over Teichmuller space. The canonical family of projective structure on P(X)x X > P(X) is induced by a relative projective structure. Let P s(X) be the set of pairs (s, a) such that s
f
S and a is a
projective structure on X(s). PROPOSITION 1. (i) There is a unique structure of a complex manifold on P s S given by (s, a)
1>
s is
analytic, and analytic families of projective structures on X given by section of p are induced by relative projective structures on X. (ii) The action of Qs(X) on P s(X) over S given by
((s, q), (s, a))
1>
(s, q+a) makes P s(X) into an analytic affine bundle over
S, under the analytic vector bundle Qs(X). Proof. The proposition is clearly local in S. Suppose a relative projec
tive structure a on X can be found (even locally over small subset of S ).
263
THE MONODROMY OF PROJECTIVE STRUCTURES
Then the map (s, q)> (s, q +a(s)) is a bijection Qs(X) .... Ps(X). Give P sCX) the induced structure; with this structure, P s(X) clearly satisfies (ii). To see that it satisfies (i), we need to know that if q is a section of Qs(X), then the family of projective structures induced by a(s) + q(s) is induced by a relative projective structure. Let (Ui,ai) be a relative projective atlas defining a. equation S(ai(s), /3i)
=
On Ui(s), the
q(s) is an analytic differential equation of third
order depending analytically on s, whose solutions will exist in Ui(s) by Lemma 3, (iv) if the Ui are sufficiently small, and will depend analytically on s if initial conditions are chosen analytic in s . But this can be done, for instance by picking (locally in S) a section S .... X of
TT
and requiring /3i(s) to coincide with ai(s) to order 2 along the section. Clearly the /3i form a relative projective atlas with the desired properties. Thus we are left with showing that over sufficiently small open subsets of S, X carries a relative projective structure. This may be shown by appealing to the universal property of Teichmuller space and the simultaneous uniformization theorem of Bers [1 ]; we shall prove a slightly more general result. LEMMA
5. Let
TT:
X .... S be a proper and smooth family of Riemann sur
faces of genus at least 2, with S a Stein manifold. Then X admits relative projective structures. Proof. Let (Ui, H ( ,TT*~£x;s)
and the first term is zero because H 1 (X(s), n the last term is zero because S is Stein.
®2 ) =
0 by RiemannRoch;
264
JOHN H. HUBBARD
Therefore refining the cover if necessary, we may assume that there are sections qi of n~5s over ui such that
Solutions ai of the differential equations S(ai, ¢i)
=
qi chosen so as to
satisfy some analytic initial condition such as to agree with ¢i to order 2 along some section S . U i of
TT,
will then form a projective atlas for
X , perhaps after further refining the cover to make them injective on
Q.E.D.
fibers. COROLLARY.
(i) The family of projective structures on the family of
Riemann surfaces p*X> P 5 X which is a on the fiber over a induced by a canonical relative projective structure a 5 (X). (ii) The space P 5 X has the following universal property:
f
P 5 X is
The map
which associates to any analytic mapping f: T > P 5 (X) the projective structure f* a 5 (X) on the family of Riemann surfaces (p of )*X is a bijection of Mor (T, P 5 (X)) onto the set of relative projective structures on (pof)*X. The proof is left to the reader.
4.
Infinitesimal deformations and Eichler cohomology In this paragraph, we shall carry out an infinitesimal deformation
theory for projective structures analogous to the KodairaSpencer theory for complex structures. Let U be a Riemann surface, a be a projective structure on U and
x
an analytic vector field on U. Choose a oneparameter family of maps
cPt : U > U with ¢ a in the direction
0
= id and ¢ 0 = x, and define the Lie derivative of
x
THE MONODROMY OF PROJECTIVE STRUCTURES
265
Clearly L)((a) is an analytic quadratic form on U, and we leave it to the reader to prove that if l; is a projective coordinate on U and
x
=
x(l;)iL,
at;
then L (a)= xm(l;)d£; 2
x
.
In particular, the Lie derivative
does not depend on the family rPt that was chosen. Let Aa be the subsheaf of the sheaf of germs of analytic vector fields which is the kernel of the morphism 1/J
f>
L)((a); we then obtain an
exact sequence of sheaves
0. Aa . TU
~
0® 2

0
which will be important; TU stands for the sheaf of germs of vector fields on U (as opposed to the tangent bundle TU ), and the Lie derivative is surjective because of the formula that computes it in a projective coordinate. REMARKS. In a projective coordinate l;, sections of Aa are exactly those vector fields which can be written p(l;)iL with p a polynomial of
at;
degree at most two; such vector fields are called infinitesimal automorphisms of a because the flows they generate send a to itself. The sheaf Aa is locally constant of rank 3, an example of what topologists call a local system. In particular, it may be thought of as the sheaf of germs of sections of a covering space, with fiber isomorphic to
C3 with the discrete topology. Let X
>
S a family of Riemann surfaces, s 0 ( S and X 0 the surface
above it. Suppose a is a relative projective structure on X which induces the projective structure a 0 on X0 . We shall describe a linear map Ts S> H 1 (X 0 ,A of a.
o
ao
) which measures the infinitesimal deformation
Let (U i• ai) be a relative projective atlas on X; by restricting S and refining the cover
'U
= {Ui I we may assume that for each i and all
s ( S the maps ai(s): Ui(s) .... P 1 are homeomorphisms onto some open set ViC P 1 . Define c/Ji(s) = ai(sr 1 0 ai(s 0 ) and c/Ji,j(s) = c/Ji(s) 1 ocjJj(s),
266
JOHN H. HUBBARD
cb·1,)·(S) is defined in an open subset of u1. n UJ· which will in
where
elude any given point for s sufficiently near s 0 . The maps cbi,j(s) satisfy the following two identities:
cb·1,)(s)oc/;J.J' k(s) = cb·1' k(s)
the first obviously and the second because (Ui, ai) is a relative projective atlas. Since cbi,j(s 0 ) is the identity map Ui n Uj ... Ui n Uj, the derivative ds 0cbi)v) = Xi)v) is a vector field on Ui
n Uj
for any v
f
Ts 0 S, and
the derivatives of the identities above give:
x·1,)·(v) + x·J' k(v) = x·1' k(v) L x·. (a 0 ) = 0 . 1,)
The first identity says that x(v)
=
lxi,j(v)l is a cocycle in
C 1 ('U, T X 0 ), and the second that it is in fact in C 1 ('U, A
ao
infinitesimal deformation of a at s 0
by ds (a)(v) 0
=
) . Define the
the cohomology class of x(v). We leave it to the reader .
to prove that the class does not depend on the projective atlas that was chosen. The map ds (a) has the following properties: 0
(i) It commutes with change of basis, i.e., if f: T ... S is a map with f(t 0 )
=
s 0 and we give f*X the relative projective structure £*a, then
dt (f* a) 0
=
ds (a) odt f. 0
0
(ii) The map i* ods a: Ts S > H 1 (X 0 , T X 0 ) obtained by composing 0
0
ds 0a with the map H 1 (X 0 , Aa 0 )> H 1 (X 0 , TX 0 ) induced by the inclusion
THE MONODROMY OF PROJECTIVE STRUCTURES
267
Aa C T X 0 is the KodairaSpencer map classifying the deformation of the 0
complex structure of X at x 0 . (iii) Let X 0 , a 0 be any compact Riemann surface with a projective structure, and let a be the canonical relative projective structure on P(X 0 )xX 0 .P(X 0 ).
Then Ta 0 P(X 0 )=Q(X 0 ), and da 0 a:Q(X 0 ).
H 1 (X 0 , Aa ) is the "connecting homomorphism" coming from the long 0
exact sequence associated to the short exact sequence (1). Part (i) follows immediately from the construction, (ii) is clear since the cocycle x(v) is a definition of the KodairaSpencer map [7], (iii) is a computation we shall leave to the reader. Now let us examine the universal case; let M be a compact surface of genus at least 2 , TT:
EM >eM
eM
the Teichmiiller space modelled on M and
the universal Teichmuller curve. We shall omit the subscript
M in the sequel.
Let p:
PeE. e be the canonical projection and give p*E the
canonical relative projective structure a given by the corollary to Proposition 1.
2. Let 8 0 be a point in
PROPOSITION
surface above it and a 0
f
P(X 0 ).
e,
X 0 = rr 1 (6 0 )
the Riemann
Then
is an isomorphism. Proof. Consider the diagram
0 T
ao
P(X 0 )
0
268
JOHN H. HUBBARD
The top line is induced by the inclusion of P(X 0 ) as the fiber of p above 80 (cf. Prop. 1), the bottom line is extracted from the long exact sequence associated to the short exact sequence (1 ), the left vertical map is the isomorphism of the corollary to Lemma 4 and the right vertical map is the KodairaSpencer isomorphism. The lefthand square commutes by properties (i) and (iii) of d
and the righthand square by property (ii).
a
ao
Q.E.D.
The proposition now follows from the five lemma.
REMARK. Proposition 2 identifies the tangent space to the space PeE of "all projective structures on all Riemann surfaces" exactly in the same sense that the KodairaSpencer isomorphism identifies the tangent space to the space 0 of "all Riemann surfaces" as Te 0 = H 1 (X 0 , TX 0 ). 0
5.
The space Hom (1, G) Let 1
be the fundamental group of a surface, given by generators
I= lal' ... ,a 2 g! subject to the one relation g
II [ai, ai+g]
1.
i=l
Clearly the set Hom (1, G) may be identified with the subset of G 2 g defined by the analytic equation f
=
1 where f : G 2 g > G is given by
g
f(a1 , ... ,a2 g)
=
H [ai, ai+g]. This gives Hom(1,G) the structure of an 1=1
analytic space. LEMMA 6. With this analytic structure, Hom {1, G) has the following universalproperty: foranyanalyticspace S, morphisms S>Hom(1,G) correspond bijectively to morphisms S x 1 > G which are analytic, and whose restrictions to ls ~
X
REMARK. In this lemma 1
1
are group homomorphisms for all
S t
S.
is considered as a discrete analytic space.
The proof is trivial and left to the reader. In particular, the analytic
269
THE MONODROMY OF PROJECTIVE STRUCTURES
structure on Hom (r, G) does not depend on the chosen presentation. We may therefore expect that there is an intrinsic description of the local structure of Hom (r, G), in particular of its tangent space, etc. The object of this paragraph is to give such a description. Let Hom*{r, G) C Hom (r, G) be the open set of representations with noncommutative image. For any representation p: r .... G, we may consider g as a rmodule by y ·
t =Ad p(y)(c;};
we shall denote this rmodule gp. Recall [6]
that a derivation 0: r .... gp is a map satisfying o(y1y2) = o(y1)+y1. o(y2) and that those derivations of the form
ot (y) = t y · t
are called principal
derivations. Call Der(r, gp) the space of derivations and IDer(r, gp) the subspace of principal derivations. A classical description of H 1(r, g P) is Der (r, g p)/IDer (r, g p); this is the description we shall use. The tangent space to G at any a is g (in two different ways); we shall use the local chart g . G of G near a given by PROPOSITION 3.
t 1> exp (t) a.
The space Hom*(r, G) is a submanifold of G 2g.
For any p (_ Hom*(r, G) the map Der (r, gp) .... g~g given by
o 1> oil
is an isomorphism of Der{r, gp) onto TPHom{r,G). Proof. A computation which begins
)(1 )( )(1+)( 2 2 e 2 =e +0(ix 1 \2 +1x 2 1) showsthatthe derivative of f at a= (a1 , ... , a 2 g) f G 2 g in the direction ~ = (t1 , ... , t 2g) f g 2 g is and ends using e
daf(~) =
l
g
n i1
i=1 j=1
[aj, aj+g]. ((1 aiai+gai 1). ti + (ai [ai, ai+g]). ti+g).
270
JOHN H. HUBBARD If a= PI~ for some p
E
Hom (r, G), essentially the same computation
shows that d 0 f(l;) = 0 is the necessary and sufficient condition for
I;: ~ .... g P to extend (obviously uniquely) to a derivation 1 Thus all we need to prove is that d 0 f: g ~g p
E
>
>
g P.
g P is surjective if
Hom *(1, G). The basic fact (left to the reader) is that if r 1 , r 2
not commute, the linear map gpx gp> gp given by ctl,t'2)
!>
E
G do
(1rl)t"l +
(1r 2)t" 2 is surjective. This result gets applied twice. First suppose that for some i, 1
S i S g,
a i and a i+g do not commute. Then since
the images of the t"i and t"i+g already fill out the image of d 0 f. If each a i commutes with a i+g the expression for the derivative of
f simplifies to
2,
g
((1 a i+g) · t"i +(a i 1) · t"i+g)) i=l
and the result is clear since for some i , j , a i and a j do not commute.
Q.E.D. REMARK. This result is a special case of the following more general results: If 1
is a group of finite presentation, G is a Lie group with Lie
algebra g, then Hom (1, G) is an analytic space, its Zariski tangent space at p is Der (r, gp) and the equations defining locally Hom (1, G) in Der(f', gp) may be chosen to have values in H 2(1, gp). In our case, this boils down to the fact that if the image of p is not commutative, H 2(1, gp)
=
0, which may be proved by Poincare duality.
PROPOSITION 4. (i) The group G acts freely on Hom*(r, G), and the quotient Hom*(I', G)/G has a unique structure of an analytic manifold such that the projection Hom*(!, G)> Hom*(!, G)/G is analytic.
THE MONODROMY OF PROJECTIVE STRUCTURES (ii) For any p
f
271
Hom*{r, G), the derivative of the inclusion
G .... Hom*(r, G) given by a r. a
g P .... Der (r, g P) given by
~
o
p
o
a 1 at 1 ( G is the map
.... 8 ~. In particular the tangent space to
Hom*{r, G)/G at the image of p is canonically isomorphic to H 1 {r, gp).
Proof. The fact that G acts freely follows from the fact that commuting is an equivalence relation on nontrivial elements of G. This may for instance be seen by observing that a 1 I= 1 and a2
I= 1 commute if and
only if they have the same fixed points in P 1 . Similarly, if p 1 and p 2 are not conjugate, they have neighborhoods U 1 and U 2 such that no element of U 1 is conjugate to an element of U 2 , since the fixed points of a nontrivial y
f
G vary continuously with
y. Therefore the graph of the equivalence relation is closed, and the
quotient is Hausdorff. The existence and uniqueness of the analytic structure follows from the analyticity of the action of G, and the derivative in (ii) is computed from
6.
e~pe~ = e~p·~p+O (1~1 2 ).
Hejhal's theorem Let
17:
X .... S be a family of Riemann surfaces with a relative projec
tive structure a. Suppose S is contractible and let r = If
Q.E.D.
17
17 1 (X)= 17 1 (X(s)).
admits analytic sections, there are relative developing maps
f: X .... P 1 , i.e., analytic maps which, restricted to X(s), are developing maps of a(s). This is just a matter of picking an analytic normalization, for instance requiring that f should agree to order 2 with a relative analytic chart along a section. REMARK. The requirement that
17
admit analytic sections is too strin
gent. In fact, there are relative developing maps if S is contractible and Stein. Indeed, the space of all developing maps of the X(s) forms a principle analytic bundle under G over S, and so is trivial by Grauert's theorem if S is Stein and contractible. This applies in particular to the universal family over
PeS .
272
JOHN H. HUBBARD Clearly if f: X_, P 1 is a relative developing map, the associated
Pf: S _,Hom (r, G) which associates to each s
E
S the monodromy homo
morphism of f(s) , is analytic. Let F : PeS _, Hom (r, G)/G be induced by the above construction, for the universal family of projective structures parametrized by PeS. REMARK. The global existence of F does not require Grauert's theorem, because we have divided by the action of G. It does require the contractibility of PeS, so that a universal covering space
;*'E
induces a
universal covering space p*S over each fiber of p*S _,PeS. THEOREM.
The map F is an analytic local homeomorphism.
The fact that F is analytic follows from the fact that F lifts locally (and even globally by Grauert) to an analytic map PeS_, Hom (r, G). By the corollary to Lemma 2, the image of a monodromy homomorphism is never commutative, and so both the range and the image of F are manifolds, whose tangent spaces we know. PeS be a projective structure on X 0 ; let f: X 0 _, P 1 be a developing map for a 0 and p: r _, G its monodromy homomorphism. Let a 0
f
The theorem will now follow from
Approachable Approachable isomorphism. The two tangent spaces look similar; they are in fact canonically isomorphic by the classical theorem of algebraic topology which says that one way to compute the cohomology of a group r
with values in a
rmodule is to compute the cohomology of a K(r, 1), with coefficients in the associated local system in the sense of the following lemma. LEMMA7.
and the map
Thegroup r
X0 x
gp
f>
acts on
Xox
gp by y·(x,t)=(y·x,p(y)*O'
Aa 0 given by (x, t)
isomorphism on the quotient.
f>
(u(x), u*f*t) induces an
THE MONODROMY OF PROJECTIVE STRUCTURES
273
The proof is immediate and left to the reader. We cannot unfortunately use the canonical isomorphism without explicitly constructing it. There are many ways to do this; the one we shall use here is adapted to our knowledge of the two spaces, one via Cech cocycles and the other via derivations. It is possible to compute Cech cohomology using a generalization of an open cover: an etale cover. The "open sets" are manifolds Ui and immersions Ui >X, whose images are required to cover X. The intersections Ui
n Uj
must be replaced by the fiber products Ui XX Uj, and
similarly for multiple intersections. Moreover, Leray 's theorem still applies: if the Ui as well as all their fiber products are cohomologically trivial (for whatever sheaf we may be considering) the Cech cohomology for that cover is the cohomology of the sheaf (either in the Cech sense of direct limit over all covers, or via resolutions, or whatever, which are all isomorphic). We shall apply this to the cover consisting of a single open set X > X. The map
X Xx X Xx X X .•.
Xx
X=
X
X
rn given by (x,
yl, ... ,
Yn)
I>
(x, y 1 (x), · · ·, Yn(x)) and the identification of Lemma 7 give isomorphisms cn(x, X; Aa)
=
g~n. In fact the complex is the classical inhomogeneous
bar complex [6] whose first two differentials are
In particular, the kernel of d 1 is formed of the derivations r .... gp and the image of d 0 is formed of the principal derivations. In our case, X 0 is a K(r, 1) so Leray's theorem applies to guarantee that the cohomology of the complex is in fact H 1 (X 0 , Aa ) . 0
LEMMA
6'. The derivative d
Approachable
F is the isomorphism H 1(X 0 , A
ao
ao
) ....
274
JOHN H. HUBBARD
Proof. Choose an analytic curve a(t) in PeE; let ft be a relative developing map and Pt the corresponding monodromy homomorphisms. Define (as in the construction of §4) a family of analytic maps cPt : Ut .... X(t) which: a) are analytic isomorphisms onto their images, and analytic in t ; b) are defined in subsets Ut
C


X 0 which fill out X 0 as t becomes
small;

c) satisfy f 0 =ftocPt in Ut, and c/Jo=identityof X 0 . Then Jt a(t)'t=O is represented by the Cech cocycle for the cover X 0 which is, on the component .fy = fft (c/Jtl
X0 x lyl
of
X0 xx
oyoc~Jt)'t=O.
0
X 0 , given by
Using ft o y = Pt(y)ft the expression above may be written .fy = Jt (f 01 0 Pt(y)of 0 )'t=O, where the entire expression f 01 opt(y)of 0 is defined in ut. If we write fft Pt(y)lt=O = .f ~ (it is best to think of .f ~ as a vector
field on P 1 ), then differentiating the expression above gives .fy = f~,f~. This is the identification of Lemma 7.
Q.E.D.
REMARKS. Some obvious questions, unsolved to the author's knowledge, are: What is the image of F? What do the fibers of F look like, and their projections in TeichmUller space? It is known [8] that F is not injective, but it is injective on fibers [3]. BIBLIOGRAPHY [1] L. Bers, Simultaneous Uniformization, Bull. A.M.S. 66 (1960), 9497. [2] C. Earle, On Variations of Projective Structures, these proceedings. [3] Gunning, R. C., Lectures on Riemann Surfaces, Princeton University Press, 1966. [4] ____ ,unpublished notes.
THE MONODROMY OF PROJECTIVE STRUCTURES
275
[5] Hejhal, D. A., Monodromy Groups and Poincare Series, Bull. A.M.S. 84 (1978), 339376. [6] Hilton, P. J. and Stammbach, U., A Course in Homological Algebra, SpringerVerlag, 1971. [7] Kodaira, K. and Spencer, T., On Deformations of Complex Analytic Structures, I and II, Ann. Math. 67(1958), 328466. [8] Maskit, B., One Class of Kleinian Groups, Ann. Acad. Sci. Fenn. 442 (1969).
HOLOMORPHIC FAMILIES OF RIEMANN SURF ACES AND TEICHMOLLER SPACES Yoichi Imayoshi
Introduction Let @5 be a two dimensional complex manifold. Denote by D the unit disc ltl < 1 and by D* the punctured disc 0 < ltl < 1 in the complex tplane. We assume that a proper holomorphic mapping rr: @5
>
D*
satisfies the following two conditions; for every t < D*, i)
the fiber St
=
rrl (t) over t is a onedimensional, nonsingular
irreducible analytic subset of @5, and ii) the rank of the Jacobian of rr is equal to one at each point of St. It is well known that the fiber space (@5, rr, D*) is differentiably locally trivial. Therefore, every fiber St has the same genus g as a Riemann surface. We shall always assume that g
~
2.
We call the triple (@5, rr, D*) satisfying above conditions a holomorphic family of compact Riemann surfaces of genus g. A fiber St satisfying above two conditions i) and ii) is called ordinary. It should be noted that, in general, the fiber over t
=
0 cannot be considered.
Further, let @5 be a twodimensional complex analytic space and
rr: @5 S
>
D be a proper holomorphic mapping. Assume that the fiber
0 = 17 1 (0)
of
6
over t
=
0 is a onedimensional compact analytic subset of
and that the triple
(SS 0 ,771SS 0 ,D*)
is a holomorphic family
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00027724$01.20/1 (cloth) 0691082677/80/00027724$01.20/1 (paperback) For copying infonnation, see copyright page 277
278
YOICHI IMA YOSHI
as stated above. In this case when the fiber
S0
is not ordinary, we call
the triple (~, 17, D) a holomorphic family of compact Riemann surfaces of genus g with a singular fiber over t = 0. In this paper, given a holomorphic family (~, rr, D*), we regard the fiber St over t (tED*) as a point in a Teichmuller space and we try to construct the fiber over t = 0 from the limit point of
l St It ED*
in the
Teichmiiller space when t tends to zero and try to construct a holomorphic family (~, 17, D) with or without a singular fiber over t can be considered as a natural extension of
=
(~, rr, D*). This
0, which
(S, 17, D)
will be called a completion of (~, rr, D*). Here the concept of the homotopical monodromy an essential role. t
=
m of (~. TT, D*), which will be defined later, plays If m is trivial or of finite order, then the fiber over
0 can be constructed by a compact Riemann surface without nodes of
genus g. On the other hand, if over t
=
m is of infinite order, then the fiber
0 can be constructed by a compact Riemann surfaces with nodes
of genus g. A precise description of the fiber t
=
0 is given in
Theorems 1, 2, and 4 in §§3, 4, and 6. More properties of the completion of (~,
TT,
D*) are given in Theorems 5 and 6.
Nishino [16] has already discussed the quite similar problem to construct the fiber over t
=
0 and obtained the same results as ours without
using the theory of Teichmiiller spaces. The author would like to express his hearty gratitude to Professor Kuroda for his constant encouragement and advice and wish to express his thanks to my colleagues Yamamoto and Sekigawa for immeasurably profitable conversations with them. The author is also indebted to Professor Bers for pointing out some errors in the original version of this paper. §1. Preliminaries First we introduce terminologies and notations which will be used throughout this paper. These are due to Ahlfors [3], [4] and Bers [6], [7],
[8].
279
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES A
Let C be the complex plane and let C be the extended complex plane. We denote by U and L the upper and the lower halfplanes in C, respectively, and by R the real axis in C . Let SL'(2; C) be the set of all complex Mobius transformations of the form
g.. z
1>
az+b b ,c, d f C d,a,
and
CZ+
ad be
=
1 ,
and let SL'(2; R) be the set of all g ( SL'(2; C) whose coefficients are all real. Let us denote by G a Fuchsian group acting on U and by Qnorm the group of all quasiconformal automorphisms w of U satisfying normalization conditions w(O)
=
0, w(1)
=
1 and w(oo)
=
oo. Here we
note that any w f Qnorm can be extended to a hqmeomorphism of U onto itself, where U is the closure of U in C. We set Qnorm(G)
=
lw fQnorm I wGw 1 CSL'(2; R)l.
Let L 00(U) be the complex Banach space of (equivalence classes of) bounded measurable functions on U and let L ""(U, G) be the closed linear subspace of L 00 (U) such that every tJ.(g(z))g'(z)/g'(z)
=
tJ.(z)
fJ.
f L ""(U, G) satisfies
for every
g f G.
We denote by L ""(U) 1 the unit ball in L ""(U) and put L 00 (U, G) 1 L ""(U, G) n L ""(U) 1 .
=
Furthermore, we mean by B/L, G) the set of all holomorphic functions c/> defined on L such that c/>(g(z)) g'(z) 2 = c/>(z)
for every
gf G
and such that its norm llc/>11
sup y 2 \c/>(z)i,
z
=
x+iy
Z(L
is bounded. The space B/L, G) is the complex Banach space of holomorphic quadratic differentials in L with respect to the Fuchsian group G.
280
YOICHI IMA YOSHI
Now we can define Teichmtiller spaces in three ways. 1) Given a reference compact Riemann surface S of genus g(;;2), a marked Riemann surface with respect to S is a compact Riemann surface S' of genus g with an orientationpreserving topological mapping f: S
>
S', which we call the marking on S'. We denote this marked sur
face by (S, f, S'). We define an equivalence relation between marked surfaces by calling two marked surfaces (S, f, S') and (S, g, S") homotopically equivalent if and only if there exists a conformal mapping h such that the diagram
s    f   s ' h
g
S" commutes up to homotopy, that is, the selfmapping g 1 oh of: S .... S is homotopic to the identity. We denote by [S, f, S'] an equivalence class of marked surfaces and by T(S) the set of all those equivalence classes, which is the Teichmiiller space of S. 2) Let G be the finitely generated Fuchsian group of the first kind acting on U induced by the universal covering group of the compact Riemann surface S of genus g. Two elements w 1 and w 2 in Qnorm are called equivalent if w 1 = w 2 on R. The equivalence class of w
f
Qnorm is denoted by [w]. The Teichmtiller space T(G) of the
Fuchsian group G is the set of all equivalence classes [w] of elements w
f
Qnorm(G) · 3) The Teichmtiller space T(G) with its complex structure can be
regarded canonically as a bounded domain in the complex Banach space B 2 (L, G) in the following way: For every 11
f
L 00 (U) 1 , there is a unique
quasiconformal automorphism w of C with w(O)
=
0 , w(l)
=
1 , w( oo) = oo
such that w has the Beltrami coefficient 11 on U and is conformal on L.
HOLOMORPHIC FAMILIES OF RIEMANN SURF ACES
281
We write w = wl1 and denote by ¢ 11 the Schwarzian derivative of wl1 in L, that is,
Nehari's theorem shows that ¢ 11 < B 2 (L, I) and
II¢)~ ;S ~, where I is
the group consisting of only the identity mapping of C onto itself. As is well known, wi11L,wlliR and ¢ 11 depend only on [w 11 ], where w 11 is the quasiconformal automorphism of U with the Beltrami coefficient 11 and w(O)=O, w(1)=1, w(oo)=oo. If 11 ¢ 11
11
is a biholomorphic bijection
of T(G) onto a holomorphically convex bounded domain in B 2 (L, G) containing the open ball of radius 1/2. From now on, we will identify T(G) with its canonical image in B 2 (L, G). Note that T(S) is canonically biholomorphic to T(G). Therefore, we can also identify T(S) with T(G). For every holomorphic function ¢
on the lower halfplane L, the
Schwarzian differential equation
Iw, z l  (w "! w ')'  12 (w "! w ')2
(1.1)
has a meromorphic solution defined in L. If 77 1(z) and 11iz) are two linearly independent solutions of the linear differential equation 27]"(z) + ¢(z)7](z) = 0,
(1.2)
then w(z) = T/ 1 (z)/TJ 2 (z) is a solution of (1.1) and every solution of (1.1) can be obtained in this way. Let ¢: o* > T(G) defined by (t) = q;t
IS
ana
lytic. In general, this mapping 1:1> is not singlevalued and it depends on the homotopical monodromy
§3. The case where

m is
m of
(@5,
77,
D*).
trivial.
First we prove the following.

m is
THEOREM 1. If the homotopical monodromy
trivial, then the map
ping 1:1>: D* _, T(G) is singlevalued and it has the holomorphic extension 1:1>: D> T(G),
so there exists a nonsingular compact Riemann surface S 0
of genus g corresponding to the point 11>(0) of T(G). Further, a holomorphic family (@5,
rr, D)
of compact Riemann surfaces
of genus g can be constructed canonically in such a way that (i)
@5 is a twodimensional complex manifold,
(ii)
the fiber St
=
;r 1(t) is ordinary for every t in D, and
(iii) the fiber St is conformally equivalent to St for every t in D*
and @5 is naturally isomorphic to
6 S0
so that the following diagram
is commutative: ~
@5
"j
D*
@5
id
·J
D
Proof. First, by a result of Earle [10], T(G) is complete with respect to its Caratheodory distance. Therefore, the holomorphic mapping 1:1>: D* > T(G) has the holomorphic extension
: D . T(G).
285
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES
Next we prove the second assertion of our theorem. We set
I = I (t, z )j t f D, z f DcPt I. Denote by g the group of analytic automorphisms of I
consisting of g(t, z) = (t, (x cPt (g))(z)) for all elements g
in G . Then the group g acts on I
properly discontinuously and fixed
~
~
point freely. Hence, if we set S = I/g, then S
is a twodimensional
complex manifold with the canonical complex structure. Denote by [t, z] ~
the point in S
represented by (t, z) ( I. The mapping
rr:
~
S .... D ,
which carries [t, z] into t, is holomorphic and the rank of Jacobian of
rr
is equal to one at each point of
S.
The fiber St =
rr 1 (t)
over t is
ordinary for every t ( D and St is conformally equivalent to St for every t phic to
D* by its construction. Furthermore, S is naturally isomor
f
S  S0
so that the diagram in Theorem 1 is commutative.
§4. The case where

:ffi is of finite order.
In this section, we prove the following. THEOREM

2. If the homotopical monodromy :ffi is of finite order, then
for the mapping 0
T>0
lim cPt exists in T(G) and it coincides with bO
~
c/J 0 =lim cPt is a fixed point of t>0
t/1 0 .
It is obvious that
m by Lindelof's theorem. ~
Next, we will construct a holomorphic family (S, ii, D) satisfying the second assertion of the theorem. We put
and
~here
T f
m= Dt/1
en(r)
!J., z
f
Dt/lr' g
f
G, n
f
Z, en(r) = (exp en:))
T,
w
f
and [wn o g]*(z) is the conformal bijection of Dt/1
induced by wn
o
N(G) with onto
T
g as (1.4). Then gn is an analytic automor
phism of ~ and the group g of all these automorphisms acts properly discontinuously on ~. Now we recall the canonical ringed structure of the quotient of a bounded domain ~ in CN by a discrete subgroup [' of Aut(~), where Aut~) is the group of analytic automorphisms of ~ . Then [' acts on
~ in properly discontinuous fashion and the orbit space X
=
~/1 with
the canonical quotient topology is a locally compact Hausdorff space. Let 11:
~
>
X be the canonical projection. We define a ringed structure ~ on
X as follows: If V is an open subset of X and if f is a complexvalued continuous function on V , then f
f
~V if and only if f
o 11
is
287
HOLOMORPHIC FAMILIES OF RIEMANN SURF ACES
holomorphic on rr 1 (V) C ~. It is well known that the ringed space (X,~) defined as above is a normal complex analytic space. Therefore, if we introduce the above canonical ringed structure on ~
6
~
=
I/g, then 6 is a twodimensional normal complex analytic space.
The projection 17: 6 ... D sending [r, z] into r P is holomorphic and the fiber st
=
n: 1(t)
is ordinary and conformally equivalent to st for every
t < o*.
If S 0 is the Riemann surface corresponding to the point ¢ 0 in T(G) and if g 0 is the conformal automorphism group of D¢ 0 induced by g, then g 0 induces an automorphism group ro of so and the fiber
§0
= ;  1 (0)
is isomorphic to S 0 /r0 .
Moreover, by its construction, 6
is analytically isomorphic to
~ S0 such that the diagram in Theorem 2 is commutative. §5. Deformation spaces of Riemann surfaces with nodes To consider the case where
:lR is of infinite order, we will first ex
plain deformation spaces of Riemann surfaces with nodes, which is due to Bers [9]. A compact Riemann surface with nodes of (arithmetic) genus g C;:;: 2) is a connected compact onedimensional complex analytic space S, on which there are k
=
k(S)
(~
3g3) points P1 , ... ,Pk, called nodes,
lj has a neighborhood isomorphic to lz 1z2 = Ollz 1 1< 1, lz 2 1< 11 with Pj corresponding to
such that (i) every nodes
the ana
lytic set
(0, 0),
(ii) the set S 
I P1 , .. ·, Pk I
has r (;:;: 1) components I
1 , .. ·,
Ir called
parts of S , where each Ii is a Riemann surface of genus gi , compact except for ni punctures, with 3gi  3 + ni:;;; 0 and n 1 + · ·· + nr
=
2k,
and (iii) g = (g 1 1)+ · .. + (gr1) + k+ 1. The condition (ii) implies that every part carries a Poincare metric and the condition (iii) is equivalent to the requirement that the total Poincare area of S equals 4rr(g1). From now on, g ( ~ 2) is fixed and the letter S , with or without subscripts or superscripts, always denotes a surface with properties (i) (iii).
288
YOICHI IMA YOSHI
If k(S) = 0, then S is called nons in gular; if k(S) = 3g 3 , then S is called terminal. A continuous surjection a: S'> S is called a deformation if, for every node P
f
S, a 1 (P) is either a node or a Jordan curve avoiding
all nodes and if, for every part ~ of S, a 1 1 ~ is an orientationpreserving homeomorphism. Once and for all we choose an integer v(> 3) which will be fixed throughout the following discussion. Two deformations a: S'. S 0 and {3: S"> S 0 are called equivalent to each other, if there exists a homeomorphism f of S' onto S" such that a = {3 of and if f is homotopic to a product of vth powers of Dehn twists about Jordan curves mapped by a into nodes, followed by an isomorphism. We denote by
=
< S', a, S 0 > the equivalence class of a
deformation a: S'> S 0 . Given a compact Riemann surface S 0 with nodes of genus g, the deformation space X(S 0 ) consists of equivalence classes < S', a, S 0 > of all deformations a: S' .... S 0 . To every node P
f
S 0 , there belongs a distinguished analytic hyper
surface
(CX(S 0 )) consisting of all f X(S 0 ) such that a 1 (P) is a node of a 1 (S 0 ). If P 1 , ... , Pk are all nodes of S 0 , we denote by X 0 (S 0 ) the set X(S 0 )  < P 1 > U .. · U < Pk >, that is, X 0 (S 0 )
=
I< S', a,
S 0 >1S' is nonsingular!.
Every deformation {3 : S 0 .... S 1 induces an allowable holomorphic mapping {3*: X(S 0 )> X(S 1 ) which sends
f
X(S 0 ) into
f
X(S 1 ).
Let r(S 0 ) induced by all topological orientationpreserving selfmappings of S 0 and let
r 0 (S 0 )
be its subgroup induced by all automorphisms of
S 0 . Then the group r(S 0 ) is discrete and the subgroup r 0 (S 0 ) is finite
and is the stabilizer of < id > ( X(S 0 ) in r(S 0 ). Let Rg be the Riemann's moduli space of nonsingular compact Riemann surfaces of genus g and let Mg be the moduli space of compact Riemann surfaces with nodes of genus g. Let TT: T(G) .... Rg and II: X(S 0 )> Mg be the canonical projections. Now we can prove the following lemma.
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES LEMMA 1.
The holomorphic mapping
j :D
has a holomorphic extension
J : D*
289
> Rg sending t into [St]
>Mg.
Proof. To prove this lemma, we use the following theorem (Kobayashi
[12], Kobayashi and Ochiai [13]). Let :J) be a complex space and
THEOREM.
group of holomorphic transformations of
:D
r
a properly discontinuous
such that M = :D;r is an
open subset of a complex space Y . Assume that (1) the pseudodistance dM' is a distance;
(2) the closure
M of M in Y is compact;
(3) given a point p (
aM
and a neighborhood U of p zn Y, there
exists a smaller neighborhood V of p in Y such that
Let X be a complex manifold and A a locally closed complex submanifold of X . Then every locally liftable holomorphic mapping f : X A > M extends to a holomorphic mapping
f:
X > Y .
We apply this theorem to the case where :J) Y
=
Mg and f
=
J.
=
T(G),
r
=
Mod (G),
So we have to see that three conditions (1), (2) and
(3) in the theorem are satisfied in this case. If p, q
f
Rg and p, q
f
T(G) such that rr(p)
where the infimum is taken over all p
f
=
p and 11(q)
=
q, then
T(G) satisfying 11(p) = p. Since
T(G) is a bounded domain in C3 g 3 and since Mod (G) acts properly discontinuously on T(G), the pseudodistance dM' is a distance. Since Mg is compact, the condition (2) is satisfied obviously. Given a point [S 0 ] on aRg and a neighborhood U of [S 0 ] in Mg, there is a neighborhood N of
< id > in X(S 0 ) such that
N is stable
under r 0(S 0) and such that N/r0(S 0) is a neighborhood of [S 0 ] in Mg.
290
YOICHI IMAVOSHI
n N = ¢ for every y t r{S 0) Take a neighborhood N 0( «; N) of < id > in
(See Bers [9].) We may assume that y(N)
f'0(S 0 ) and U = NIf'0(S 0 ).
X(S 0) such that N 0 is stable under f'0(S 0) and such that V = N 0 /f'0(S 0 ) is a neighborhood of [S 0 ] in Mg. Let a: S > S 0 be a deformation and let a*: T(G)> X(S 0 ) be an allowable mapping which sends [S, f, S'] into 0 and small, then there exists a unique loxodromic
Mobius transformation hi s. , which conjugates the r'i into r'i, has '
1
(j
the multiplier si and has fixed points in ~j and ~e
and
e being
as before). Set s = (s 1 , ···, sk). If lsi =max lsi I is small, the group H 0 s generated by H and by all elements hiS· with si '
'
Kleinian group.
I= 0 is a
1
Let s be as before and let W be a quasiconformal automorphism of
C
such that W leaves 0, 1,
group and such that
Wl~o
=
fixed and WoHo,s oWl is a Kleinian
is conformal. Then
WI~/
j
=
1, ···, r, defines
292
YOICHI IMAYOSHI
an element rj of the Teichmtiller space T(Hj), which we represent as a bounded domain in
c
3g·3+n· J J . If si ~ 0' set ti = ai ai where ai is
the repelling fixed point of Wohi '
S· 1
oWl and ai is the fixed point of
Wof'i o w 1 in W(~j). If si = 0, set ti = 0. Then the point (T
'
t)
= (T
determines the group Hr,t
1'
• •• T
t
' r' 1'
•• • tk) '
L
'""
C3g3
W oHo,s oW 1 completely. We denote by
=
Xa(S 0 ) the set of all (r, t) for which a group Hr t is defined. Here a
'
represents the choices made; the group Hj and the subgroups Then Xa, bydefinition,there T ,t T ,t exists a homeomorphism gr: s > sr such that a =a of 1 og and rr,tr rr,tr r r gr is homotopic to a product of vth powers of Dehn twists about Jordan curves mapped by a r r into nodes, followed by an isomorphism. Hence, T ,t if this product of Dehn twists is denoted by dr, then [S, fr, Sr] = [S, dr a g~ 1 ofr, S r r] in T(G). We can assume that a: S . S 0 is locally T ,t quasiconformal and for any sufficiently small neighborhood
1\,
i = 1, · · ·, k,
of Pi in S 0 , we can also assume that there exists a quasiconformal mapping hr: S . S r r such that [S, fr, Sr] = [S, hr, S r r] and hr = T ,t T ,t
drog~ 1 ofr on Sa 1 (8), where O=o 1 U···Uok. (SeeBers[S].) Let
TT :
U . U /G = S and if:
tions, where
U is
U . U/Hj
= !j be the natural projec
the upperhalf plane in the zplane, let Llj be a con
nected component of rr 1 oa 1 (!jo) and let l'ij = rr 1 (Sj, 0')
rr 1 (!jo).
Since
and rr 1 (!j, 0) are conjugate by a, we can lift a to a quasi
conformal mapping A: Llj . l'ij such that Gj and
I\
are conjugate by A. ~
Let Wr = W¢
be the quasiconformal automorphism of C defined by
r
[S, hr, S r r] and let Wj r = Wj ¢· be the quasiconformal automorphism ~ r,t ' ' J,r of C defined by [!j, Fj,r• !j,r]. Then
v.],r
= W oA 1 oW: 1 r J,r
:
w.],r(l'i.). W (fl.) J r J
is a conformal mapping, because Wr and Wj,r oA have the same Beltrami coefficient on Llj . We can assume that Wj,r converges to the
w1. 0 : z
~ uniformly on any compact subset z +1 of U as r . 0 , because F j ,r converges to identity as r . 0 . If we set
Mobius transformation
'
r>
Wj, 0 (1'ij) = ~j,o• then IVj,rl for r ~ (0, 1) forms a normal family on l'ij,o and hence IWrl is also a normal family on Llj. Now we can prove that Gj, 0 assume that
v1.'n r
=
X¢ 0 (Gj) is not degenerate: We may
converges to a holomorphic mapping
v1., 0
uniformly
294
YOICHI IMA YOSHI
on any compact subset of ~j,o as rn _, 0. Suppose that Vj,o is a constant mapping with a value c . There exist two loxodromic elements 7j and ji in H j and a point
~ in ~ j such that ii((), ji(() are con
tained in ~j and iiY f, }iii. Since Hj is conjugate to
Gj
by A, we
seethat,if TJ=A 1 r,A, y=A 1 jiA and TJ 0 =x~ 0 (TJ), Yo=X~ 0 (y), then ., 0 and y 0 are of infinite order and TloY 0
1
YoT/o. If we set iin =
X~. (TJ), Yn=X~. (y), Tin=X~ (TJ), Yn=X~ (y) and ~n= J,rn J,rn rn rn
W1. r (~), then we have Tin o v 1. r (~n) = 'n
'n
v 1. r
'n
o i'in(~n). By taking limit of
this equality, we obtain ., 0 oVj, 0 (() = Vj, 0 a.,((), which implies ., 0 (c) =c.
Similarly, we have y 0(c) =c. Three properties obtained
above that ., 0 , y 0 are of infinite order, ., 0 oy 0
1
y 0 o., 0 and ., 0 (c) =
y 0(c) = c contradict the discreteness of G 0 . Therefore, Vj,o is a univalent holomorphic mapping. Using above functions V j, 0 and co~tracting _each oj to the _poin~ Pj , we can con~truct a conformal_ mapping Vj,o of U onto Dj,o = Vj,o(U) suchthat Vj,o conjugates Hj into for any g 0 < G0

Gj, 0 ,
Gj, 0 .
Then Dj, 0
Fuchsian and Dj,o is an invariant component of
Q(G 0)
Therefore, G 0 =
x~
fixed points of G 0 on
and,
we see g 0(Dj, 0 )noj,o = ~ and Vj,o induces
an isomorphism of Ij = U/Hj onto Dj, 0 /Gj, 0 . Hence is a component of
cQ_ 0(G) corresponding to c/> 0 is a regular tends to zero
bgroup and x cf>t (G) converges to x cf> 0 (G) uniformly as t through D 1 .
If
mis regarded as a mapping from
fixed point of
m in
T(G) to T(G)' then cf>o is a
T(G)' where T(G) is the augment space of T(G)
in the sense of Abikoff [1]. Proof.
By LindelOf's theorem, it is sufficient to prove that cf>t converges
to c/> 0 for a fixed argument of t as r = It I > 0. We may assume arg t = 0 and will prove that c/>r converges to an element c/> 0 in aT(G) as r
>
0, where r is a positive real number.
For a convergent subsequence lcf>r l of lcf>rl with its limit c/> 0 n
as rn . 0, we have cf>o image J(O) = [S 0 ) of t
t
=
aT(G). In fact, assume. c/> 0
t
0 by the holomorphic mapping
t
T(G)
T(G), then the
j:
D
>
Mg in §S
is contained in Rg, that is, S 0 is a nonsingular Riemann surface. Hence there exists a sufficiently small neighborhood U 0 of cf> 0 in T(G) such that ,jif5cf>t) Since
t
U 0 for a sufficiently small t
t
D and for every integer f.
m is of infinite order, this contradicts the discreteness
This implies c/> 0
t
of Mod (G).
aT(G).
The facts stated in §S show that, for any convergent subsequence lcf>r l of lcf>r l with its limit cf>o n
t
aT(G) as r .... 0, x cf> (G) converges rn
to a regular bgroup G 0 = X...t.. (G) and that D 0 U !parabolic fixed points 'PO of G 0 on D 0 l!G 0 is isomorphic to S 0 . For another convergent subsequence lcf>r 'l of lcf>rl with its limit cf> 0't aT(G) as rn'__, 0, G0 = X...t.. '(G) n
"'O
is also a regular bgroup and D 0 ' U !parabolic fixed points of G 0 ' on D0 'l/G 0 ' isisomorphicto S 0 , where D 0 '=0(G 0 ').:\(G 0 '), O(G 0 ') is the region of discontinuity of G 0 ' and .:\(G 0 ') is the invariant component of G 0 '. From the univalent holomorphic functions Vj, 0 , j
=
1, · · ·, r, con
structed in the proof of Lemma 2, we can construct a conformal mapping h: O(G 0 ). O(G 0 ') such that h conjugates G 0 into G 0 '. Hence, by a theorem due to Abikoff [2] and Marden [14], the mapping h is a complex
296
YOICHI IMA YOSHI
Mobius transformation. Hence G 0 is conjugate to G 0 ' in SL'(2; C). On the other hand, by Bers' theorem (6], every boundary group of G is conjugate, in SL'(2; C), to precisely m1 other boundary group of G, where m is the index of G in its normalizer in SL'(2; R). Since m is finite and since the mapping 1 (O, 1 ): (0, 1) . T(G) sending r into fi>r is continuous, we have f/> 0 = f/> 0 '. Therefore,
1> 0
1>r
converges to an element
< aT(G) as r. 0.
By Lindelof's theorem, it is obvious that f/> 0 is a fixed point of
:m.
This completes the proof. Further we have the following. THEOREM

4. If the homotopical monodromy :liT is of infinite order, then
a twodimensional normal complex space @3 can be canonically constructed in the following way: (i)
(@3, ;;., D) is a holomorphic family of compact Riemann surfaces
of genus g with a singular fiber over t (ii)
=
0,
if S 0 is the Riemann surface with nodes corresponding to
¢ 0 < aT(G) in Theorem 3 and if 10 is an automorphism group of S 0 m
duced by
m,
then the fiber
§0 = ;  1 (0)
to S 0!i0 , and (iii) @3 is naturally isomorphic to
over t
S S0
=
0 of
6
is isomorphic
so that the following
diagram is commutative: @3       @ 3
Til
7Tj D*
id
'"+
D
Proof. We use the notations in §5. Let f* be an element of Mod (S) corresponding to
:ffi
of Mod (G). Since the mapping ] : D . Rg sending
t into (St] has a holomorphic extension
J: D . Mg
with J(O)
=
[S 0 ]
and since there is a neighborhood N of < id > in X(S 0 ) such that N is stable under the finite group 1 0(S 0 ) and N/10(S 0 ) is a neighborhood of
297
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES
[S 0 ) in Mg, there exists a positive integer p such that f P is homotopic to a product of vth powers of Dehn twists about Jordan curves mapped by a into nodes. Set 8 =(I( I< 1) and 8* = 8IOI in the (plane and let K: 8 . D be the mapping sending ( into ( P. Consider the holomorphic family (S, n, 8 *) constructed by (@3'
TT,
D*) from the relation t = ( p.
Then the analytic mapping K: 8 * . X(S 0 ) sending ( into < S(, a ofz 1 , S 0 > is singlevalued. Hence K has a holomorphic extension K: 8 . X(S 0 ) with K(O)
=
< id >.
We denote by H(() and U(() the Kleinian group H(r, t) and its components U(r, t), respectively, determined by the point K(() in Xa(S 0 ). Now we can canonically construct a completion
S of
S
=
(r, t)
as follows:
We set
Then, by its construction,
S1
becomes a twodimensional complex mani
ill: sl> 8* is the mapping sending ((, [z)) into (, then , , (S 1 il 1 8 *) is a holomorphic family of compact Riemann surfaces of
fold. If
genus g. Let and 'I' be the analytic mappings (manyvalued) of
(S 1 , il 1 , 8 *)
(S, TI, 8 *) and that
=
into T(G), respectively. We may assume
cS, n, 8 *)
'I' on 8 * and that
and
cS 1 , TI 1 , 8 *)
homotopical monodromy for a certain positive integer
so sl u !n(O)/H(O) =
p.
have the same We set
with the images of all elliptic vertices removed I
and
S=S 1 U !U(O)/H(O)
with the images of related elliptic vertices identified!.
Then, by its construction,
S0
becomes a twodimensional complex mani
fold. We can naturally define a locally compact Hausdorff topology on such that
S0
is an open dense subset of
S,
S
By Cartan's theorem on the
continuation of normal complex spaces, we can induce a normal complex structure
:R
on
S such that the restricted structure :RIS ..... ..... ... 0
same one given on
S0
and
S  S0
to
S0
is a proper analytic subset of
is the
.....
S.
(The
YOICHI IMA YOSHI
298
functions which separate the points of
S0
are obtained from the automor
TI : S. 11
phic forms constructed by Bers [9].) Then the projection ing
(~, [z])
into
~
(S, TI, 11)
is holomorphic and
send
is a completion of
(S, II, 11 *). ~
Finally we construct a completion @3 of @3 as follows: For an element w < N(G) with an automorphism
y1
of Lemma 1,
Yt
of
=
s1.
m,
the automorphism [m]* of F(G) induces
By the similar reasoning to that in the proof
has a holomorphic extension
81 is the inverse mapping of y1 , 80 : S0 • S. We can prove that the
S0 . S.
y0 :
Similarly, if
then § 1 has a holomorphic extension
y0
mappings
and
80
are not constant
on each part of S 0 as follows: Let P1 , ···, Pk be the nodes of S 0 . Denote by
~~
the fiber of
S1
over
~ < 11 *.
Yo
I£
has a constant value
q 0 < S 0 on a part ~e of S 0 , then there exist a small neighborhood A of q 0 in such that
S and a skmall neighborhood bj of y 0 (~e .u bj) is contained in A ]=1
Pj in S 0 , j and such that
=
1, ···, k,
~,. n A s
is
homeomorphic to a disc or an annular domain for every small ~ < 11 *. We k
can take a sufficiently small neighborhood B of ~f .~ 1 bj in ~
~
that y 0( "~
n
B) is contained in A and
~
"~
nB
~
S
such
]
is homeomorphic to
k
bJ. for every small ~ < 11 *. Therefore, ~,. n B must be schlicht s for every small ~ < 11 *. Then ~~ n B has at least three boundary curves ~f U
]=1
c 1 , c 2 , c 3 such that each of them is not contractible to a point in ~~. On the other hand, the images ~~1
n
y0(c 1 ), y0 (c 2 ), y0(c 3 )
A with ~~1 =Yo(~~). Since ~~1
n
are contained in
A is homeomorphic to a disc
y0(c 1 ), y0(c 2 ), y0 (c 3 ) contradicts the fact such that y0
or an annular domain, at least one of the curves is contractible to a point in ~~ . This 1
induces an isomorphism of ~~ onto ~~ 1 . Similarly, on each part of S 0 . If
y0
is not constant
is not holomorphic at some node of S 0 , then § 0 is constant
on a certain part of S 0
y: S. S. y8 = § y =
~
o0
,
which implies
y0
has a holomorphic extension
Similarly, § 0 has a holomorphic extension §: id on
S, y
is an automorphism of
S.
S. S.
Since
299
HOLOMORPHIC FAMILIES OF RIEMANN SURFACES
r
~
Therefore, the automorphism [w]* of F(G) induces a finite group of automorphisms of
S.
Then the quotient space
normal complex space by a Cartan's theorem. If
~
=
S;f
becomes a
~
17 is the projection of @;
onto D, then by its construction, (@;, iT, D) is a completion of (@;, rr, D*) and
s0
=
"l (0) is isomorphic
phism group of S 0 induced by
so~r0'
~0
r.
where
r0
is a finite automor
Thus we have Theorem 4.
§7. An extension theorem THEOREM 5.
Let (@;,
TT,
D*) be a holomorphic family of compact Riemann ~
surfaces of genus g and (@;,
17, D) the completion of (@;, rr, D*) con
structed canonically as above and let @;' be the image of @; by the inclusion map of @; intq @; . Then, for a complex manifold X and for a locally closed complex submanifold A of X, every locally liftable holomorphic mapping f: X A > @;' extends to a holomorphic mapping
f: X> S.
~
Proof. From the construction of @; , this theorem can be proved by the
same reasoning as that in the proof of Lemma 1. THEOREM 6.
Let (@;, rr, D) be a holomorphic family of compact Riemann
surfaces of genus g with a fiber S 0 over t
=
0 and (@;,
ii, D) the com
pletion of (@;S 0 , rr, D*) con~tructed canonically as above. Then the inclusion mapping j : @; S 0 > @; induces a bimeromorphic equivalence
j:@; ....
s.
Proof. The set Sing (S 0 ) of singular points of S 0 has at least codimen
sion 2, that is, Sing(S 0 ) is the set of finite points. By Theorem 5,
~S 0 has a holomorphic extension ) : @;Sing(S 0 ) .... ~. The graph GJ of ) is the set ! (p, )(p)I p @; and
w: @; x@; > @; ,
the mappings
300
YOICHI IMA YOSHI
W[G? and J
ping
j: S
W[G~ are proper.
Hence j extends to a bimeromorphic map
J
>
S.
This completes the proof of Theorem 6.
DEPARTMENT OF MATHEMATICS COLLEGE OF GENERAL EDUCATION OSAKA UNIVERSITY TOYANAKA, OSAKA, 560 JAPAN
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W. Abikoff, Moduli of Riemann surfaces, in "A crash course on Kleinian groups," Springer Lecture notes, No. 400(1974), 7993. , On boundaries of Teichmiiller spaces and on Kleinian groups: III, Acta Math. 134(1975), 211237. L. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, in "Analytic functions," Princeton Univ. Press (1960), 4566. , Lectures on quasiconformal mappings, Van Nostrand Math. Studies #1 0 (1966 ). L. Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215228.     , On boundary of Teichmiiller spaces and Kleinian groups: I, Ann. of Math. 91 (1970), 570600. , Uniformization, Moduli and Kleinian groups, Bull. of London Math. Soc. 4(1972), 257300. , Fibre spaces over Teichmiiller spaces, Acta Math. 130 (1973), 89126. _____ , Spaces of degenerating Riemann surfaces, in "Discontinuous groups and Riemann surfaces," Ann. of Math. Studies 79, Princeton Univ. Press, (1974), 4355. C. J. Earle, On the Caratheodory metric in Teichmiiller spaces in "Discontinuous groups and Riemann surfaces," Ann. of Math. Studies 79, Princeton Univ. Press, (1974), 99103. P. A. Griffiths, Complex analytic properties of certain Zariskiopen sets on algebraic varieties, Ann. of Math. 94(1971), 2151. S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker Inc., New York, 1970. S. Kobayashi and T. Ochiai, Satake compactification and the great Picard theorem, J. Math. Soc. Japan 23 (1971), 340350. A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. 99 (197 4), 383462. B. Maskit, On boundary of Teichmiiller spaces and on Kleinian groups: II, Ann. of Math. 91 (1970), 607639. T. Nishino, Nouvelles recherches sur les fonctions entieres de plusieurs variables complexes: [V], Fonctions qui se reduisent au polynomes, J. Math. Kyoto Univ., 15(1975), 527553.
COMMUTATORS IN SL(2, C) Troels J 0rgensen * As usual, we denote by r the trace function on SL(2, C). With use of the two wellknown identities r(xy) + r(xy 1 )
1)
r(x)r(y)
and 2)
r(xyx 1 y 1 )  2
=
[r(x)r{y)] 2  [r(xy)2][r{xy 1 )2],
we will prove the following: PROPOSITION.
If two elements
x
and y with equal traces generate a
nonelementary discrete subgroup of SL{2, C), then
Under the assumptions of the proposition, one can show that lr(xyx 1y 1 )2\ attains a global minimum, and that this minimum is less than ~. I do not know its exact value. The method used below would allow us to obtain a constant slightly larger than ~, but not the best possible constant.
*supported by the National Science Foundation MCS 7800949.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00030103$00.50/1 (cloth) 0691082677/80/00030103$00.50/1 (paperback) For copying information, see copyright page 301
302
TROELSJ!DRGENSEN
If no hypothesis is made about the traces of x and y , then the trace of the commutator may be arbitrarily close to 2. This was shown in [2]. Let us put a
\r(xy)21
{3
\r(xy 1 )2\
y
=
o= Assuming that r (x)
=
\r(xyxly 1 )21
lr 2 (x)41
.
r (y), we obtain from 2)
3)
and from 1), with use of the triangle inequality, 4)
a+{3?_o.
Now recall the following result: LEMMA. If A and B generate a discrete subgroup of SL(2, C), then
unless BAB 1
f
{A, A 1 !, in which case the subgroup is elementary.
This was proved in [1]. When applied to x and y , one obtains with use of 4) the inequality
5)
a+{3+y?_1.
If instead the lemma is applied with xy or xy 1 as A and y as B, then one sees almost immediately that 6)
a 2 + 4a + y ?_ 1
and 7)
{3 2 + 4{3 + y ?_ 1
0
With this information available, we are ready to prove the proposition.
COMMUTATORS IN SL(2,
By
symmet~y,
303
we may assume that
8)
a ~
If a
C)
~ ~, then 6) shows that
If a > ~ and y ~
yy
~
>
k.
y
{3.
k, then we deduce from 3) and 5) and 3) and 5) that
(3 8 > V 8 > contrary to 8). This finishes the proof. UNIVERSITY OF MINNESOTA
REFERENCES [1] T. J 0rgensen, On discrete groups of Mobius transformations, Amer. Math. 98(1976), 739749. [2]
, Comments on a discreteness condition for subgroups of SL(2, C), Can. J. Math. 31 (1979), 8792.
J.
TWO EXAMPLES OF COVERING SURF ACES T. Jprgensen, A. Marden, C. Pommerenke*
1.
Introduction We will present two classes of Riemann surfaces which are, in a
sense, opposite extremes:
1. Surfaces S which cover nothing, 2. Surfaces S which cover themselves. By a surface S we mean a maximal Riemann surface. That is, one for which there is no embedding S < S' as a proper subset such that the inclusion of the fundamental group
77 1
(S) ...
77 1
By a cover we mean an analytic covering
(S') is injective. 77:
S ... S 0 such that (i)
every point q < S 0 is the center of a closed disk ~ such that each component of 177 1 (~)! is compact (in short, closed arcs can be lifted except at branch points), and (ii) the branching, if any, is constant on each fiber l77 1
(q)!, q < S 0 . We do not admit the trivial covering S 0
=
S and
77 =
id.
Example (2) requires and example (1) becomes more difficult to find if we stipulate that
77 1
(S) must have infinite rank.
The examples will be constructed in the context of fuchsian groups acting on hyperbolic 2space H, which will be realized as the upper half plane or unit disk D depending on which is more convenient under
*Authors
supported in part by the National Science Foundation.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00030513 $00.65/1 (cloth) 0691082677/80/00030513$00.65/1 (paperback) For copying information, see copyright page 305
306
T. JII'>RGENSEN, A. MARDEN AND C. POMMERENKE
the circumstances. The connection with maximal surfaces arises through the fact that a surface is maximal if and only if the fuchsian covering group in its universal covering surface is of the first kind.
2.
Maximal fuchsian groups
Let
r
r
be a fuchsian group acting on a particular realization of H .
is called maximal if the augmented group is not discrete for
each Mobius transformation t acting on H with t
I r . Maximal groups
are necessarily of the first kind. Finitely generated ones were investigated principally by L. Greenberg [2]. He showed that for most signatures, the groups which are maximal form a dense open set, and the others lie on subvarieties representing lower dimensional Teichmtiller spaces. In particular for finite n :;:: 4, most free groups of rank n are maximal. In contrast to Greenberg's analysis however, we will exhibit fairly explicitly maximal groups of rank n . At the same time our analysis works for the infinite rank case, whereas dimensional arguments break down. THEOREM
1. For every n, 4 :s; n :s;
~0 ,
group rn which is free of rank n' and
there exists a maximal fuchsian
u;rn
is an (n+l)punctured
sphere.
The proof depends on a property of groups close to the group representing the triply punctured sphere. PROPOSITION.
Suppose G = is a fuchsian group generated by
parabolic transformations a, {3. There exists a universal constant e > 0 such that if (1)
then no group of the form , n :;:: 2 , is discrete.
The hypothesis of the proposition is equivalent to the condition that HI< a, {3> be a twice punctured disk with a{3 determined by a simple
loop surrounding the two punctures (see §4). Let a(a{3) denote the
307
TWO EXAMPLES OF COVERING SURFACES
length of the geodesic in the Poincare metric in the free homotopy class of this loop. Condition (1) is equivalent to the relation, a(a{3) < e* ,
(2) where
As is customary, the notation try will denote the trace (a+d) of the normalized Mobius transformation y(z) = (az+b)/(cz+d), adbe= 1. In general this is determined only up to sign. For our work here we will always choose the matrices representing parabolic transformations to have trace +2 and then we will not encounter ambiguities of sign. The symbol a 1 /n designates that parabolic transformation with the same fixed point as a such that (a 1 /n)n = a. We will prove the proposition after proving the theorem. Proof of Theorem 1. We will first construct a maximal group ['00 of rank
N 0 . Set x 0 = 0 and choose an infinite sequence of disjoint closed intervals [x 2 k, x 2 k+l] on the positive real axis, k = 0, 1, ... , with x 2 k+l  x 2 k < 1/2. Space the intervals far enough apart so that the disks Dk = {z: lz(x 2 k+x 2 k+ 1 )/21 < 11 are mutually disjoint. .Set 0 = C\U{xil. Now push the endpoints of each interval [x 2 k, x 2 k+l] close enough together so that the length of the simple geodesic in 0 (in the Poincare metric of n) separating x 2 k and x 2 k+l from all other xj has length < e*. This can be fulfilled by estimation in Dk: the Poincare metric of Dk \{x 2 kl U lx 2 k+l! is larger than that of 0 restricted to Dk. Finally, we require that the length of these geodesics approach zero, as k
> oo.
Let ['oo denote a universal covering group for 0, acting in the unit disk D. Choose a point 0 say, and fix a point Let
p
f
o*
0 on the negative real axis, the point z f
D over 0. We may assume
o*
=
1,
is the origin.
denote the Poincare fundamental polygon for [' with center at
o* .
308
T. J0RGENSEN, A. MARDEN AND C. POMMERENKE
We claim that the interior of
P
projects onetoone on the region
n0 c n
obtained by slitting the complex plane along the positive real axis [0, + oo). This is true because
n
is invariant under the reflection z
1+
z
which
is therefore an isometry in the Poincare metric. Consequently, the geodesic from 0 in
n
cross the real axis the interior of
0
to any point in the upper or lower half plane does not It follows that the lift of
no
aP
ano
from o* is exactly
P.
We can describe
by travelling around
0
We see that
aP n ao
is the union of a countable number of parabolic fixed points of roo and one point ( which is the limit point of these fixed points.
P
is sym
metric about the diameter of D through (. We could have imposed the additional requirement that the
REMARK.
segments [x 2k, x 2k+l] approach + oo so fast that the length of the geodesics in lz: of
n'
belonging to the free homotopy classes of the circles
lz I= (x 2kl +x 2k)/2!, P would approach (
are uniformly bounded above. Then the sides nontangentially.
Now we are ready to show that roo is maximal. Start by fixing a free set of generators {y2k, y 2k+l!, k = 0, 1, ···, with the following property: y 2k, y 2k+l are induced by simple loops in
n
n
from 0, contractable in
to x 2k, x 2 k+l respectively, and so oriented that y 2k+l y 2k is homotopic in to a simple loop surrounding the segment [x2k' x2k+l]' sepa
n
rating it from all other xj. Thus the group Gk
=
satisfies
the hypothesis of the proposition. Suppose that roo is not maximal. Then there is a Mobius transformation t
I r such that the extended group r*
=
oo. This cannot occur in a finitely generated group. REMARK.
If we had imposed the additional condition on Q suggested in
the remark above, then y*(~) at (
from arising.
P
could not contain the internally tangent disk
for metric reasons. This fact alone would prevent Case 3
310
T. J0RGENSEN, A. MARDEN AND C. POMMERENKE
Finally, to complete the proof of Theorem 1, we will show how the argument above can be modified to yield the existence of maximal free groups of any rank n > 4 . The case of odd n (an even number of punctures) is easiest. Choose the pairs [x 2 k, x 2 k+l] close together as above so as to obtain the two generator groups Gk satisfying the hypothesis of the proposition. In addition, position the (n+1)punctures so that there is no conformal automorphism of
n
(this cannot be done if there are only four punctures).
Only Cases 1 or 2 can arise for the corresponding group f'n and both of these are excluded for the reasons cited above. For the case of even n we proceed in the same way except that an additional observation is necessary. Let Yn+l be a parabolic transformation corresponding to the unpaired puncture. Assume that for one of the paired transformations y 2 k, say, it is true that ty 2 kc 1 has the same fixed point ~ as ayn+l a 1 , some a
f
f'n, but is not a power of it. Let
o be a generator of the parabolic subgroup of Then (3
= a 1oa
['* =
< f'n, t > fixing ~.
has the same fixed point as Yn+l and Yn+l
=
(3m for
some integer m . We claim that (3f'n(3 1 C f'n. Consider for example y 2 j. If {3y 2 j(3 1 I f'n then (3y 2 j(3 1 has the same fixed point as a 1 yn+l a! 1 , some a 1
f
f'n, but is not a power of it. Therefore, for some integer d,
or
and y 2 j is a power of (3 0 f ['*. Since < (3 0 , y 2 j+l > is not discrete, this is impossible. We conclude that (3f'n(3l C f'n but this is not possible either.
3.
The group for the triply punctured sphere
In this section we will prove the proposition. The proof is based on the following results.
TWO EXAMPLES OF COVERING SURFACES
LEMMA
311
1. Given M > 0 there exists N > 0 such that for any noncyclic
fuchs ian group G
=
< x, y > with two parabolic generators x, y satisfying
ltr xy I < M, the group < x 1 /n, y > is not discrete for any n > N. LEMMA
2. Suppose G = is the group of the triply punctured
sphere with the parabolic generators a(z)
=
z +2, f3(z)
=
z/(2z + 1).
Then (i)
G 1 = contains G as a norma I subgroup of index two,
and a 'hf3 is elliptic or order two, (ii) G 2
=
contains G as a normal subgroup of index six,
aV.f3 is elliptic of order three, and G 2 is conjugate to the modular group. The cases (i) and (ii) are the only discrete groups of the form ,
n
~
2.
Proof of Lemma 1. By conjugating G we may assume x : z y: z
f>
f>
z +a ,
z/(2z+1). Replacing x by x 1 if necessary we may assume
a > 0. According to Siegel [S] or to [4], when G is a nonelementary discrete group, a2'1/2. Thusif a=(2trxy)/22M 1 the group < x 1 /n, y > cannot be discrete.
Proof of Lemma 2. When a = 2 , the only possibilities for adjoining roots are • n = 2 , 3 or 4 since a 2' n/2. The case n = 3 cannot occur because then a 113 f3 would be an elliptic element of trace 2/3. This is not the trace of an element of finite order. The other cases are as indicated.
Proof of the proposition. It is known that for E
1/Eo there is a group with O
z/(2z + 1). Because xkyk is a simple hyperbolic ele
ment, ak > 2. In the notation of Lemma 2, lim xk =a. There are only a finite number of possible n for which isdiscrete(Lemma1). Since >, forlarge k
312
T. ]0RGENSEN, A. MARDEN AND C. POMMERENKE
only the cases n
=
2 and n
groups. For otherwise
=
4 have any chance of yielding discrete
would not be discrete (Lemma 2) and
< xk 1 /n, yk > would contain elements arbitrarily close to the identity as k . "". This is impossible by [5] or [4]. But the same line of reasoning shows that the cases n
=
if, for example, the group
2, n
=
4 cannot occur either, for large k. For
< xk y,, Yk > were discrete for infinitely many k,
it would contain elliptic elements of arbitrarily high order and hence elements arbitrarily close to the identity as k . "", since tr xky,Yk = 2 ak and ak . 2. We conclude that for sufficiently small
E,
the proposition
is true. REMARK. Instead of the proposition, another approach is to observe that, with the normalization of the generators x, y of < x, y > as in the proof of Lemma 1, if 2 < a < 4 then x 1 /ny is elliptic of trace 2 2a/n, n > 2. Consequently, if a is known to be transcendental, no group n
< x 1 /n, y >,
> 2 is discrete. In the context of our example r"", this can be
arranged for each Gk as follows. Set up
r""
as above but introduce the
parameter A= x 1 x 0 , keeping the other points xi fixed. As A decreases, the geodesic length ak of the simple loop in
n
surrounding
[x 2 k, x 2 k+l] also strictly decreases. The values of A for which exp (ak) is an algebraic number for some k;: 1 are countable. Hence for all except a countable number of values of A, the traces of all y 2 k+ 1 y 2 k are transcendental.
4.
Conformal automorphisms of surfaces The following is a rather striking application of Theorem 1. Without
the condition that the surfaces involved are maximal, it was obtained by Greenberg [1]. COROLLARY. Let be an abstract ngenerator group, 4 :S n :S X 0 . There exists a maximal Riemann surface S whose group of conformal
automorphisms is isomorphic to . The surface S can be taken to be a covering surface of the (n+1) or (n+2) punctured sphere.
TWO EXAMPLES OF COVERING SURFACES
313
Proof. Working with our maximal free group in of rank n , there is an
isomorphism where N is the normal subgroup of in generated by the "relations" of
. We must, however, ensure that N I= {id.!. Therefore if is itself free of rank n < ~0' use rn+l instead of rn' or if is free of rank ~0 ,
choose the isomorphism to send into ['00 , not onto. Then N
will be a nonelementary group (one having more than two limit points) and therefore, having the same limit set as rn' will be a group of the first kind. Let S be the maximal surface
S
=
H/N.
The group of conformal automorphism Aut S of S is determined by the normalizer of N in the group of all Mobius transformations acting on H. This group is discrete, hence because of maximality it is precisely in or f'n+l . That is, Aut S ::: f'n/N or ['n+/N (if is free of rank n ). REMARK.
Aut S is exactly the group of cover transformations of S over
the (n+l) or (n+2) punctured sphere. Thus is an ngenerator finite group if and only if S is a finite sheeted cover of the (n+l )punctured sphere.
5.
Groups conjugate to a proper subgroup
In this section we will investigate fuchsian groups F for which tFC 1 < F f~~ some Mobius transformation t (the symbol < denotes proper inclusion). This phenomenon does not appear for finitely generated groups nor, according to Heins [3 ], for groups without elliptic elements which are of divergence type (i.e., the quotient surface does not support a Green's function). Here we will construct large families of such groups. Since we allow groups with elliptic elements, the examples presented are more general than those suggested in the introduction. According to Heins [3], the question for covering surfaces was originally raised by H. Hop£.
314
T. JORGENSEN, A. MARDEN AND C. POMMERENKE
Without the requirement that F be a group of the first kind, trivial examples abound. For instance let C 1 denote the semicircle lz: [z1121=114, Im z::::Ol and C 2 the semicircle lz: \z3/21 = 114, Im z 2: 0 l. Let A 0 denote a Mobius transformation sending the upper half plane to itself and the exterior of C 1 in it onto the interior of C 2 . Let t denote the translation z
t>
z + 2 and set Ak
tk A 0ck, k
=
2: 0. Then
the group F generated by all Ak, k ? 0, has the property that tFC 1 is a =
G0 * < t > .
Conversely, if G is a fuchsian group of the first kind with a decomposition G = G 0
*< t >
where t £ G is parabolic, then there exist groups F
of the first kind with G
< F, t > and tFC 1 < F.
=
Proof. If tFC 1 < F, the groups tkFck,  oo < k < oo, are nested; the larger the k, the smaller the group. Set H = Uk=oo tkFck. Then H satisfies tHC 1 Set G
=
=
H while tn
< H, t >
=
I
H for all n
f 0
0
< F, t >. Then G is also a fuchsian group of the
first kind (because H is a normal subgroup) and satisfies, (i)
G
(ii)
the commutator subgroup [G, G] lies in H,
=
ltnh: h£Hl, G/H"" ,
(iii) for n
f 0 and h £ H , tnh is not elliptic or the identity.
These are true because in any word in G, the letter t can be moved to the left end by virtue of the fact tHt l
=
H.
If G is finitely generated it has a standard generator system of hyper
bolic elements lai, bil elliptic lei!, and parabolic lpil satisfying the relation II[ai, bi]llei llpi
=
1. By (ii) the commutator [ai, bi] £ H and by
(iii) ei £H. We can assume one Pi is t. Therefore if this relation is to hold, there must be at least two parabolic transformations pi. This
315
TWO EXAMPLES OF COVERING SURFACES
means the surface H/G has at least two punctures, one of which determines t. Draw a simple arc on this surface from one of these punctures to the other. This arc determines a splitting of G as required. If G is not finitely generated, then the surface H/G has a puncture
determined by t and at least one other ideal boundary component. Connect these two by a simple arc. In the same way as above, this determines a splitting of G. To prove the converse we begin by constructing the fuchsian group H C G which is the free product of the groups tkG 0ck, "" < k < "",
Because H is a (nonelementary) normal subgroup of G, it has the same limit set and therefore is also of the first kind. Define the projection Pk: H follows. For x pk(x)
=
t
tjG 0 Cj, j
I=
>
tkG 0 tk which is a homomorphism as
k, set pk(x)
=
id.; for x < tkG 0 ck, set
x , and extend to a homomorphism of all H.
Let N be any proper normal subgroup of G 0 , for example, N
=
lid.!.
Define
The group F is not elementary and it is a normal subgroup of H. Consequently F isofthefirstkind. Furthermore, tFC 1 z + 1 the geometric picture in
the upper half plane is quite nice: Each ek has its fixed point at the
316
T J0RGENSEN, A. MARDEN AND C. POMMERENKE
north pole of the circle of radius 1/2 centered at z = k. The group F can be constructed as above taking N =lid. I, or as follows: F=lhfH:thetotalnumberof ek's with k, t hyperbolic but not necessarily simple, we can construct as above a group F with tFC 1 < F. On the other hand, suppose G is a surface group of genus g + 1 ;::: 3. Cutting the surface along a simple, nondividing loop gives rise to a representation of G as an HNN extension of a subgroup G 0 by a hyperbolic transformation t. The presentation of G 0 looks like
where 1 :::: i ~ g. A homomorphism ¢: G 0
>
Ig onto the surface group of
genus g, Ig = < ai, hi : TI[ai, hi]= 1 >, is determined by setting c/J(ai) = ai, c/J(bi) = hi and c/J(x) = c/J{C 1 xt) = 1. Let c/Jk denote the corresponding homomorphism tkG 0 ck .... Ig. Consider the free product with amalgamation, H
=
We can extend each c/Jk to a homomorphism c/Jk : H c/Jk(x) = id. for x ( tjG 0 tj, j
>
!g after requiring
I= k. Now form the normal subgroup,
F = {hfH: c/Jk(x)=id.
for all
Again, tFC 1 < F and F is of the first kind.
k can even be a happy, closed surface. UNIVERSITY OF MINNESOTA
REFERENCES [1] L. Greenberg, Conformal transformations of Riemann surface, A mer. ] . Math. 82 (1960), 7 49760. [2]
, Maximal groups and signatures, in Discontinuous Groups and Riemann Surfaces, L. Greenberg, ed., Annals of Math. Studies 79, Princeton Univ. Press (1974).
[3] M. Heins, On a problem of H. Hopf, .Jour. de Math. 37(1958), 153160. [4] T. ] !lirgensen, On discrete groups of Mobius transformations, A mer. .J. Math. 98(1976), 739749. [5] C. L. Siegel, Uber einige Ungleichungen bei Bewegungsgruppen in der nichteuklidischen Ebene, Math. Ann. 133(1957), 127138.
DEFORMATIONS OF SYMMETRIC PRODUCTS George R. Kempf* Let C be a smooth complete algebraic curve of genus g over an algebraically closed field k . If n > 2g 2 , any universal abelian integral
J : c(n) ~ 1 n
from the nth symmetric product c(n) to the Jacobian
J of C isalocallytrivial pr_bundle,where, r=ng. If one deforms jective bundle
J
n
.J as an abelian variety, one may ask whether the promay also be simultaneously deformed along with .J .
The main result of this paper is that such a deformation is impossible even in first order unless the deformation of J comes from a deformation of C, if C is not hyperelliptic. We first show that any deformation of the variety c 0, k = r(e, 0c) ~ r(e, ~c(c)). Let
be the product sheaf on en. Thus, as r(en,
m)
®
77:
0c(c)
1 e(n) be a principal Gbundle. Then p~ if
TT
77
is a trivial bundle over the product en if and only
is a trivial bundle OVer e(n).
323
DEFORMATIONS OF SYMMETRIC PRODUCTS Proof. Clearly, if that p~ TT: Y' = y
X
is trivial, then p~ TT is trivial. Conversely, assume
TT
c
cnPn >en is trivial. Let
T:
en > Y' be a section
of p~ TT. Let r' be another section of p~ TT. Then, r '= a· r, where a: c en is equivariant. If we can show that the image r(Cn) is in
variant under the action of Sym(n) on Y', then r(Cn)/Sym(n) Y' /Sym(n) = y TT
will be isomorphic to c(n) via
TT:
c;
y > c (')x(D 0 ) > (')X(D 0 )\ D > 0 of sheaves on X . 0
Next, I will explain what these abstract ideas mean in this particular case of interest. Consider the divisor Dn on C x c(n) from section 1.
=
For each point d = c 1 +···+en of c(n), Dnld is the divisor D [c 1 ] + · ·· + [en] on C. Thus, one may confuse points of c(n) with effective divisors of degree n of C. This confusion does not lead to any infinitesimal ambiguities because A. The characteristic mapping, :td: Tangent space of cCn) at d ....
r(C, (')c(D)\D), is an isomorphism of ndimensional vector spaces. One may even globalize the characteristic mappings to get A'. The characteristic mapping :X:® (n)> 11 *((') (n)(Dn)ID ) gives c C cxc n an isomorphism of locally free (.') (nfmodules of rank n.
c
In this case,
>H 1(C, (')c) is the linearization of the ,c : c(n)> Pn = 177£H 1 (C, (.') )\deg 7]=nl.
B. the mapping t/Jd :® (n) abelian integral More globally,
f
c
"n
C*
B'. we have a commutative diagram of homomorphisms of (.') (nfmodules,
c
326
GEORGE R. KEMPF
where 8 is the boundary in the long exact sequence of direct images R*rr
c
*
of the sequence
The following special case will be very useful. C'. If n
=
1, we have a commutative diagram
The dual 1/J~: H 1(C, t)c( ®k ()c phism H 1 (C,
>
e~ 1 =We induces the usual isomor
t)c( __::::__, l(C, we) on the global sections of these sheaves.
If g > 0, Blc and 1/Jic are injective at each point c of C. Next we will reverse the roles of C and c 0.
Proof. By the diagram
*
and the last lemma, OCc must be injective if
1/lc is injective. In point C, we have seen that 1/lc is in fact injective. Thus, to prove the lemma, we need only check that r(c(n), x) .
A universal deformation of X is a formal deformation s : ~
that any formal deformation s': ~,> morphism lls
:f'> :f.
:f'
> :f
such
is induced from s by a unique
Clearly, if s is a universal deformation of X , then
must be an isomorphism. We have another criterion for the universality
of a given deformation. CRITERION. If s : ~
>
:f
a) the linear mapping b) c)
is a deformation of X such that llo
is an isomorphism,
rex, E>x) is zero, and :f is a smooth formal scheme,
then s : ~>
:f
is a universal
deformation of X. The most wellknown case of a universal deformation is when X is a smooth complete curve C of genus g > 1. Then it is well known that there exists a universal deformation s :
e . . :r
of c ' where
:r
is a
smooth formal scheme. In fact, if C is a compact Riemann surface,
:f
is the formal completion of Teichmuller space at a point corresponding to C. With our universal deformation s : construct a deformation sn : en ....
:r
e . . j"
of the curve C, we may
of the product en by taking en
=
e xj" e ··· xj" e.
Furthermore, we have a deformation s: ecn> .... j" of
c(n) where e
=
en/Sym(n). A key result of this paper is
THEOREM 4.2. If C is a nonhyperelliptic curve of genus > 2, then s: cCn) .... j" is a universal deformation of the symmetric product c.
336
GEORGE R. KEMPF
Proof. We will use the above criterion to check the universality of c {3(m)
=
zero in P0 and s 0 o {3 =a.
Thus, we have an mmorphism m X
Po
Furthermore, using the addition + {3: m xj" ( + {3)
m xj"
0
pn . . m xj" pn,
(m X
s) by n
D(a,
e(n) > m xj" pn
assumed that {3(m)
xj"
be a morphism such that
sn: m xj"e(n)> m xj" pn pn
>
pn
using
we have a misomorphism,
m.
Trivial, one may check that D(a,
s :c (n) > pn n
m:
as we have
zero.
The converse is usually true. THEOREM
5.2. If C is a nonhyperelliptic curve of ~enus > 3, then
any deformation of the inte~ral
f : c(n)> Pn n
has the form D(a, {3),
where these morphisms a and {3 are uniquely determined by the deformation.
a.
given by addition by {3. Denote the composition
is a deformation of =
Po
340
GEORGE R. KEMPF
Proof. Let F :
:R "
>
J.
~ be the deformation of
m "
In Theorem 4.2, we have
n
determined that any deformation of the symmetric product c(n) has the form m xj" ecn) .... m where a: m .... j" is a unique morphism. If we replace S:
e
>
:R . . m
j" by m X S : m Xj"
e
>
m, We may assume that the deformation
of c(n) is the s(n): ecn) .... m of the first part of this section.
Next, consider the deformation ~ .... m of Pn . As we may take a section
of ~
T
m and pn is a principal homogeneOUS space under an
>
abelian variety, the Theorem 6~ 14 of [9] implies that ~ .... m possesses a unique structure of an abelian mscheme with identity
T.
Thus, ~ is a
group scheme over m and we may now apply Proposition 5.1 to the mmorphism F: ecn) .... ~. Therefore, F factors through an mmorphism
1/J:
Pn
>
~. As 1/J must be a deformation of the identity of Pn, 1/J is an
misomorphism. Thus far, we have shown that our deformation must have the form K
0
D(a, zero): m xj"e(n) .... m xj" pn' where
K
is an '"isomorphism
m xj" Pn, which is a deformation of the identity of Pn. By Corollary 6.2 0
f [9],
K
must be translation by a unique j" morphism {3 : m ....
Po'
which
vanishes at the point m of m. Therefore, our deformation is in fact
D(a, {3). Q.E.D. Now, I may explain why I wrote this paper. Let n > 2g2. Then,
s :c(n) .... pn n
is a locally trivial projective bundle. If one chooses a
point in Pn, one may identify Pn with the Jacobian of C. I was trying to answer the question: Given a deformation (f variety, when can the projective bundle bundle
G > Cf ?
n
>
j( of J as an abelian
be extended to a projective
n
As the bundle
f : cCn) .... Pn,
J
G > (f
" j("
would be a deformation of the integral
the Theorem 5.2 implies that we must be deforming J as
a Jacobian variety, when C is not hyperelliptic. PRINCETON UNIVERSITY AND THE JOHNS HOPKINS UNIVERSITY
DEFORMATIONS OF SYMMETRIC PRODUCTS
341
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
H. Cartan, Seminare: Families d'espaces complexes et fondements de la geometrie analytique, Secretariat mathematique, Paris, 19601961. P. Griffiths, Some remarks and examples on continuous systems and moduli. J. Math. Mech. 16 (1967), 789802. A. Grothendieck, Seminare de geomehie algebraique, l.H.E.S., BuressurYvette, 19601961. _____ , Fondements de la geometries algebraique (Extracts du Seminare Bourbaki) Secretariat mathematique, Paris, 1962. R. Gunning, Riemann surfaces and generalized theta functions, SpringerVerlag, Berlin, 1976. G. Kempf, Toward the inversion of abelian integrals I, to appear. A. Mattuck, Symmetric products and Jacobians, Amer. J. Math. 83 (1961), 189206. A. Mayer, Rauch's variational formula and the heat equation, Math. Ann. 181 (1969), 5359. D. Mumford, Geometric Invariant Theory, SpringerVerlag, Berlin, 1965.
[10] · Lectures on curves on an algebraic s'urface, Princeton University Press, Princeton, 1966. [11] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Annals of Math. 67 (1958), 328466. [12]      , A theorem of completeness of characteristic systems of complete continuous systems. Amer. of Math. 81 (1959), 477500. [13] J.P. Serre, Groupes algebraiques et corps de classes. Hermann, Paris, 1959.
REMARKS ON PROJECTIVE STRUCTURES Irwin Kra and Bernard Maskit* In this note the authors continue their previous investigations of deformations of Fuchsian groups [10, 14]. Throughout this paper [' denotes a finitely generated, purely loxodromic (including hyperbolic) Fuchsian group of the first kind operating on the upper half plane U. Then [' operates discontinuously and freely on U, and S = U/1 is a closed Riemann surface of genus g :;: 2. We denote the space of holomorphic quadratic differentials on S by B 2 (f'). We do not differentiate between differentials on S and automorphic forms for [', so that we regard ¢
f
B 2 (f') as a holomorphic function on U , with ¢(Az)A'(z)2 = ¢(z),
for all
A
f [',
and for all
z
f
U .
We use the Nehari norm [20] 11¢11 = sup ll¢(z)l (2 Im z) 2 ! . ZiU
(We will on occasion need to regard U as the unit disc, in which case 11¢11 =sup li¢(z)\(1lz! 2 ) 2 ! ziU
*Research
.)
supported in part by NSF grant MCS 7801248.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00034317$00.85/1 (cloth) 0691082677/80/00034317$00.85/1 (paperback) For copying information, see copyright page 343
344
IRWIN KRA AND BERNARD MASKIT
Given ¢
f
B 2 (r), we can find a locally schlicht meromorphic function f
on U with !f, z l
¢(z) [9, pp. 376377]; here
=
differential operator !f
'
!f, ·I is the Schwarz ian
1 (f")2 (f")' f' 2 f'
·I = 
The function f will be unique once we normalize by requiring f(z)
=
(z c', 4> b, 5> c, 4'> b', 5'> c', 1'. a, 6 > d, 7> e, 6'> d', 7'> e', 8> d, 9> e, 8'> d', 9'> e'. As before, this map extends to a local homeomorphism of U onto gates
r*
~
which conju
onto G . The local homeomorphism is replaced by an analytic
local homeomorphism using a variation of the complex structure through quasiconformal mappings.
§3. A problem We can regard our second example from a slightly different point of
351
REMARKS ON PROJECTIVE STRUCTURES
view. We start with the Fuchsian group r surface
s of genus
finite index in
f,
2' and we observe that there is a subgroup r ' of and there is a uniformization (ll, G) of S, so that
f:
the covering map
representing the closed
U . tl conjugates the proper subgroup r
onto all

of G. One observes that this can occur only if f is not injective. PROBLEM.
For which uniformizations (ll, G) of a Riemann surface S
can one choose a proper subgroup
r
of the Fuchsian group
ing S, so that the cover map f : U . tl conjugates
r
l~
represent
onto all of G?
§4. Generalizations of the examples Our two examples involve low genera and low index
[f : rJ;
these
examples can easily be generalized to higher genera and higher index. In fact it was only because of the low genus that we included the first example. If we start with a deformation r:P, f,
e: r. G,
where S
=
U/r
is a branched covering of S( = f(U)/G = ll/G), then we let G 0 be a torsionfree normal subgroup of finite index in G (Selberg [21 ]). One easily sees that r;e 1 (G 0 ) coverings u;r0
.
=
r/r0 is isomorphic to G/G 0 , and so the
u;r and ll/G 0
.
ll!G have the same number of sheets.
It follows that f projects to a covering U /r0 . tl!G 0 which has the
same number of sheets as the covering U/r. tl/G. Of course, since G 0 is torsion free, the covering U/r0
.
tl!G 0 is unbranched.
Ex'cept for genus 2, the generic Fuchsian group is not contained in any other Fuchsian group (Greenberg [6]), and so our examples of nontrivial coverings are exceptional. Of course, once we have such an example r:/J, f,

e:
r . G, then we can obtain a family of such examples

by varying r, the extension of r. In fact, we can regard T(r), the Teichmliller space of r, as a submanifold of T(r), and every point of this submanifold admits a deformation which yields a nontrivial covering.
§5. Theorem 1 In this section we prove Theorem 1; as we have already remarked, it
352
IRWIN KRA AND BERNARD MAS KIT
cP for which f is a cover map is bounded.
suffices to show that the set of
Our proof makes use of a generalization of the Kraus [13] Nehari [20] theorem. LEMMA 5.1.
Let f be a locally schlicht meromorphic function on the
unit disc U, and let
cP = If, ·I.
(a) If f is schlicht on every nonEuclidean disc of radius
II¢ II : :;
6 tanh 2
(b) If
o.
\\¢\\ :S k,
disc of radius
o,
then
and k > 2, then f is schlicht on every nonEuclidean
~ log~:~·
(If k < 2 then, by Nehari's theorem [20],
f is schlicht in U . )
We actually only need part (a) of this lemma. To prove this part, we set A(z) = o* z , where
o* = tanh 0.
Observe that f 0 A is Schlicht in
the unit disc, and hence \¢(0)\ =\If, Ol\ =\If, A(O)l\ = \lfoA, O}A'(Or 2 \
(5.1)
:S \\lfoA,·l\\\o*\ 2 :::;6(8*)2; where we have used the Cayley identity, and Kraus' theorem [13] which
cP is the Schwarzian of a function that is schlicht in the unit disc, then \\¢\\ :S 6.
asserts that if
Now for any z /, 0, in U, we let T z be some nonEuclidean motion mapping 0 to z. Then, using the Cayley identity, the invariance of the Poincare metric and (5.1), we obtain (5.2)
llf,zll(1\z\ 2 )2
llfoTz,OII\T~(o) 2 \(1lz\ 2 ) 2
\lfoTz,Ol\ :::;
6(o*r 2
Combining (5.1) and (5.2) we obtain
\1¢\1 :S 6(8*) 2
=
6 tanh 2
o.
.
353
REMARKS ON PROJECTIVE STRUCTURES
The proof (although not the present formulation) of part (b) appears in Kra [10]. It is based on Nehari's theorem [20]. We turn now to the proof of Theorem 1. We have already observed that f is a possibly ramified cover map and

that hence fop
=

p of is a· possibly ramified covering of S. Let f' be
the corresponding cover group (the Fuchsian model of G). Since f' C f',
f
must be a finite extension of f'. We cover
S by a finite number of
opensets {Uj}, sothatforeach j, (pof) 1 (Uj) isadisjointunion {U ja l of open circular discs, each of which !s precisely invariant under a (possibly trivial) finite cyclic subgroup of f'. It is easy to see that f i~
schlicht on each Uja. Since we can cover a fundamental domain for
f' by a finite number of the Uja, we have shown that there is a 8 > 0 so that for every z
f
U , f restricted to the nonEuclidean disc of radius
8 about z is schlicht. The above remarks hold for any given covering map f, and, of course,
8 depends, not on f , but on S . Since f' has at most finitely many extensions, we can in fact find a 8 independent of S. (This 8 still depends on S .)
§6. The radius of univalence In this section we obtain another (more geometric) lower bound on the radius of univalence of cover maps f (producing an alternate proof of Theorem 1). We may assume that f is not univalent. We let
8 = 8(1) =
inf
p(z, Az)
> 0,
ZfU
Aff'\{1} where
p
is the nonEuclidean distance function on U . The quantity
8(1) is the length of the shortest closed geodesic on S.
(1) If f is onetoone, then f is clearly univalent in every nonEuclidean disc of radius :S 8(1).
354
IRWIN KRA AND BERNARD MASKIT

(2) If f is an unramified nsheeted cover, then f is univalent in every


disc of radius :S 8(1), where 1

is the cover group of S. But it is easy
to see that 8(1) 2: 8(r) ' and from RiemannHurwitz n :S g  1
(g
=
genus of S ) .
Thus f is univalent in every disc of radius :S 8(1)/(g 1).

(3) Finally, if f is ramified we pass to a torsionfree finite index subgroup G 0 of G. Let 1 0 the index n of 1 0 in 1
=
e 1 (G 0 ).
Then 1 0 is a subgn;mp of 1 and /
equals the index of G 0 in G .. Clearly
As before, we need only estimate the number of sheets k in the covering f: L'./10
>
D/G 0 . Since the following diagram commutes
T
f

f
L'./1     D /G
and each of the vertical maps is nsheeted. we see that k is the number of sheets in f: L'./1 >DIG. But it is well known that k
=
Area(L'./1) Area(L'./1)

where 1
is the branched universal cover group of S ( = full preimage of
G via the map f). Now Area L'./1
=
2rr(2g 2), and Area (L'./1) 2: rr/3
(see, for example, Kra [12, pp. 7678]). This last estimate follows from the fact that G cannot be a triangle group ( D would have to be simply connected for G to be a triangle group). We conclude that in this case f is univalent in every disc of radius :S 8(1)/12(g 1).
355
REMARKS ON PROJECTIVE STRUCTURES
§7. Convergence of cover maps In this section, we discuss one example of convergence of cover maps in B 2 (1) and prove Theorem 2; our discussion is based on the classification of geometrically finite Kleinian groups with an invariant component (Maskit [18], [19]). We start with the situation described in the hypotheses of Theorem 2. We can find a simple loop w 0 on S, so that 0(A 0 ) f G corresponds to the lifting of w 0 (Maskit [17]); there is, in fact, a set of simple disjoint loops w 0 ,w 1 ,· .. ,wk on S, andasetof"integers" 1 ::; aj ::;
oo,
(1) If aj
Ax
Rn+l: xn+l > O! which
models hyperbolic nspace. Let G be a discrete subgroup of M(n+1) acting discontinuously in R~+l, and, perhaps, on some (open) set fl(G)
c Rn =
aR~+l . Then the limit set A(G) of the group G lies in Rn
and either coincides with Rn or is nowhere dense in Rn. However, for n
2' 2 we may start from discrete groups acting in Rn; their continua
tions in R~+l (obtained in the wellknown way) will be discontinuous there. For the elements A
1::
M(n+1) there is well defined the value IA'(x)l
=
ldAxl/ldxl ,
which is the linear expansion at the point x ; it is independent of the direction. We assume that mnA(G) > 0.
364
S. L. KRUSHKAL'
In A(G) one may form (using the axiom of choice) a fundamental set e A which contains one point of each orbit Gx for almost all x fA( G), or rather, for all x f A(G), different from the fixed points of the elements of G (see [3]). As it has been shown in [3], there is a condition which guarantees that this set is nonmeasurable. If we pass from R~+l to the unit ball IxI < 1 , then the abovementioned condition is ( *) the intersection of isometric fundamental polyhedron with A(G) has zero measure. More generally, we have PROPOSITION
2. If G satisfies condition(*), every subset e
C
eA
with positive exterior measure is nonmeasurable. It is not difficult to construct the Kleinian groups which have measur
able eA. We shall not dwell upon them here. Henceforth we consider only these groups for which Proposition 2 is valid.
For an arbitrary set E C A(G) we may define the multiplicity function k(x; E)
=
card E
n Gx , x f A(G) .
It is clear that if E 1 C E, then k(x; E 1 )
_'S
k(x; E). For the measurable
set E the function k(x; E) is also measurable, and we have the decomposition
where
We then have THEOREM
2. If E is a measurable subset of A(G) having positive
measure, then almost everywhere on E the function k(x; E)= the intersection E points.
oo,
that is,
n Gx is either empty or consists of an infinite set of
365
SOME REMARKS ON KLEINIAN GROUPS
This is in sharp contrast with the fact that any compact set F C R~+l U U(G) can intersect only a finite number of images A(P G) where P G is a fundamental domain for the group G in R~+l U U(G). Such rigidity must, of course, raise the possibility of the existence of nontrivial automorphic forms and deformations with supports on A(G). For many forms this question is completely solved by the following theorem. THEOREM
3. For
p
f
Z\ I0 I the equation
c,b(Ax)IA '(x)l P
=
c,b(x)(A f G, x fA( G))
has, in the class of measurable a. e. finite functions on A(G), only the zero solution.
In particular, if p+q
I= 0 every measurable Gform c,b(z)dzPcrzq(z £C),
concentrated on A(G), assumes only the values 0 or
oo.
§3. The rigidity of deformations Theorem 3 does not embrace the automorphic functions c,b(x)(p = 0) and, for n
=
2 (in C), also the measurable forms c,b(z)dzPdzP, p
f
Z,
with supports on A(G). These include the Beltrami differentials c,b(z)dz/dz which are of special interest and connected with planar quasiconformal mappings. The question of the existence of nontrivial differentials of such type presents a special case of the general rigidity problem of quasiconformal deformations of Kleinian groups in Rn, n ~ 3. This is connected to the rigidity problem for Riemannian manifolds with negative curvature. Precisely, one questions whether there can exist quasiconformal automorphisms f : Rn . Rn (n
2: 2) deforming G into an isomorphic Kleinian group which
is not conformal on A(G). In particular, whether any quasiconformal automorphism of Rn compatible with the discrete nondiscontinuous group G, must be reduced to a Mobius one (that is, the composition of a finite number of inversions in spheres). Such deformations are, for instance, obtained as the boundary values of quasiconformal automorphisms of the halfspace.
366
S. L. KRUSHKAL'
The affirmative answer to the latter question would mean that every two discrete subgroups G 1 , G2 from M(n+1), n ~ 2, which are conjugate by a quasiconformal automorphism of R~+ 1 are also conjugate in M(n+1). For groups whose quotients have finite volume, this result has been obtained by Mostow [4]. However, B. Apanasov [1] has recently shown that the rigidity theorem fails in the general case. INSTITUTE OF MATHEMATICS SIBERIAN BRANCH OF THE USSR ACADEMY OF SCIENCES NOVOSIBIRSK
REFERENCES (1] Apanasov, B. N., On Mostow's rigidity theorem, Dokl. Akad. Nauk SSSR, t. 243, No. 4 (1978), 829832. [2] Knapp, A. W., Doubly generated Fuchsian groups, Mich. Math. J., v. 15, No. 3 (1968), 289304. [3] Krushkal', S. L., On a property of limit sets of Kleinian groups, Dokl. Akad. Nauk SSSR, t. 225, No. 3(1975), 500502; t. 237, No. 2(1977), 258. [4] Mostov, G. D., Strong rigidity of locally symmetric spaces. Princeton, Princeton Univ. Press, 1973, 195 p. (Ann. of Math. Stud., No. 78).
REMARKS ON WEB GROUPS Tadashi Kuroda, Seiki Mori and Hidenori Takahashi
1.
Let G be a finitely generated nonelementary Kleinian group and
let Q(G) be the region of discontinuity of G. The limit set A(G) of G is the complementary set of Q(G) with respect to the extended complex plane
C.
For a (connected) component n* of Q(G), we denote by Gn
*
the stabilizer subgroup of G for n*, that is, Gn = !y. 4. By using Lemma 2, we can prove the following characterization of web groups. THEOREM
1. Given a Kleinian group G, the following four propositions
are equivalent to each other: i)
G is a web group.
ii)
G is finitely generated and A0 (G) = L 2 (G) ~ cP.
iii) G is finitely generated, L 2 (G)
~
cf> and M(A) = S(G) for every
A f L 2 (G).
iv) G is finitely generated, L 2 (G) ~ cP and M(A) = M(A') for arbitrary two A, A' f L 2 (G). Proof. First we suppose i). We note A 0 (G) f, cP. If L 1 (G) is not empty,
then we can find a sequence that
n
I an 1;=l
of distinct separators of G such
(an)= {AI f L 1 (G). The properties c) and d) of separators of G n=l imply an C (a1 ) for every n. By the definition of separators of G, there is a component On of !l(G) such that an = annj for some component Unj of C  fin. Furthermore, the definition of web groups implies that ann is a quasicircle. Hence an = ann. Since A I nn' we see that nn is the component of C  an not containing A. This fact holds for every n and we have 0
1
= On for every n. This contradicts the fact
=!AI. Hence i) implies L 1 (G) = So i) implies ii).
cb.
0()_
n (an) n=l Therefore we have A 0 (G) = L 2 (G)f,cf>.
371
REMARKS ON WEB GROUPS
Next we prove that ii) implies iii). It suffices to show that S(G) C M(A) for every A < L 2 (G) under the condition ii). Take an arbitrary a< S(G) and choose a point z < O(G) which lies in a component of
C
a
not containing A. Obviously a< S(G; z, A). Suppose that a is not the maximal separator in S(G; z,A) for A, that is, suppose a~ a(z,A). Take a point z' < Q(G) in a component of
C
a(z, A) containing A and
choose a separator a'< S(G; z', A) of G. Then a(z, A), a and a' satisfytheconditionfor a 1 , a~,
a;
inLemma2andwehave L 1 (G)~r/J.
Hence ii) implies a= a(z, A) and a< M(A), which proves S(G) C M(A). The proposition iv) is immediately obtained from iii). Now we prove that iv) yields iii). For the purpose we assume that under the condition iv) there exists a point A < L 2 (G) with M(A) ~ S(G). Let a be a separator of G belonging to S(G) M(A) and let z < Q(G) be a point which lies in a component of C  a not containing A. Then
a~ a(z,A). Clearly, in a component of
C a(z,A)
not containing a,
there is a separator a'< M(A). By Lemma 2, there exists a point A'
R
=
nU
is
U/GR be the projection,
then the pullback go TTl is a nonconstant ADfunction on R' = TT(D
n U)
whose real part vanishes on aR'. Thus R X 0 AoD, which proves the theorem. q.e.d. §2. The case of finite genus For surfaces of finite genus, it is known that
FUCHSIAN GROUPS AND OPEN RIEMANN SURFACES
381
Hence Theorem 1 implies the following THEOREM 1 '. Suppose R is of finite genus. If R belongs to class
0 AD, then R is of the type I. This theorem can be improved as follows. Now we shall introduce new classes of Riemann surfaces. Let OAD,n (1
S: n
S oo) denotes the
class of Riemann surfaces on which there are no nonconstant ADfunctions which are at most nvalent, i.e. take every complex numbers at most n times. We may consider 0 AD,""
=0 AD .
Every surface of positive genus
belongs to 0 AD, 1 , while for planar surfaces 0 AD, 1 is identical with the class OSD introduced by AhlforsBeurling, and 0 AD ~ 0 AD, 1 (cf. [1 ], [12 ]). Generally 0 AD C 0 AD ,n+ 1 C 0 AD ,n for every n . One can show a surface belonging to
as follows. ExAMPLE 1. Let R* be a twosheeted covering surface of genus g over the whole Riemann sphere
C = C U l ool,
and rr: R*
>
C
be the pro
jection. Take a compact set E on C which does not contain the branch points of R* and belongs to the class NSB ND. Put R
=
E=
rr 1 (E) and
R* E. Then R ~ OAD, for there exists a nonconstant ADfunction
f on
C E
and f
o
rr belongs to AD(R). Next we show that
R < 0 AD ,n for every n . Otherwise there is a nonconstant mvalent function F < AD(R), and F can be extended to a homeomorphism of R* onto another compact Riemann surface s* of genus g, because
E
has
a planar neighborhood on R* and of class NSB. s* is an msheeted covering surface of
C
and S* F(R) belongs to the class NSB. On
the other hand we can find that s*  F(R) is not in the class N SB, as the area of F(R) is finite. This is a contradiction. THEOREM 3. Suppose R is a Riemann surface of genus g(O 2= g < oo).
YUKIO KUSUNOKI AND MASAHIKO TANIGUCHI
382 Then,
(1) if R belongs to OAD,g+l, then R is of the type I. In particular, every planar surface of OSD is of the type I . (2) There exists a planar surface of the type I which does not belong to the class 0 8 D ~ 0 AD . Proof. Suppose that R is of the type II. We consider, as in the proof of

Theorem 1, the open Riemann surface R with the same genus g as R . Let p be an interior point of
R R
and consider the divisor 8
=
pg+l.
Then by RiemannRoch theorem on open Riemann surface R (Kusunoki [5]) there is a nonconstant meromorphic function f such that f is the multiple of 1/8 and df is (exact) canonical. It is known that f is at most (g+1)valent on R and lldfll is finite outside of a neighborhood of lpl. Thus the restriction of f on R is a nonconstant ADfunction on R, hence R \ 0 AD ,g+l , which proves the first statement. The second statement comes from the following example. EXAMPLE 2. Let ~
=
liz I < 1! and E be its countable subset such
that E does not cluster in ~ and the closure E contains a~' (for example, E
=
lzn,k1;,k=l, where zn,k
=
(1
~) · exp(v12~k)).
Then
R = ~ E is the required for the statement (2). For the sake of the completeness we include the proof. First R
I
OSD,
because R admits a nonconstant schlicht ADfunction z . Now suppose
=
that R is of the type II, and consider an abstract Riemann surface R (U U D)/GR as in the proof of Theorem 1. Since R is planar, by the
classical uniformization theorem we find that there exists a conformal mapping of
R onto a
bounded domain in the complex plane. We denote by
R' the image of U/GR (which we may regard as R) by . Then aR' contains a compact analytic boundary arc b', the image of a free boundary arc b for GR . Each point of E is clearly removable for , thus can be extended to a conformal mapping of
~
to a bounded domain
which contains R' densely. Moreover, by Caratheodory theorem the
~,
FUCHSIAN GROUPS AND OPEN RIEMANN SURFACES
383
conformal mapping  1 can be extended to a continuous mapping from
LS. Let 8' be a simply connected subregion in R' corresponding to D' n U, D' being a smaller disk in D such that aD' is orthogonal to b. Let 8 =  1 (8') C ~. We note that 8 does not con~' U b' into
tain any point of E and that on a~. If p 1
I=
p2 , 8
as
terminates at some points p 1 and p 2
would contain an infinite number of points of E,
which contradicts with the above. Hence p 1
p 2 , but this is also im
=
possible by RieszLusinPrivaloff theorem. Consequently R must be of the type I. q.e.d. §3. Other classes of Riemann surfaces I)
The classes of Riemann surfaces we shall be concerned here are those
related implicitly with the compactification theory. For any positive integer n we denote by 0~ 0 the class of Riemann surfaces R such that dim HD(R)
T g such that the fiber above each
point in T g is the Riemann surface represented by that point [9]. In this paper, however, we shall have no need for the additional structure on X0 nor for the global construction of V. We shall only be interested in any small neighborhood T of o in T g and in purposes, TT: ~
>
X=
TTl(T).
For our
T is more conveniently constructed as the complete,
effectively parameterized deformation of X0 in the KodairaSpencer sense (13], [12], [14]. Since the tangent space to T g at o may be
GENERALIZED WEIERSTRASS POINTS AND RINGS
393
identified with H 1(X 0 , ®), where ® is the tangent sheaf to X 0
[5,
p. 131), TT: V .... T g is locally complete and effectively parameterized. So the constructions from [9) and [12) coincide locally. One can also prove this coincidence from the universal property in [9, Theorem 3.1, pp. 781 For x 0 f X0
,
let dn(x 0 ) be the maximal order of a zero at x 0 of an
h f f'(X 0 , tl(,!l)). Recall that Dn = dim f'(X 0 , tl(,!l)). Observe that since the singular points of a subvariety are nowhere dense and since the number of generalized Weierstrass points of order n is bounded in terms of g [4) the first condition on h in Theorem 2.1 below is the generic case. THEOREM 2.1. Let X 0 be a Riemann surface of aenus g > 2. Let TT: ~
>
T be the complete effectively parameterized deformation of X 0
Let x 0 £X 0
.
.
Fix n~1. Supposethat dn(x 0 ):2:Dn. Let G={Xf~,
x near x 0 1dn(x) = dn(x 0 )l. In case n;::: 2, dn(x 0 ) = 2n(g1) and d 1 (x 0 ) = 2g2, G coincides with G 1 = !xf~, x near x 0 ld 1 (x) = d 1 (x 0 )l. In
ail other cases, G is a submanifold of ~ of dimension 3g3+Dndn(x0 ). TT: G .... T is a finite proper map. If there does not exist an h
with a zero of order dn(x 0 )

1 at x 0
,
f
f'(X 0 , tl(Kn))
then G is transverse to X 0
,
TT(G) is a submanifold of T and TT: G .... TT(G) is a biholomorphic map. If
there exists h f r(xo, tl(,!l)) with a zero of order dn(xo) 1 at xo' then G is not transverse to X0 and TT(G) is sinaular or some points in TT(G) parameterize Riemann surfaces with more than one point x near x 0
with dn(x) = dn(x 0 ). Proof. We shall first consider the case n ;::: 2, dn(x 0 ) = 2n(g1) and d 1 (x 0 ) = 2 g 2 . By taking tens or products, we see that G 1 C G . Let (x denote the point bundle at x on X . Let 
denote equivalence of line
bundles. Then x f G if and only if ,!I  (;n(g1). Then (K(; 2 g)" 0. Since d 1(x 0 ) = 2g 2 , K(; 2g  0. Let K be the line bundle over X 0
which restricts to the canonical bundle on each fiber. By [3, Theorem II, p. 208) or [19, (4.7), p. 52), for each h f f'(X 0 , tl(K)), there exists
394
H
f
HENRY B. LAUFER
r(X:, 0, q 1 q2
f
W2 , and h
f
Mg such that h(Tt 1/J 1 )
=
f
W1 ,
1/J 2 , and similarly for t < 0.
We may assume by Corollary 2.6 that W1 and W2 contain ¢ 1 and ¢ 2 with simple zeroes, that ¢ 1 and ¢ 2 have closed horizontal trajectories in the homotopy classes a 1 and a 2 and nondividing closed vertical trajectories in the classes {3 1 and {3 2
Mg so that g({3 2 ) = {3 1 . Mapping by g, we may assume the vertical trajectories of ¢ 2 are in the class {3
.
Find g
f
{3 1 as well. Then ¢ 1 and ¢ 2 determine projective foliations a 1 = [F¢ 1 ] and a 2 = [F¢ 2 ] and ¢ 1 and ¢ 2 determine the same class {3 = [F]. By Proposition 3.2, rn r13 n fixes two classes =
a1
of foliations [F1 n] and [G 1 n] and for particular representatives the
436
HOWARD MASUR
quadratic differentials ¢ 1n
=
[F2n) and [G 2n) and ¢ 2n
=
(F1n, F2n) _, ¢ 1 . Similarly r~/f"t fixes (F2n, G2n) _, ¢ 2 . Moreover, [F1n) and
[F2n] are attractive fixed points for rn r13 n and rn r13 n resp. in the a1
a2
sense of Proposition 3.2. Since the map p : Q _, PR 8 is continuous, [F1n]_,[F¢ 1 ], [F2n]>[F¢ 2 ] andboth [G 1n] and [G 2n]_,[F] as n>oo. By Proposition 2.5 there are neighborhoods V1 of [F¢ 1 ], V2 of [F¢ 2 ] and V3 of [F] in
:f
such that if [H 1 ] f V1 , [H 3 ] f V3 there
exist H 1 f[H 1] and H 3 f[H 3 ] suchthat (H 1 ,H 3)CW 1 . Similarlyfor [H 2 ] f V2 and [H 3] f V3 there exist H 2 and H 3 such that (H 2 , H 3)fW 2. Pick n large enough so that [F1n] f V1 , [F2n] f V2 and both [G 1n] and [G 2n] f V3 . Let U 1n C V1 , U 2n C V2 and U 3n C V3 be disjoint compact contractible neighborhoods in
:f
of [F1n] , [F2n] and both
[G 1n] and [G 2n] resp. Since [F1n] and [F2n] are attractive fixed
rn r13 n resp., Approachable Approachable
a2
for positive k large enough,
k
and (r~ 1 rjt) (U 2n)CU 1n. Since [G 1n] and are repulsive and U 3n is a neighborhood of both, for large positive k n n k n n k (ra 1 rj3) (U 3n)CU 3n and (ra 2 rj3) (U 3n)CU 3n. L et gnk
=
n n)k n n)k ( ra//3 (ra 2 r/3 and hnk
=
n n)k n n)k (ra//3 (ra//3
Then
gnk(U 1n) C U 1n' g~~(U 3n) C U 3n' hnk(U 2n) C U 2n and h~~(U 3n) C U 3n · By the Brouwer fixed point theorem gnk has fixed points in U 1n and U 3n and hnk has fixed points in U 2n and U 3n. Proposition 2.5 and
3.1 imply there is a quadratic differential qnk in W1 whose orbit T t(qnk) is fixed by gnk. Similarly, there is a Tt(Ci';;k) is fixed by hnk. However, fnk
=
Ci;;k
f W2 whose orbit
(r~ 2 rj3n)k conjugates gnk and
hnk; in fact, mapping the fixed points of one to the fixed points of the other. Therefore, fnk(T tqnk)
=
Ts
Ci';;k.
There are two possibilities. The
first is if gnk and hnk have finite order and are conformal mappings preserving qnk and ~. Then the orbits are fixed pointwise and there exists t such that fnk Ttqnk
=
~. We may not be able to guarantee
both positive and negative t as n and k vary in this case. However,
DENSE GEODESICS IN MODULI SPACE
437
since Mg acts properly discontinuously [6], [10] on T g, there can only be finitely many n and k such that for all but finitely many n and k,
~k
~k
is conformal. This means that
and hnk are hyperbolic and act
as translates of the orbits. Here the numbers Ank of Proposition 3.2 are greater than one and in fact .
oo
again by the discontinuity of Mg. For
each integer m, (hnk)m · fnk(qnk) is on the orbit of
&;k.
Composing
with (hnk)m has the effect of translating fnk(qnk) along the orbit. For each m then there is a t such that Tt((hnk)m · fnk(qnk)) = ~. By taking lml large enough we can insure t both positive and negative. The theorem is proved. UNIVERSITY OF ILLINOIS AT CHICAGO CIRCLE
REFERENCES [1]
[2] [3] [4] [5] [6]
L. V. Ahlfors, On Quasiconformal mappings, Journal D'Analyse Mathematique, 4(1954), 158. L. Bers, Quasiconformal mappings and Teichmiiller's theorem, in Analytic Functions. (R. Nevanlinna et al., eds. ), Princeton University Press, 1960.      , An extremal problem for quasiconformal mappings, Acta Math. 141: 12, 7398 (1978). G. D. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publications, 9 (1927). A. Douady and J. Hubbard, On the density of Strebel Differentials. Inventiones Math. 30 (1975), 175179.
C. Earle and J. Eells, A fibre bundle description of Teichmiiller theory. J oumal of Diff. Geometry 3 (1969), 1945. [7] G. Hedlund, The dynamics of geodesic flow, Bulls. AMS, 45(1939), 241260. [8] J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221274. [9] H. Keynes and D. Newton, A Minimal nonuniquely ergodic interval exchange transformation. Math Z. 148(1976), 101105. [10] S. Kravetz, On the geometry of Teichmiiller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. 278(1959), 135. [11] H. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case. Israel Journal of Math. 21 nos. 23 (1975), 133144.
438
HOWARD MASUR
[12] H. Masur, The Jenkins Strebel differentials with one cylinder are dense, Comm. Math. Helv. 54(1979), 179184. [13] K. Strebel, On Quadratic differentials and extremal quasiconformal mappings, Univ. of Minnesota. Lecture Notes. (1967). [14] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, to appear. [15] A. Fathi et al., Travaux de Thurston sur les surfaces, Asterisque 6667 Societe mathematique de France, 1979.
AUTOMORPHISMEN EBENER DISKONTINUIERLICHER GRUPPEN Gerhard Rosenberger
Einleitung A. Se i G = < s 1 , · · · , s m, a 1 , · · · , ap Is~ 1 = · · · = s;m [ap_ 1 , ap]
=
=
s 1 · · · s m [a 1 , a 2 ] · · ·
1 >, yi ::;, 2 sowie m ::;, 3 und m 2  . ~
1=1
t
> 0 falls p
=
0,
1
eine Fuchssche Gruppe mit kompaktem Fundamentalbereich. In [15] hat Zieschang gezeigt, class jeder Automorphismus von G induziert wird von einem Automorphismus der freien Gruppe vom Rang p + m (vgl. auch [17]). Hier erweitern und erganzen wir dies en Satz, indem wir unter anderem zeigen: Jeder Automorphismus von G wird induziert von einem Automorphismus der freien Gruppe vom Rang r, wobei r der Rang von G ist. Dies beinhaltet gleichzeitig einen neuen Beweis fiir den erwahnten Satz von Zieschang. B. Diese Arbeit verwendet die Terminologie und Bezeichnungsweise von [8] und [16], wobei < ... I .. ·> die Gruppenbeschreibung durch Erzeugende und Relationen bedeutet. Unter < a 1 , ···,am> verstehen wir die von a 1 , ... ,am erzeugte Gruppe. Wie in [8] und [16] gewinnen wir oft aus einem System lx 1 , ... , xn! durch freie Ubergange (NielsenTransformationen) ein neues und bezeichnen es mit denselben Symbolen. Es bedeute: [a, b].
=
aba 1 b 1 den Kommutator von a und b. (a, {3)
den grossten gemeinsamen Teiler von a, {3
f
N.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00043917$00.85/1 (cloth) 0691082677/80/00043917$00.85/1 (paperback) For copying information, see copyright page 439
440
GERHARD ROSENBERGER
§1. Vorbemerkungen Sei G = H 1
* H2 A
freies Produkt der Gruppen H 1 und H 2 mit
Amalgam A= H 1 n H 2 . Ferner sei in G eine Ordnung und eine Lange L wie in [8] und [16] definiert. Die Ordnung geniige der Bedingung, class vor einem Produkt von Restklassenvertretern 11 .. · lm von A in den Hi nur endlich viele Produkte 11 .. · lm_ 1 1 liegen, wobei 1 ein Restklassenvertreter a us demselben Faktor Hi wie lm ist. Aus Satz 1 von [16] und Korollar 2 von [8] erhalten wir SATZ 1.1.
1st !xl' .. ·,xnlCG einendlichesSystem, soist !x 1 , .. ·,xnl frei aquivalent zu einem System !y 1 , · · ·, y n l, fiir das einer der folgenden
Faile vorliegt: (i)
q Ei Fiir jedes w f gibt es eine Darstellung w = TI Yv.'
i=1
Ei = ±1, Ei = Ei+ 1 falls vi= vi+ 1 mit L(yv)
1
:S L(w) fiir i = 1, ... ,q.
1
(ii)
q
E·
EsgibteinProdukt a= TI Yv1., aid, Yv.fA(i=1, .. ·,q) und i~1
1
1
in einem Faktor Hi ein Element x ~A mit xax 1 fA. (iii) Einige der Yi liegen in einer zu H 1 oder H 2 konjugierten
Untergruppe von G, nicht aile liegen in A, und ein Produkt in ihnen ist zu einem von 1 verschiedenen Element aus A konjugiert. BEMERKUNGEN:
1) Der freie Ubergang kann in endlich vielen Schritten so gewahlt werden, class !y 1 , ... ,ynl ktirzer ist als !x 1 , ... ,xnl oder die Langen der Elemente von ! x 1 , 2) 1st ! x 1 , (i): L(yi)
· · ·,
.. ·,
xn l erhalten bleiben.
xn l ein Erzeugendensystem von G , so folgt a us Fall
:S 1 ftir i = 1, ... , n.
Im weiteren benotigen wir noch die folgende, geringfiigige Verallgemeinerung von Satz 1 aus [6]: SATZ 1.2.
Sei H = , m ~ 2, yi
~
2. Sei
!x 1 , ... ,xnl C H(n · Sei !x 1 , · · ·, xp+m 1 1 ein Erzeugendensystem von G. 1st x 1
13·
konjugiert zu einem sj J, so ist (/3j, Yj) = 1 nach Satz 2.1. 1st x 1
13·
konjugiert zu einem aj J , 1 S j S n , mit 1 S /3j < aj, /3j [aj , so ist /3j = 1 ebenfalls nach Satz 2.1. 1st m = 2 und y 1 = y 2 = 2, so ist x 1 nicht konjugiert zu einer Potenz von a , da sonst das freie Produkt * 1. Es ist m 2: 3 , denn ftir m = 2 erhalten wir mit Satz 1.1 und Satz 1.3 sofort einen freien Ubergang von lx 1 , · · ·, xp+ 1 ! zu einem System la 1 , · · ·, ap, y 1 1 (mit Hilfe von Satz 2.1 ergibt sich sofort, dass kein xi, 2
~
i
~
p+1, in
derselben zu H 2 konjugierten Untergruppe liegen kann wie x 1 ). Sei nun m > 3 . Das freie Produkt Y·
.
* * den Rang p + m 2 hat. Mit derselben Begriindung kann es nach Satz 1.2 nicht sein, dass einige der xj (2 ~ j ~ p + m1) in einer zu K 1 konjugierten Untergruppe von G liegen und ein Produkt in ihnen zu einem von 1 verschiedenen Element von B konjugiert ist. Da andererseits wieder eine
447
EBENER DISKONI'INUIERLICHER GRUPPEN
Situation (iii) von Satz 1.1 eintreten muss, liegt also Fall (c) von Korollar 1.5 vor; d.h. es gibt einen freien Ubergang von lx 1 ,
... ,
xp+m 1 1 zu einem
System lx 1 ,z 2 , .. ·,zm_ 1 ,za 1 z 1 , .. ·,zapz 1 1, z cG. Dieswiderspricht aber unserer Annahme. Also gilt Behauptung (2.5). Dam it konnen wir xm_ 1+i = ai, i = 1, · · ·, p, annehmen. Wir betrachten wieder die alte Faktorisierung G = H 1 * H 2 und verkiirzen A
lx 1 , ... ,xm_ 1 ,a 1 , ... ,apl bzgl. dieser Faktorisierung. Tritt eine Situation (iii) von Satz 1.1 ein, an der einige der xi, 1
S i S m1,
beteiligt sind, so trittanalog wie bei (2.5)einer der
folgenden Falle ein: (2.8): Es ist m ungerade, y 1
= .. · = Ym
= 2, und es gibt einen freien
Ubergang von l x 1 , · · ·, xm_ 1 , a 1 , · · ·, ap I zu einem system
l zs 1 s 2 z  1 , .. ·,zs 1 smz
1 ,a , .. ·,ap, I 1
Zf
G.
(2.9): Es ist m gerade, alle Yj gleich zwei his auf eines, das ungerade ist, und es gibt einen freien Ubergang von lx 1 , ... ,xm_ 1 ,a 1 , ... ,apl zu einem System ly 1 , ... ,ym_ 2 ,za 0 z 1 ,a 1 , ... ,apl, ::;.1, z 3 werdenin [S] gegeben. 2) Fur p = 0, m = 3 gilt eine Aussage wie in Korollar 2.11 im allgemeinen nicht (vgl. [2] und [10]). Hier sind die Erzeugendenpaare durch Satz 4 von [10] gegeben. SATZ 2.12. Sei p = 0, m ;:> 4 und Rang (G)= m2. Dann gibt es genau
eine NielsenAquivalenzKlasse minimaler Erzeugendensysteme. 2 =sY=s ···S =1> m>4 Konkreter: Sei G=(sk) = cksj Jck_ 1 , ck j d1, ... , ml, so ist Yk = Yj und
l3j
f
G,
= ±1 (mod Yj ). Nun folgt fiir p = 0,
m > 4 und r(G) = m1 die Aussage von (3.1) aus Korollar 2.11. 5) Sei nun m:;:: 3, p = 0 und r(G) = m 2. Dann folgt die Aussage
von (3.1) unmittelbar aus Satz 2.12. q.e.d. BEMERKUNG. Fiir .}1 + Y~ + .}3 > 1 falls m = 3 ist G endlich, und die Aussage von Satz 3.1 ist im allgemeinen nicht richtig. 1st
..!.
Y1
+
..!. Y2
+
falls m = 3, so ist die Aussage von Satz 3.1 richtig. In [2] und [10] benotigten wir aus beweistechnischen Grunden die Voraussetzung
l
Y1
+ .!._ + l < 1 falls m = 3 . Y2 Y3
ABTEILUNG MATHEMATIK DER UNIVERSITAT DORTMUND POSTFACH 50 05 00 4600 DORTMUND 50
..!. =
Y3
1
EBENER DISKONTINUIERLICHER GRUPPEN
455
LITERATUR
[1]
Automorphisms of the Fuchsian groups of type R.N. Kalia, G. Rosenberger, (0; 2,2,2,q; 0). Comm. in Alg. (6) 11 (1978),
[2]
A. W. Knapp,
Doubly generated Fuchs ian groups. Mich. Math.
[3]
W. A. D. N.
Combinatorial group theory. New York: Wiley 1966.
11151129.
[4] [5] [6] [7]
Magnus, Karrass, Solitar, Peczyns ki,
15 (1968), 289304.
J.
Eine Kennzeichnung der Relationen der Fundamentalgruppe einer nichtorientierbaren Flache. Diplomarbeit, Bochum 1972. Uber Erzeugendensysteme von Fuchsschen Gruppen. N. Peczynski, Dissertation, Bochum 1975. Uber Erzeugende ebener diskontinuierlicher Gruppen. N. Peczynski, G. Rosenberger, lnventiones math. 29(1975), 161180. H. Zieschang, The isomorphism problem for onerelator groups with S. J. Pride, torsion ist solvable. Trans. Amer. Math. Soc. 227
(1977), 109139.
[8] [9]
G. Rosenberger,
Zum Rang und Isomorphieproblem fur freie Produkte mit Amalgam. Habilitationsschrift, Hamburg 1974.        · Zum Isomorphieproblem fur Gruppen mit einer definierenden Relation. Ill. J. Math. 20 (1976),
614621 [10] ________ , Von Untergruppen der TriangelGruppen. Ill. J. Math. 22 (1978), 404413. [11] ·       · Uber Gruppen mit einer definierenden Relation. Math. z. 155 (1977), 7177. [12]       · Alternierende Produkte in freien Gruppen. Pacific J. Math. 78 (1978), 243250. [13] G. Rosenberger, Eine Bemerkung zu den NielsenTransformationen. Mh. Math. 83 (1977), 4356. F. Tessun [14] 0. Schreier, Uber die Gruppen Aa Bb = 1 . Abh. Math. Sem. Univ. Ham b. 3 (1924 ), 167169. [15] H. Zieschang, Uber Automorphismen ebener diskontinuierlicher Gruppen. Math. Ann. 166 (1966 ), 148167. [16] , Uber die Nielsensche Kurzungsmethode in freien Produkten mit Amalgam. lnventiones math. 10(1970), 437. [17] H. Zieschang, Flachen und ebene diskontinuierliche Gruppen. E. Vogt, Lecture N.otes in Math. 122, Springer 1970. H.D. Coldewey,
REMARKS ON THE GEOMETRY OF THE SIEGEL MODULAR GROUP Robert I.
J. Sibner
Introduction
1.1. Thinking of a torus T (with modulus r) as the quotient of the universal cover C by the group of translations
! L 1 , Lrl where Laz
=
z +a,
the various types of conjugate holomorphic involutions of T can be obtained as projections of involutions of C as follows: (i) If Re r reflect C in the line y
= r /2
=
0,
. The quotient surface is orientable and has
two boundary components (i.e. a cylinder). (ii) Again for Re r translate by Ly, and reflect in y
= r /2.
=
0,
The quotient surface is a Klein
bottlecompact and nonorientable. (iii) For Re r translate by Ly, and reflect in y
=
=
~, one can again
r /2 . Now one obtains a nonorientable
surface with one boundary component (a Moebius band). Note that a Moebius band can also be obtained for lrl = 1 by reflecting in the line through the origin and the point r . Given the above information, Bers once observed that lrl Re r
=
=
1 and
~ are the defining conditions for the boundary of the standard fun
damental domain of the elliptic modular group and that Re r
=
0 is an
axis of symmetry. This paper is an attempt to make precise this observation and to obtain a corresponding statement about the period matrices for surfaces of arbitrary genus.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00045707$00.50/1 (cloth) 0691082677/80/00045707$00.50/1 (paperback) For copying information, see copyright page
457
458
ROBERT J. SIBNER
1.2. We define an involution J of the Siegel generalized half plane and show that any period matrix of a surface conformally equivalent to its conjugate is equivalent (modulo the Siegel modular group) to its image under
J . Moreover, the Siegel fundamenta 1 domain F is invariant under the action of J , as are the various types of boundary sets. An extended modular group is formed by adjoining ] to the Siegel modular group and a fundamental domain is obtained for this group. We show that every surface which is conformally equivalent to its conjugate
has a period matrix located on the boundary of this fundamental domain. A symmetric Riemann surface is a surface which admits a conjugate holomorphic involution and as such is conformally equivalent to its conjugate. More precise information on the location of symmetric surfaces has been obtained and will appear in [2]. II. The modular group 2.1. We denote by Hg the Siegel (generalized) half plane [3] consisting of g x g complex matrices Z
=
X+ iY with Y positive definite. The
group of (holomorphic) automorphisms of Hg is the symplectic group Sp (g, R), whose elements have the form
with real g x g matrices A , B , C , and D satisfying MTMt = T where
and
2.2. A fundamental domain D for a group G acting discontinuously on Hg is the closure of an open connected subset of Hg with the following two properties: (i) D contains a point in every Gequivalence class and (ii) no two points in the interior of D are equivalent by a nontrivial ele
ment of G.
459
GEOMETRY OF THE SIEGEL MODULAR GROUP
As is well known, the Siegel modular group (of degree g) 1
=
Sp (g, Z)
C Sp (g, R), consisting of symplectic transformations with matrix coeffi
cients A , B , C , and D having rational integral entries, acts discontinuously on Hg and has a fundamental domain F which we now describe
[3]. Denoting by IIWII the absolute value of the determinant of a matrix W, and by L 1 [W], ... , Lq [W] the homogeneous linear functions of the elements of W which arise in the Minkowski theory of reduction of quadratic forms, the Siegel fundamental domain F is the set of all Z < Hg satisfying (a) IICZ+DII2: 1 for all modular transformations Z> (AZ+B)(CZ+Dr 1 (b) Lr[yl] :;> 0 (c)
21
~ by ii) above. Let such an interval define the annulus A in question, so that x fA C N(y) and log (dy) varies through
""2~
in A , by construction.
vi) Since A C N(y), diameter A and diameter y A< 8 by i) and Corollary 2. vii) Since dy "" 1/Ar 2 on N(y) and thus on A, variation
logdy2~ on A impliesmodulus A~(logr 22 /r 1 2 =2~ iff log r 2 /r 1 =~ ). Similarly yA has radii dr 1 e 2 ~ and dr 2 (for some d) because we know dy on A. Thus the modulus of yA is
log
Now we pass from the sphere to the plane. LEMMA 4. If e
is an absolutely continuous isomorphism of the plane
(relative to Lebesgue measure) carrying B to B', then a subset A C B with a proportion TJ of area is carried to a subset e(A)
=
A' C B' of pro
portion at least TJ' of area where TJ' = 1d(1TJ) and d is the maximum ratio of area distortion at various points of B. Proof. By an affine scaling we can assume area B = 1, the low value of area distortion is 1, and the high value is d. The worst case occurs when 1 occurs on all of A and d occurs on all of the complement of A. Then, LEMMA 5.
TJ' = area A'/area B' = TJITJ+d(1TJ) ~ 1d(1TJ).
Let X be a set in the plane of positive measure and let TJ
and ~ be given positive numbers. Consider sectorial boxes of shape ~,
Then there is a 8 > 0 and a subset X' of X of positive measure so that each box of shape ~ and diameter < 8 containing a point of X' also contains at least the proportion TJ of X.
474
DENNIS SULLIVAN
Proof. The class of sectorial boxes of shape 1'1 are generated by simi
larity transformations from one of them. Thus Lebesgue's theorem concerning density points is true using these instead of round disks. (See
E. Stein "Singular Integrals ... " pp. 11, 12.) So for almost all x there is a largest positive portion of X in boxes containing x of diameter
ox ox
is a positive measurable function which has to be greater
than some
o> 0
on a set X'C X of positive measure. This proves the
lemma. Now consider a conformal transformation of the plane ad be= 1,
c,£0
and sectorial boxes BL'1 of shape 1'1 < rr/2 centered at d/c. LEMMA 6. If A C BL'1 is any subset with the proportion 71 of area, then the variation on A of the real and imaginary parts of log y'z
is at least
21'1(1 e 21'1(171)). Proof.
i) On a unit square the function (x, y) . x has variation at least
71' on any subset whose proportion of area is at least 71'. ii) Introduce the variable ee
=
z +d/c so that the variation of
1log y'z = log  = 2 log (Z+d/c) + constant on A C BL'1 is just the (cz+d) 2 variation of 2e on a corresponding subset A' of a square in the tplane of side 1'1 .
logy '(z)
2 {; + constant
c
475
THE ERGODIC THEORY AT INFINITY
iii) The ratio of area distortion of exp at different points of the
e 2 ~. By Lemma 4 the proportion of A'= exp 1 (A) in the square is at least r/ = 1e 2 ~(17]).
square is at most
iv) Applying i) the result follows. Now we are ready to prove the nonexistence of invariant measurable line fields for groups of finite solid angle. i)
Choose a small number, rr/2 > ~ > 0 and a set of positive measure
X in the plane where the hypothetical invariant line field varies only in
~~ . ii) Choose 0 < 7] < 1 so that 1 e 2 ~(1 77 + e 2 ~(17])) > 1/2. iii) Find X'C X of positive measure satisfying a o'> 0 uniform
an interval of inclinations of length
density relative to X, 7J, and sectorial boxes of shape ~ (as in Lemma 5). iv) Choose a point of density of X' and stereographically project the action of [' on the plane to a sphere resting on this point. v)
Let Y denote the intersection of X' with a ball B' about this
point sufficiently small so that the distortion of stereographic projection on 2B is as close to 1 as we need for the following. Let
o = min (o',
radius B'). vi) Relative to
o, ~
and Y (put Y on the sphere) find the element
y and the concentric annulus A satisfying Lemma 3 (and put A back on
the plane). In A choose a sectorial box B of shape
 ~ containing x
and centered at the pole of y (possible because we know the variation of log dy on A). vii) Since the diameters of B' and y B' are less than
o'
(even o)
they each contain the proportion 7J (at least) of X . (x 0 on
orbit of some smaller ball. Now consider the identity
The righthand side is uniformly bounded wrt N. The lefthand side is at least e I gy(yp) as N
r
+ "".
We conclude that "B = P77 8 or 1T 8 (y)
Thus almost all paths starting at y hit
r
B when
l
yfr
gy(yp)
=
= oo.
dJ.Lx ii) For fixed s 0 and x 0 the function of x, d (s 0 ) is a
llxo
Pharmonic function (Lemma 2), whose boundary values are + oo at s 0
1.
481
THE ERGODIC THEORY AT INFINITY
and zero at other points of the sphere. One sees by a standard limiting procedure [K] that gx(y)
~(s 0 )
is the corresponding Green's density for
the random walk conditioned so that the limit at
oo
is s 0 . By Fubini's
theorem and i) for almost all s 0 , almost all paths starting at y and conditioned to end up at s 0 must also hit [' B . Now if for one of these s 0
!
yen 2 . ' ' ,......,. ,......,.
Note that if Rk,oQk,j and Qk,lQk, 2 are members of a subsequence with coinciding limit, then R 0 must be arbitrarily close to A(yk), and
..
so [R 1 R 2 [ must go to zero. Hence we assume that vk,j > 0. (Recall
,....
,......,.
that by the choice of subsequences l Qk 1 Rk 1 ! and l Qk 2Rk 1 !, every remaining k will be represented in the respective subsequences lvk 1 ! I
I
I
1
'
or lvk 2 !, both with limit zero.) ' Consider the arrangement of the isometric circles l(y) and I(y 1 ) and the axis A(y) of the hyperbolic transformation y(z) = az +db. Compucz+ tations show that the distance D between either endpoint of the axis of y and the nearer of the endpoints of the isometric circles under the axis
1  _E_, of y satisfies D > · [ckl 2[ck[
E
> 0, where
E.
0 as the length of
the geodesic determined by y goes to infinity. Hence, if any subsequence of l [ck[ l remains bounded, len I :::; M, then, for lkl > K, Dk ;=:
[c1k 1  21 ~k/>0,
e>O, and
Dk2:~{1~)>0
for lki>K. ByLemma2,
503
ON INFINITE NIELSEN KERNELS
as k
+  oo,
~j)
vk,j (ordinate of the intersection point Qk,j of A(y) and
must go to zero. Since Dk;:::
h(1 ~) , Qk,j
,.........
cannot collapse to
Wk 1.. Hence for iki > K, the arc Wk 1Qk 1Vk 1. (a side of the fundamen' ,,, ~ tal polygon) must be contained in l(y)(I(y 1 )). But then Wk hQk h Vk h, ' ' ' hI= j, must be an arc outside I(yk 1 )(I(yk)), which is tangent to and outside a circle of radius
h centered at least ,......
~
(1 V units away from
,......
Wk,i. Since y is conformal, and Rk, 0 Qk,j, Qk,jRk, 1 are distinct, the image of ak,j , atj with vertex Qk,h also must go to zero, and hence vk h
,......
0. Then Qk 1 Qk 2 must approach the x axis, with small ' ' ' Euclidean length. This clearly cannot be done via a sequence of convex +
fundamental polygons. Hence {len II must be unbounded.
,.........
We now must consider a sequence of transformations {ykl where the radii of I(y) and I(y 1 ) become arbitrarily small. Suppose {Wk,jQk,jVk)
!w;
has a subsequence which remains inside or on the isometric circle I(yt 1 ). Since 1 1 .... 0, ckl
'
1.Qk 1Vk 11 must collapse to a point, and ' '
Vk 1. and hence Vk h must be arbitrarily close to points on {A(y)nRI. ·~ ' +1 ~ If {Wk 1Qk 1Vk 11 remains outside (or on) I(yk ), then {Wk hQk hvk hi '
,
,
+l
'
,
,
must be inside (resp. on) I(yk ) and the same argument holds. The corresponding situation exists for the neighboring y (by the remark following
..
Lemma 2). A computation then shows that IR 1 R 2 1 must go to zero. WELLESLEY COLLEGE WELLESLEY, MASS. 02181
REFERENCES
[1] Bers, L., Nielsen Extensions of Riemann Surfaces, Ann. Acad. Sci. Fenn., 2 (1976), 1722. [2] Keen, L., Canonical polygons for finitely generated Fuchs ian Groups, Acta Math., 115 (1966), 116. [3] , Intrinsic Moduli on Riemann Surfaces, Ann. of Math., 84 (1966), 404416. [4] Meschkowski, H., Noneuclidean Geometry, Academic Press, New York, 1964.
HYPERBOLIC 3MANIFOLDS WHICH SHARE A FUNDAMENTAL POLYHEDRON Norbert J. Wielenberg*
1. Introduction Let G be a discrete, torsionfree subgroup of PSL(2, C). Then the quotient under the action of G of the region of discontinuity of G in the closure of hyperbolic 3space is a 3manifold, possibly with boundary. The boundary is U(G)/G where U(G) is the ordinary set of G in C U
Iool.
Marden [2] has discussed group constructions which topological
ly consist of gluing two boundary components of the same or of different 3manifolds. This is of particular interest if each component of each lift of a boundary component is a euclidean disc. Geometrically the constructions then consist of gluing totally geodesic surfaces which are incompressible and boundaryparallel in the 3manifolds. Algebraically, the group constructions are free products with amalgamation or HNN extensions. See [8] for link groups constructed in this way and Thurston [6] for more general examples. As corollaries of the theorems of Gromov and JI!Srgensen, the following results are given in [6]. The set of volumes of hyperbolic 3manifolds is well ordered. The volume is a finitetoone function of hyperbolic manifolds with finite volume.
Research supported in part by the National Science Foundation.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedi~s of the 1978 Stony Brook Conference 0691082642/80/00050509$00.50/1 (cloth) 0691082677/80/00050509$00.50/1 (paperback) For copying information, see copyright page 505
506
NORBERT J. WIELENBERG We will discuss some groups which are constructed by HNN extensions
and which justify the following. THEOREM. For each integer N, there is a polyhedron of finite volume
·in hyperbolic 3space which is a fundamental polyhedron for at least N different 3manifolds. In our examples, the corresponding groups are subgroups of finite index in the Picard group PSL (2, Z(i)).
2. Hyperbolic isometries As a model for the simplyconnected hyperbolic 3space we will use H3
c
=lz+tj:j=(0,0,1) and t>Ol. TheRiemannianmetricis
2(dx 2 + dy 2 + dt 2 ) and the volume element is dV = C 3 (dx dy dt). The
operation of inversion in the sphere of radius 1 with center at the origin can be written as z + tj .. (z + tj)* where
z+ tj(z+tJ')* = lz+tjl2 This is an isometry of H 3 . For c < C, we can define an orthogonal transformation Tc of R2 by T c(z) =  (c/c)z. An orthogonal transformation of R2 extends in an obvious way to an orthogonal transformation of R3 which is an isometry of H 3 , i.e., leave the tcoordinate unchanged. We remark that if A(z) = az + b with ad be = 1 , then an elementary calculation shows that cz + d A(z) = e.+ c
2 T (z + 9..)* (..1..) lei c c
We then extend A to act on H 3 by (1)
A(z + tj)
e. + c
2 T lz + tj + 9..) * (..1..) lc I c\: c
This is an isometry of H 3 and the group of all orientation preserving
507
HYPERBOLIC 3MANIFOLDS
isometries of H 3 can be identified with PSL (2, C), where A corresponds to
±(: :)
and
A~l
The circle where lcz +dl we have isometric spheres
=
to
±(~:
~~).
1 is called the isometric circle of A. So
lz+tj+~l
_!_ and lz+tj~~l
=
lei
and A~l. When c is real, note that Tc(z) tion in the plane x
=
=
=
_!_ for A
lei
~z, that is Tc is reflec
0. Recall that a parabolic, elliptic, or hyperbolic
transformation can be conjugated into PSL (2, R). We obtain the action of A on H 3 as discussed by Riley in [4]. It is possible to read off from the matrix of A how the points of the two isometric spheres are paired by A and A~l.
If c
t
0, then by (1) A consists of the composition of inversion in
the isometric sphere of A, reflection in a plane through ~ ~ and perpendicular to the line between
~ ~ and ~, a rotation leaving ~ ~ fixed
(possibly the identity), and translation of the isometric sphere of A to the isometric sphere of A~l. The reflection and rotation can be combined into a reflection in a plane through ~ ~ and perpendicular to the complex plane. Note that if A is parabolic, the isometric spheres are tangent at the fixed point. If A is hyperbolic, they are disjoint. If A is elliptic, they intersect in a circle. If A is loxodromic, the isometric spheres may be disjoint, tangent, or intersecting.
3. Some link groups If the volume of H 3 /G is finite, then H 3 /G is homeomorphic to N~e where N is a closed 3manifold and
e
=
C 1 U ... U Cn where each Ci is
homeomorphic to S 1 . (See [7] or Thurston [6].) The fundamental group of S3 ~
e is
called a link group. We refer to Riley [5] for a discussion of
representations of link groups in PSL (2, C). In particular, an antiisomorphism between a discrete finitevolume subgroup of PSL(2, C) and a link group implies that the link complement is homeomorphic to the corresponding quotient space.
508
NORBERT
J. WIELENBERG
By a cusp torus we mean a set homeomorphic to lz: 0< lzl < 11 x S 1 . Cusp tori occur in H 3 /G as deleted neighborhoods of the components of a link [2] and correspond to conjugacy classes of rank two parabolic subgroups of G. The number of cusp tori in H 3 /G is equal to the number of cycles of parabolic fixed points on the boundary of a Poincare region or Ford domain for a finitevolume group. We refer to Maskit [3] for a discussion of fundamental polyhedra for discrete isometry groups of H 3 . Let M(G) = (H 3 U O(G))/G. Recall that a surface S in a 3manifold M is incompressible if the inclusion S ... M induces an injection of fundamental groups,
11 1
(S) ...
11 1
(M).
It follows from Fine [1] that the Picard group is generated by
v =
(_~ ~)
w =
(~ ~)
0
The group G 1 generated by t 4 = t 4 , u 2 = u 2 and v is a free product G 1 = < t 4 , u 2 , v; t 4 u 2 =u 2 t 4 >. aM(G 1 ) has two components, each a 3punctured sphere. These components are covered by euclidean discs stabilized by fuchsian subgroups of G 1 , e.g.,
(See Figure 1 which shows the isometric circles of v and v 1 and the generators of F 1 and F 2 and lines paired by t 4 and u 2 . Imagine hemispheres over the circles and halfplanes perpendicular to C along the lines.) Let Ii be the hemisphere over the disk ~i preserved by F i. Then I/Fi is incompressible, totally geodesic, and parallel to ~/Fi. We
509
HYPERBOLIC 3MANIFOLDS
2
Fig. 1
can glue 2 1 /F1 to 2 2 /F2 by adding a generator s such that sF2 s 1 = F 1 and s takes the outside of 2 2 to the inside of 1 1 . By SeifertVan Kampen the result G is an HNN extension of G 1 , G = < G 1 ,s; sF2 s 1 = F 1 >. It was shown in [8] that if s is parabolic, then H 3 /G = S 3  B where B is the Borromean rings. The surface 2/F i occurs in
s3 
B as a
twicepunctured open disc which spans a component of the link (Figure 2(a)). The group presentation reduces by Tietze transformations to
(a)
(b) Fig. 2
(c)
510
NORBERT J, WIELENBERG
A loxodromic s , which can be thought of as a parabolic transforma
tion followed by a 180° rotation, gives a halfinteger twist of a spanning disc. The result is the link in Figure 2(b). (See also Thurston (6], Section 6.8.) The group presentation in this case reduces to 1 ] [ 1 1 ] 1 .
Fig. 3
The fundamental polyhedron for both groups is shown in Figure 3. It lies above the hemispheres indicated by circles and between the portions of vertical planes indicated by lines. A third group also shares this polyhedron. It is discussed in detail in (8]. The link complement is shown in Figure 2(c) and can be obtained from 2(b) by a twist along a different spanning disc. It is now fairly clear how to proceed to obtain examples which illustrate
our theorem. Start with a large fundamental polyhedron as indicated in Figure 4(a). The resulting 3manifold M has an even number of 3punctured spheres as boundary components. The boundaryparallel, totally geodesic surfaces in M can be glued in many different ways with parabolic and loxodromic transformations. The only requirement is that the additional generators conjugate the fuchsian groups which stabilize the corresponding hemispheres and interchange inside and outside of the hemispheres.
511
HYPERBOLIC 3MANIFOLDS
• • • (a)
(b) Fig, 4
If we use all parabolic generators to get a cyclic nfold covering of the Borromean rings in the obvious way, we obtain a link complement like the one in Figure 5. The point at the center indicates a component of the link which can be thought of as a straight line perpendicular to the plane
u a •
Fig. 5
512
NORBERT]. WIELENBERG
of the paper and passing through
oo.
If one of the parabolic generators is
replaced by a loxodromic, the result is a halfinteger twist along a twicepunctured spanning disk. The link complements in Figure 6 can be obtained, for example.
(a)
(c)
(b) Fig. 6
In particular, the groups indicated by Figure 4(b) can give link complements with 6, 7, 8, or 9 cusp tori. The fundamental polyhedron is the same for each of these groups but the groups are clearly not isomorphic. An arbitrarily large number of nonisomorphic groups can be seen to share a fundamental polyhedron in this way. The following considerations show that these groups are subgroups of the Picard group. Form the composition of inversion in a hemisphere of radius 1 and reflection in a vertical plane parallel to x
=
0 through the
center of the hemisphere. Following this with translation two units to the right gives a parabolic, translation two units up gives a loxodromic. The centers of the isometric spheres of v, v 1 , w, w 1 are at 1, 1, i, i and the spheres all have radius 1. By conjugation by translations, we can move these centers to any point m + ni where m and n are integers. The index of each group in the Picard group is given by the ratio of the volume of its polyhedron to the volume of a fundamental polyhedron for the Picard group. The Borromean rings are of index 24. UNIVERSITY OF WISCONSIN PARKSIDE KENOSHA, WISCONSIN 53141
HYPERBOLIC 3MANIFOLDS
513
BIBLIOGRAPHY [1] B. Fine, The structure of PSL(2, R); R, the ring of integers in a euclidean quadratic imaginary number field, Discontinuous Groups and Riemann Surfaces, ed. L. Greenberg, Ann. of Math. Studies 79, 1974. [2] A. Marden, Geometrically finite kleinian groups and their deformation spaces, Discrete Groups and Automorphic Functions, ed. W. J. Harvey, Academic Press, 1977. [3] B. Maskit, On Poincare's theorem for fundamental polygons, Adv. in Math. 7(1971), 219230. [4] R. Riley, A quadratic parabolic group, Math. Proc. Cambridge Phil. Soc. 77(1975), 281288. [5] , Discrete parabolic representations of link groups, Mathematika 22 (1975), 141150. [6] W. P. Thurston, The Geometry and Topology of 3Manifolds, lectures at Princeton University, 197778. [7] N. Wielenberg, Discrete Moebius groups: Fundamental polyhedra and convergence, Amer. Jour. of Math. 99(1977), 861877. [8] , The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Phil. Soc., to appear.
THE LENGTH SPECTRUM AS MODULI FOR COMPACT RIEMANN SURFACES Scott Wolpert* The purpose of this note is to announce results concerning a moduli problem. A complete description will appear elsewhere, [1]. Denote the Teichmiiller space for compact Riemann surfaces of genus g, g 2: 2 as T g. A point IS l of T g is the equivalence class of a pair, a Riemann surface S and the homotopy class of a homeomorphism from a base surface S 0 to S. Let Mg be the extended Teichmiiller modular group. A point of T g/Mg is the conformal and anticonformal equivalence class of a Riemann surface. Let ~(S) be the set of free homotopy classes of closed curves on S. The marking homeomorphism of
l Sl
< T g is used to identify
rr 1 (S 0 ) (resp. ~ =~(S 0 )) and rr 1 (S) (resp. ~(S)). Each element [y] of ~(S) contains a unique Poincare geodesic; denote by [y](S) the length
of this geodesic. Let 1 be the group of deck transformations for the covering of S by the upper half plane. It was noted by Fricke Klein that the information I ([y], [y ](S)) I [y] < ~ l determines 1 modulo conjugation by PGL(2; R) and S modulo Mg. Later it was asked if the set (with multiplicities) of numbers l[y ](S) I [y] < ~ l determines 1 modulo conjugation. We emphasize in the former case one is provided with considerable
*Partially
supported by National Science Foundation Grant #MCS7507403A03.
© 1980 Princeton University Press Riemann Surfaces and Related Topics Proceedings of the 1978 Stony Brook Conference 0691082642/80/00051503 $00.50/1 (cloth) 0691082677/80/00051503 $00.50/1 (paperback) For copying information, see copyright page 515
SCOTT WOLPERT
516
topological data and in the latter case no topological information is given. The set l[y ](S) I [y] £ S: I will be called the length spectrum of S and denoted Lsp (S). We have obtained the following result. THEOREM 1. A real analytic subvariety Vg of T g is defined. Let IRI, ISI£Tg besuchthat Lsp(R)=Lsp(S). Theneither IRI=ISI mod Mg or IRI. lSI
f
Vg.
The argument involves the theory of amalgamation of Fuchsian groups. Criterion are developed to identify [y 1 ], ···, [yp] £ S: as specific classes from knowledge (only) of the graphs of the functions [y 1 ](S), ···, [yp](S),
S £ T g.
Our argument involves the following results.
LEMMA 2. Let f: R
>
S be a K quasiconformal map and y a closed
curve on R. Denote by f (resp. f ') the length of the Poincare geodesic freely homotopic to y (resp. f(y) ). Then
A deformation is a continuous map of [0, 1) into T g. LEMMA 3. Let [y 1 ], · · ·, [y p]
f
S: be primitive. Then [y 1 ], · ·~, [yp] are
simple and disjoint if and only if there exists a deformation St with
[y)
0 ·is determined. Then p(log [y] (S)) < [a] (S) + c for all
§ ( T g. A deformation st is determined such that [y] (St) [a] (St) 'S p (log [y] (St)) + c .
.... 0 and
A characterization of the first eigenvalue of the (Poincare) Laplace Beltrami operator is also required. A recent result of M. F. Vigneras establishes that Vg is not empty for certain choices of the genus g, [2]. REFERENCES [1] S. Wolpert, The length spectrum as moduli for compact Riemann surfaces, Annals of Math., 109 (1979), 323351. [2] M. F. Vigneras, Exemples de sousgroupes discrets non conjugues de PSL(2, R) qui ont meme fonction zeta de Selberg, C. R. Acad. Sc. Paris, t. 287, 1978.
Library of Congress Cataloging in Publication Data
Riemann surfaces and related topics: proceedings of the 1978 Stony Brook conference. 1. Riemann surfacesCongresses. II. Maskit, Bernard. III. Title. QA333.C593 1978 515'.223 ISBN 0691082642
I.
Kra, Irwin.
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