An Introduction to Teichmuller Spaces

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Yorcut lvayosHr Departmentof Mathematics,college of GeneralEducation,osaka University,Toyonaka, Osaka560.Japan Ivfrrs.lHrxoTesrcucnr Departmentof Mathematics,Faculty of Science,Kyoto University, Sakyo-ku,Kyoto 606, Japan

ISBN 4-431-70088-9 Springer-Verlag Tokyo Berlin HeidelbergNew York ISBN 3-540-70088-9 Springer-VerlagBerlin HeidelbergNew York Tokyo ISBN 0-387-70088-9 Springer-VerlagNew York Berlin HeidelbergTokyo Tokyo1992 @ Springer-Verlag Printed in Hong Kong This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing and binding: Best-set Typesetter, Ltd., Hong Kong

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g as finite branched covering surfaces of the Riemann sphere, and determined the number of parameters of Mo by the number of degrees of freedom of the branch points. In this book, we treat moduli spaces through Teichmiiller spaces and Teichmiiller modular groups as follows. Let R be a closed Riemann surface of genus g, and let X be a marking on ft, i.e., a canonical system of generators of a fundamental group of .R. Two pairs (R,D) and (B', D') arc defined to be equivalent if there exists a biholomorphic mapping f : R--- -R'such that /.(X) is equivalent to Dt. Denote by [E,X] the equivalence class of (R,E). Such an equivalence class [R, I] is called a ma"rked closed Riemann surface of genus g. The Teichmiiller space ?o of genus g consists of all marked closed Riemann surfaces of genus g. It is verified that ?, has a canonical complex manifold structure, and it is a branched covering manifold of the moduli spaceMn.Its covering transformation group is called the Teichmiiller modular group Modo which corresponds to the change of markings. It turns out that Mn is identified with the quotient space TofModr, which has a normal complex analytic space structure. The Teichmiiller space4 h* appeared implicitly in the continuity arguments of Felix Klein and Henri Poincar6, who studied Fuchsian groups and automorphic functions from the 1880s.Robert Fricke, Werner Fenchel and Jakob Nielsen constructed Tc k 2 2) as a real (69 - 6)-dimensional manifold. Fricke also asserted that ?, is a cell. Their method was based on the uniformization theorem of Riemann surfaces due to Klein, Poincar6, and Paul Koebe: every closed Riemann surface of genus S (> 2) is identified with the quotient space H f I of the upper half-plane .I/ by a Fuchsian group f which is isomorphic to a fundamental group of .R. Then each point [R, I] in ?, corresponds to a canonical system of generators of l- . Hence we see that [.R,X] is representedby a point in R6g-0 which is called the Fricke coordinates of lR,t). Moreover, the Poincar6 metric on f1 induces the hyperbolic metric on .R, and the conformal structure defined by this hyperbolic metric corresponds to the complex structure of .R. One of Oswald Teichmiiller's great contributions to the moduli problem was to recognize that it becomes more accessibleif we consider not only conformal mappings but also quasiconformal mappings. A quasiconformal mapping means a homeomorphism which satisfies the Beltrami equatiotr ut7 = pu". A Beltrami coefficient p measures the magnitude of deformation of a complex structure or a conformal structure. Around 1940 Teichmiiller discoveredan intimate relation between extremal quasiconformal mappings and holomorphic quadratic differentials, and asserted thatTn is homeomorphic to R6g-0. He also introduced the Teichmiiller distance o\ Ts. In the end of the 1950s, Lars V. Ahlfors and Lipman Bers developed the fundamentals of the theory of Teichmiiller spaces,and they gave rigorous proofs for Teichmiiller's results. They also showed that To @ 2 Z) has a natural complex structure of dimension 39 - 3, and can be embedded in A2(R) as a bounded domain, where ,42(R) is the space of holomorphic quadratic differentials of a closed Riemann surface E of genus g. From the Riemann-Roch theorem, it is

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those which determine a given point of "(E) is a Teichmiiller mapping. Then it turns out that Q k > 2) is homeomorphic to the space Ar(F) _ofholomorphic quadratic differentials on .R. Hence, ?s is homeomorphic to R6c-0. We also show that "(.R) is complete with respect to the Teichmiiller distance. In Chapter 6, using the Schwarzian derivative, we construct the Bers embedding of "(R) into a bounded domain in ,42(.R.), the space of holomorphic quadratic differentials on ft*. Here, E* denotes the mirror image of .R. By the Riemann-Roch theorem, Az(R-) is also identified with the (3g - 3)-dimensional complex Euclidean space C3r-3. Using this embedding, we see that "(ft) has a natural complex manifold structure of dimension 3c - 3. It is also proved that the Teichmiiller modular group M odo is a discrete group of biholomorphic automorphisms of ?r, and acts properly discontinuously on "0. This shows that the moduli space Mo =Ts/Modc has a normal complex analytic space structure of dimension 3C - 3. Chapter 7 treats the Weil-Petersson metric on 4. The holomorphic tangent space of To at a point [.R,X] is identified with the dual space of ,42(R). Then the Petersson scalar product on.42(R) induces the Weil-Peterssonmetric on ?n' We give two proofs for the fundamental fact that the Weil-Petersson metric is Kihlerian. Both of them a.redue to Ahlfors. In Chapter 8, we establish a beautiful formula due to S. Wolpert, which states that the Weil-Petersson Kihler form on 4 h* a simple representation with respect to Fenchel-Nielsencoordinates. We also give two appendixes. Appendix A deals with Schiffer's interior variation from the viewpoint of quasiconformal mappings. We explain Ahlfors' construction of the complex structure for Ts, which was the first construction of its natural complex structure. We also discuss variations with respect to degenerations of Riemann surfaces. In Appendix B, we explain briefly the compactification of moduli spaces. At the end of each chapter, there are bibliographical notes of books and articles to which we referred in the text. The bibliography is not complete. There is a vast literature relating to the theory of Teichmiiller spaces.We hope that this list helps the reader to begin to explore these researchpapers. Any omissions of references,or failure to attribute theorems, reflects only our ignorance. The authors are extremely grateful to Professor Osamu Takenouchi who recommended that we write this book. They also gratefully acknowledge the generous contributions of our friends and colleagues Makoto Masumoto, Hiromi Ohtake, Hiroshige Shiga, a^ndToshiyuki Sugawa, who read the original manuscript, and made many helpful mathematical suggestionsand improvements. Yoichi Imayoshi Masohiko Taniguchi October, 1989

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Contents

Chapter 5 Teichmffller

Spaces

5.1 Analytic Construction of Teichmiiller Spaces 5.2 Teichmiiller Mappings and Teichmiiller's Theorerms 5.3 Proof of Teichmiiller's UniquenessTheorem Notes Chapter

Chapter

Analytic Theory of Teichmiiller Spaces Bers'Embedding Invariance of Complex Structure of Teichmiiller Space Teichmiiller Modular Groups Royden's Theorems Classification of Teichmiiller Modular Transformations Notes

135 144

146 r47 r52 r62 r67 'l'71

179

7

Weil-Petersson

Metric

7.I Petersson Scalar Product and Bergman Projection 7.2 Infinitesimal Theory of Teichmiiller Spaces 7.3 Weil-Petersson I\{etric Notes Chapter

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6

Complex 6.1 6.2 6.3 6.4 6.5

119 119

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8

Fenchel-Nielsen

Deformations and Weil-Petersson Metric 8.1 Fenchel-NielsenDeformations 8.2 A Variational Formula for Geodesic Length Functions 8.3 Wolpert's Formula Notes

219 219 224 226 232

Appendices A B

Classical Variations on Riemann Surfaces Notes Compactification of the Moduli Space Notes

233 243 244 253

References

254

List of Symbols

271

Index

274

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z a- Plane

ar-plane

Fig.1.1. A coordinate neighborhood(U, z) of a Riemann surface .R is a pair of an open set [/ in ,R and a homeomorphism z of U into the complex plane such that for any element (Ui, ri) of a system of coordinate neighborhoods with U nU1 I $, the mapping zoziL: zi(U nU)--- z(U nUi) is biholomorphic. This [/ is also called a coonlinale neighborhoodof r?. Such a homeomorphism z is said to be a local coordinale ot a local parameter on U of R. A coordinate neighborhood ([/, z) with p e U is called a coorilinate neighborhood around p, and z is called a local coorilinateor local parameter arounil p. Local analysis on a Riemann surface ,R is reduced to analysis on domains in the complex plane via local parameters. For example, a holomorphic funclion on ,R is a function / on l? such that f oz-L is holomorphic on z(U) for any coordinate neighborhood (U,z) of ft. A mapping f of R into a Riemann surface,9 is said to be a holomorphic mapping if wof oz-r is holomorphic for all coordinate neighborhoods (U, z) of R and (V, u) of S with /(U) C V. A biholomorphic mapping f : R --- S means a holomorphic mapping f of Ronto,S which has the -1 : S - ft. Two Riemann surfaces l? a"nd S are holomorphic inverse mapping f biholomorphically equiualenlif there exists a biholomorphic mapping between .R and S. In this case, we regard ,? and ^Sas the same Riemann surface and write R = S. We say also that R and S have the same compler slruclure. Complex structures, biholomorphic mappings, and biholomorphic equivalencemay be and are actually often said to be confortnal straclures, conformal mappings, and conformal equiaalence, respectively(see$1.5). Remark. A Riemann surface is a two.dimensional real-analytic manifold, and the Cauchy-Riemann equation implies that local coordinates determine its orienta-

crqdroruoloqlq e sl U r- C : Jr leql ferrr e qcns uI araqds r } = ? e r e q a ' I - C - O u l " u r o pe q } o t u o { 0 > r r r , - I = H aueld-}1eq I C > ^l = *H aurld-;1eqra/(ol eql pus {0 < rnurl I C > t} raddn aq1 qloq fllecrqdrotuoloqrq sderu .rn - z uollcunJ ctqdrouroloq aq;,

'z'T'ttd eueld-or

aueld-z

'euo JeuJoJ eqt al"ts er* 'ala11'uollsnulluoc ct1f,1euefq ro Pue ln)r' ,,e1sed - (n)l - z 'dleotsselC 'panle^-e13utssr "n Jo Poqlau aql fq Pelrnrlsuoc sr 1r uorlcunJ crqdrouroloq eqlJo uorl)unJ asrelur aql qllqa uo e?"Jrnsuueruelg aql sI (lxaN slql '4 - rn uorlrunJ cre.rqe3peql Jo eeeJrnsuuetuelll at{} aes sn lal 'Qlt'{o} p"n (z'3) spooqroqq3raueleurProo?o'n1fq Peusep ?) sr C uo eJnlcnJls xaldurbc V'eceJJns uueurelg e osl€ sI'3 aueld xelduroc aql '{-} go iorlecgrlcedtuoc lutod auo eql q qc$Iir{ n C = ? ataqds uuvu'ery aq; '(r'O) eleulproo, auo fluo fq uarrrSsr O uo ernlcnrls xalduroc e pooqroqq3rau 'flaurep 'e?€Jrnsuueruolg e st aueld-z xalduroa aqt ul O ul€ruoP ,traaa'1p;o 1sltg sa"BJrnS uueurarlr 3o salduruxg

'Z'T'I

'[86-Y] '[OO-V] re3ut.rdg 1e3ar5'[37-y] ueur.re3uts Pus '[ga-V] €rN pue se{re.{ '[ZZ-V] ,tqoC pue seuof '[Ol-V] Suruung '[ag-V] ratsrog '[gt-V] s.reg '[g-y] oIr€S pue sroJIqV 'ecuelsut roJ (llnsuol 'sace;tns uuerualg 'pa1e1n3u€Ir}aq u"c pue slas uado;o ;o ,{roeq1 1e.raue8eql pue s}teJ esaq} .rog 'uoll€luelro sr$q elqelunot € seq e?eJrns uueuralg ,t.ra,ta1eq1 u^{onl-llaa s-ItI '.rageara11'uor1 sr eceJrns uusruarg € letll etunsse airr slql qtla paddrnba s,tea,r1e saf,"Jrnsuu?uraru'I'I

1. Teichmiiller Space of Genus g

mapping. This R is the Riemann surface of w = t/7. (See Ahlfors [A-4]' Chap. 8; Jones and Singerman [A-48], Chap. 4; and Springer [A-99], Chap. 1.) 1/7 is also Note that the Riemann surface R of the algebraic function w = z. u,2 equation by the defined regarded as the algebraic curve Finally, we seeelliptic curves, i.e., tori from the viewpoint of algebraic curves. For any complex number ) (# 0, 1), Iet .R be the algebraic curve defined by the equation w2=z(z-1)(z-.\).

(1.1)

In other words, .R consists of all points (z,w) e C x C satisfying algebraic equation (1.1) and the point p- = (oo, oo). We can define the complex structure of ,? by the complex structure of the z-sphere so that the projection r: E e, r(z,w) = z, is holomorphic. This r? is a two-sheeted branched covering surface over the z-sphere with branch points 0, 1, I' and oo. The mapping written as u, = f : R - e, fQ,u) = w, is holomorphic. This function / is and R is a Riemann surface on which the algebraic function \rc=W]

u - {z(z _tG

-,

is single-valued.

The Riemann surface ,R defined by algebraic equation (1.1) is rega.rdedtopologically as a surface illustrated in Fig. 1.5. Take two copies of the Riemann ,ph".", St, Sz with cuts between 0 and 1, and between,\ and m (fig' 1'3)' place them face to face (Fig. 1.4), and join along their cuts (Fig. 1.5). The resulting surface is homeomorphic to the Riemann surface -R. Hence, R looks like the surface of a doughnut. We call such a Riemann surface a torus. A torus is also called an elliptic curue; Lhis name comes from the elliptic integral (see $1.4).

Fig.1.3.

t=f{ !g'!V} lo srolor?u?6lo an1sfrsf)?ruouoc n

ro t=r1VAl'llV)}

'(od'a)rv 1ec e16

r=f

'(rlunaqr) r = r-[rB]r-tfvltrsltrvlL[ 6

uoll€ler l€lueu"PunJ eql sogsll€s Ptre 6g t6, t"''rg Iy uro.r;pecnpul luql'luVJ ' "' ' llgr] ' lly] sess€lcfdolouroq eq1 fq pale.rauaEsr od lurod eseq q?lar U 1o (od'g1ttt dno.r3 leluaurePunJ eqtr '(f't ;St.f) seprs dy ql1,u uoE,tlod xaruoc e o1 crqdroruoeuoq ul€tuoP e 1aBar* ' " ' 'r€l(I}t selrnf, uerll 'g'I 'EU ul se od lutod as€q I{lI^r ug'6V Pasolcaldurts od e4e; e lurod 3uo1e g, lnt pue d snuat Jo U ac"Jrns uueuell{ Pasolc e uo 'acottns uuourery uado ve Pe11eo y 'snua! allug Jo ac"Jlns uuetuerll pesolc € sI st a?"Jrns uueruarg lceduroc-uou u^rou{-lle^{ sl U'I snuaS;o $ snro} e Pue /tJa^a a?€Jrnsuu€r.uaru l€qt lceduroc '6 snuaE;o sr araqds uuerualg aql 'f snua6 to ecottns uuDut?NAPesop e PelIe? sl g'I '3lJ uI sB solPu"q d qtl,u alaqds e o1 crqdrouroeuoq ec€Jrnsuueuary Y

sa?BJrns rruBtuaru pasolc '8'I'I

'e'r'tt.r

'r'r'ttJ

sef,"Jrns uu"uall|I'I

l. Teichmiiller Space of Genus g

Fig. 1.6. (g = 3)

Fig. 1.7. (g : 3)

L.L.4. Lattice

Group

Representations

of Tori

We shall represent a torus as the quotierrt space C lf of the complex plane C by is a single-valuedmeromorphic a lattice group l-. since ur(z) = fiG4Q=U we can consider the complex (1.1), by equation defined function on the torus R p = (z(p),u''(p)) on 'R, th9 point any For paths on ft' along integral of Ilw(z) of algebraic function u(z) branch a selecting by defined elliptic integral @(p) is setting by and and a path from oo to z(p),

o(d=

f2lP)

J*

dz ,/r(t-l)(z-))

(1.2)

saprs Eurr(;rluapr fq peul"lqo aceJJnse ss Pazllear sl J/C eceds luarlonb sq; '[z] sasselc ecuele,rrnba lle Jo st$suoc.7 ,{q C P JIC eceds luarlonb aq;

'e'r'ttJ

'z fq paluasardar sselc acuale,rrnbeaqf [z] fq alouaq '(z)L= / {lla J ) L lueluele ue slslxe eral{}JI J raPun Tualoar,nba fes e6'a1ozau*rvut s l u r o do. / $ 1 Iz =(z)L uollelslr"rl€ are I 1 " t 1 1 C , z ' z '9 dnorS ursrqd qlr^r pagtuapl sl J ) qlu+rvrn, =,L fra,ra'1ce; u1 3o (9)ry -Jouroln€ arlrtpue aqt JoJ dnorS e?I11"1Y dnodqns 3 se papreSar sl ,7 u Jo 'U 'U roJ ilno.r,6acqyoye PIag raqunu I€er J qcns II€c alA fpeaurl ere faql '0 < (z)Llr)L)u1 ,$sr1eszv pue rl spollad aql ra^o luepuadaput '; aq? ecqs 'E, 1o pouad e Pall€? sI J Jo luauale fraag Jo slueuela dq raqlo qf,ee tuo+ rasrp qlrqAr senlel fueur .{1a1tugulseq (d)P uotlcun; eql t€q} ees a/tr '{Z>u'*l'ou*rYut}

= J 3ur11a5'flarrtlcadsar

,k__4$_4 - f zp

or ,l

z=iv

(u- z)(r- z)zf pue -rp

of ,l

,=tt

fq paluasarder are g'I '3rg ur Ig 'd pue -d Eurutof q1ed e uo (Iy salrn? pesol? eldtuts aq1 Suop O Jo sanle eq;, spuedep 11 '{lanbrun peururelap lou s-IO 1e.r3a1uratldqle slq} Jo enp^ eqtr 'U uo ea.r3aP;o I

g__,la_at zp l€IlueraSlp crqdrouroloq aq1 auII € s€ (6'1) pleSal o1 alenbape arour sr II'tlroureg 3o'q1ed e 3uo1e'1er3a1ur

-ralep sr (z)np

'1er3a1ur ql"d aq1 3uo1e uoll"nulluot cltfleue dq paunu Jo enle aql pue 9.1'31g ur -d lurod aq? o? spuodsarroe oo eJal{^r se)"Jrnsur"urerll'I'I

l. Teichmriller SPaceof Genus I

8

,4 with A' and B with 8' in the lattice of Fig. 1.8 by the translations 7r1,tr2, respectively. Now, we define a complex structure of C / f . Let r : C + C / f be the projection, i.e., "(r) - lzl fot z € C. Introduce the quotient topology on C/f , which is open if the inverse image r-r(Lr) is is defined as follows: asubset U otC/f open in C. It is verified that C/f is a connected topological space. For any two points [o],[6] € Cf l,we can take neighborhoods7o,V6 of a,b with r(I/") n r(%) - {. Since z is an open mapping, this shows that C/iis a Hausdorffspace. Moreover, for any point [c] e C/f , taking a sufficiently small neighborhood vo of a, we see that n gives a homeomorphism of v" into C/f . Let Uo = r(Vo) and zo: (Jo - Vo be a homeomorphism with zo(lzl) = z. Then (t/",2o) gives a coordinate neighborhood around lalin C/f - Thus C/f becomes a torus, i.e., a closed Riemann surface of genus I such that the projecgives an example of tion zr: C --- Clf is holomorphic. The triple (C,r,C/f) universal coverings, considered in $2.1 of Chapter 2. As is known in the theory of elliptic functions, the mapping [@]: r? '--' C lf sending a point p e R to a point [O(p)] e C/l- is biholomorphic. Hence we see that a torus defined by equation (1.1) is representedby a Riemann surface c/lr for a lattice group l-. In Chapter 2, we shall show that every torus is represented by a lattice group l- in c (see the corollary to Theorem 2.13). conversely, it is known that such a Riemann surface C /f is always biholomorphic to an elliptic curve defined by algebraic equation (1.1). For details, we refer to Ahlfors [A-4], Chap.?; Clemens [A-21], Chap. 2; Jones and Singerman [A-48]' Chap' 3; Siegel [A-98], Chap. 1; or Springer [A-99], Chap.l.

1.2. Teichmiiller Space of Genus 1 Let us construct the Teichmiiller space of genus 1.

L.2.1. The Moduli

Space of Tori

we use the fact that every torus is represented by a Riemann surface c/f, where ]- is a lattice group on c as in $1.4 (see the corollary to Theorem 2.13). On performing the transformation z r* zf 4,if necessary'we may assume from the beginning that the generatols ?r1and 12 Lor I a,re the ca"nonical ones I and r with Imr ) 0, respectively. Now, consider a lattice group f"={j=m*nrlm,n€Z}, where r € H = {r € C I Imr > 0}. As wa.sseen in $1.4, the lattice group I} corresponds to a subgroup of ,Aul(C), and the Riemann surface R, = Cf l, is Notice thal cf f, has the a torus. Denote by r, the projection of c to c/f,. group. structure of an additive

'(z'dtsalH =tw '(Z'Z)lSd fq g;o eceds 'sl teqt luarlonb eql qtl^\ PeUIluePtsl I,f41}€tll sarTdurrI'I uraroeqJ'IrolJo sassBl?af,ual -earnba rrqdrotuoloqlq IIe Jo tas eqt ''a'r'r.uoqto aeods,Ppout eqt aq rW P"l 'g reddn aql aue1d11eq 'ilno.r,6Jolnpou eq1 ;l ,L fra.rrg

3o ursrqdrourolne crqdrouroloqlq e s\ (Z'7,)'IS1

( l P + t ' c- , ' ' l = ( z ' 7 , ) r s d Q ) L| I\ t = " 9 - p o p u e z ) p ' ? ' q ' "ll '?7+=! o

) dnorS aq1 1ec a,u 'aaog

tr sl ,U -

'l'(P - ([z])/ fq ua'rt3 + n)) ,'A I t Surddeu crqdroruoloqrq e uaql 'splotl (8'I) ;t 'f1asra,ruo3 'I = cq - pD s^eq a1ll -lD + tcl '0< (, ,nD;q ,rwr _d

f = (t[or-;f

',! - (,t){or-/ pu" esuls 'I+ = ?q - pe l3q1 aas e^a ' (eJoureqlrr\{ 'srafielur eJ€ suotlela.reqt urorJ /p Pue ,?' ,9' ,D alaqlvr ,tP*'rP -" rQ* ,'t,o 'r-! o, lueurnS.reetues eql 3ut{1ddy 1aBen 'Ptt? - " ' 9*te ur€?qo ein 'a.ro;ereqa 'sre8alur ar€ P Pue

'? 'q'D aleqlr

'P+tc=n=(1)l 'q+tP=1a-(,t')l aleq a^{'ecua11'? rapun 0 = (0U o1 luap,rrnba are (1)/ pue (,-r.)rfqloq snq1, '0= d ecueq pue'0 = (0)1 leql erunsse,teur eaa'.re,roaro141 '(9'6 eufrua1 ,lc ra = (,)! lc) 0 + lc pue sreqlunu xelduoc 5rc ! pue ereIIA\'d + '(7'6 rue.roaql 15) s€ ualllr^a sr / uaqa 'j t, o. 'crqdrouroloqlq q .f esn€ceg 'l ,!)Lo! qanf 3 - C,! Eurddeurcrqdrouroloqe'sr 1eq1 P ! Wle lo"o-wrll 'pelcauuoc t(1durs sr acurg 1i slsrxe eraql teql salldtq ureroeqf fuorPouoru aq1 '1srtg 'ig oluo ,"A /oo"l4' Io / Surddeu crqdrouroloqlq e q arerll l€t{} alunss? 'c 'q 'o e.taym 'I = cq - p,Dqwn sta,aTut arD p PUD ,Plrc _ t , g*tp uoxlnpt, tf, fi,1uopuD fi Tuapamba frllocttlilloutopqrq erD 'tg puo at17filsr7os p puv t lt .r,edilneql w / pao t squr,oilony fiuo rol '11- tuaroaql uol onl 'g auold-{1ot1

(e'r)

'Z'I I snue5;o aoedg rall+urqrral

1. TeichmiillerSPaceof Genusg

l0

It is known that the quotient spaceHf PSL(2,2) is a Riemann surface (cf. $2.4 of Chapter 2) and that a fundamental domain (cf. $a.2 of Chapter 2) for PSL(2,2) is the shaded area in Fig. 1.9. Intuitively, we get the Riemann surface bV identifying the sides of this fundamental domain under the H/PSL(2,2) transformations z > z +1 and z e -lfz as is illustrated in Fig. 1.9. Hence we see that the moduli space of tori is biholomorphic to the complex plane. For more details, see, fot example, Ahlfors [A-4], Chap.7; and Jones and Singerman [A-48], Chap. 6.

Fig.1.9. Remark. A torus given by equation (1.1) depends on a complex parameter ,\(f 0, 1), which is denoted by ,Sr. It is well known that two such tori 51 and S1, are biholomorphically equivalent if and only if there exists a linear fractional transformation which takes the set of branch points { 0, 1, }, oo } of Sr to the set of branch points {0, 1,^',m} of ,91, (see,for example, Clemens [A-21], Chap. 2.7). Thus we see that 51 and 51, are biholomorphically equivalent if and only if )/ is equal to one of the following numbers:

.\.

1 + , ,r

1-),

't r-)'

l-1 )

'

l ,\-1

Now, let G be a finite group of order 6, generated by cr()) = 1/) and - {0, 1 }. This fact Sz(\) = I -.\ which are analytic automorphisms of D = C quotient spaceof Dby G (cf. $2.a shows that ML= D/G,where Df G means the of Chapter 2). Moreover, we find a biholomorphic mapping F : D lG ---+C, which is defined uy r([.1]) = /(.\) with

/()) =

(,\2-^+l)3 t2() - 1)2

r ueaaleq a)uaraJrp aql l€rll reprsuo) uec e1t{snql'sdno.r3 arr11e1ueemlaq I * ,'J:/ rusrqdourosreqt o1 spuodsauoc q?rq^,r'[((,r)tg)/] = ([(,r)tg])V pun = ([(,r)ty])V reqr qcns (od''A)ru * (od','g1rv: { ursrqd.rourosr l((,4trtll u€ sarnpur "A - ,"A:/ Surddeu ctqdloruoloqlq e^oqe aql leql ees e.tretueg 'l(,t)tV) srole.reua3 '{ go tua1s,{sle)ruou€) € seq qclqm (od' ,tA')r)Lqtrw ] [(,r),g] p a g r t u a p rs 1 , 1 . ( f p e p u r r S ' . { 1 e , r r 1 c a d s a r ' [ ( r ) t g ]p u e [ ( " r ) t y ] o t , p u e I s P u e s qclqlr\acuapuodseiloraql repun (od'"A)to o1 crqdrouroq sl iI ueql'(odtrU)I, Jo srol€reua3 go urals,ts letluoue? e a;rt3 ['g] Ptn [Jy] sasselcddolouroq aqa 'od ?urod es€q qtl^{'U uo (-r,)tg pue (z)ty sa,rrn, pesoll aldurts eulure}ap ''U '.{1a.l.r1radsar'C pue'I Pu" 0 uea^l}aq sluaru3asaql ul r pue 0 uae^4,laq dno.r3 leluetu"PunJ aql Jo lurod aseq € sB [0] - od a4ea'dno.r3 eql uorJ luelua?els e^oqe eql rePlsuoc el!\ eql Jo 1utod,r,ret.rr. 1o (od''g)rl I"lueu"punJ

'01'T'tIJ

+

,,1 -

'(Ot't'3t.f pu€ .{1a.,rr1cedsa.r',r. aas) I o? (,r)/ pue (1)1 '{q spuas (1)//z teql /$oqs ol lu?Icgns sr 1r lB ',S] eleq e.rlrsnqa 'ctdolouoq er€ rd '16' Pue / feqt flrsea a,rord ,Sl = [/ uec a \ 'aue1d aql ul {s!p lrun aq} o1 crqdrouroeuoq $ U urorJ fg pue fy 1e Eur -1e1apfq peurc?qo uretuop aql aculs 'uorlrusap aqt fq [td'S] = [6',9] teqt atoN 6 snuag;o aeedg rallgurqrral 't'I

9I

1. Teichmriller SPaceof Genus 9

16

we find a qua.siconformal mapping /o homotopic to / (Bers [26] or Lehto [A68], Chap.5, Theorem 1.5). This fo is not necessarily smooth; however, there exists a real-analytic quasiconformal mapping homotopic to f, (the Corollary to Theorem 6.9). Finally, we define a canonical group action on the Teichmiiller space ?(R). Let Mod(R) be the set of all homotopy classes [o] of orientation-preserving diffeomorphisms ar: .R * -R. We call Moil(R) the Teichmiiller modular group or the mapping class group of .R. Every element [ar] acts on ?(R) by [r].([S, /]) = [S, f or-'] for any [S, /] e "(n). We call every lw)* a Teichmil,ller moilulor transformation. Let Mo be the moduli space of closed Riemann sarfaces of genus g, i.e., the set of all biholomorphic equivalence classes [S] of closed Riemann surfaces ,9 of genus g. since for a,n arbitrary closed Riemann surface .s of genus g there exists an orientation-preserving diffeomorphism of R onto ,S, the moduli space M, is identified with the quotient space T(.R)/Mod(R) of "(i?) by the action of. Mod,(R). Therefore, we can study the moduli space Mo via the Teichmiiller space ?(.R) and the Teichmiiller modular group Mod(R). In Chapter 6, we shall see that "(E) has a (3c - 3)-dimensional complex manifold structure and that M od(R) acts properly discontinuously on "(8) as a group of biholomorphic automorphisms. In particular, the moduli spare Mo has a (3g - 3)-dimensional normal complex analytic space structure.

1.4. Quasiconformal

Mappings and Teichmiiller

Space

Let us reviewthe Teichmiillerspace?(,R) constructedin the previoussection from the view-point of the theory of quasiconformal mappings. L.4.1. Deformation

of complex

structures

and Beltrami

coefficients

For a point [S, /] g ?(,R), we want to compare the complex structures of ft and s. Take a coordinate neighborhood (u,z) on I and a coordinate neighborhood (lz,to) on ^5with f (U) C V, and set F = ?r,ofoz-l. Then

p-

Fz F'

is a smooth complex-valued function defind on iur open set z(Lr) in the complex plane. Note that it is independent of the choice of a local coordinate u.'. Since ] i, .n orientation-pr"r"ruing diffeomorphism, the Jacobian of F, i'e., lF,l' F' is l&12 i" positive-definite on z(U). Thus we have lpl < 1 on z(U). Further, biholomorphic on z(U) if and only if F = 0 on z(U)' We call y' the Belt'rcrni coefficient of / with respect to (U , z). It should be noted that a Beltrami coefficient of / depends on the choice of a local coordinate z on R. How it depends is shown as follows: take coordina*e

. l ( o ) ' r l --I

i(g;ffi=(o)>r

u asdrlla srr{}Jo srxe rounu aqt o} srxe roleur aq] Jo ol]er el{l

'l"l(l(o)'rl - r)l(o)"/l + r)l(o)"/l i l(o)zl 5 l,l(l(o)'tl sarlrlenbeutaqt ,tg '(tt't '8t"f) aueld-rn aqt ul asdrlla ue o1 aueld-z aql ur 0 raluec qt-ra elcrlc e spues 7 deur 't > r€aurteqt 'raaoero141 l$)"t /(O)ttl = l(g)r/l pue O * @)"1 1eq1saqdunqcrq,ra

'o< - .l(o)"/l = (o)/r .l(o)"/l sagsrles0 - z le (6)f uerqocel s1t 'urstqdrouroagrpSutrlraserd -uorleluerro ue sr / acurs '0 - z Ie / ;o uorsuedxa ro1,te; eql Jo tural rapro lsrg eqt aq z(iltt + z(g)'t = G)l 1a1 'aue1d-rnxalduroc eqt ul /O uIPruoP e oluo aueld-z xaldruoc aql ul 6 urSr.roeq1 Surureluor O ureluop € Jo tuuqd.rouroagrp Surarasard-uorl€luarroue sr / l€ql etunssearra'spooqroqq3reualeurpJoo?Surraprs -uoc 'srql easoI 'sluer)lgeoc tusrtleg;o Surueeurcr.rlatuoa3eq1 ureldxa a.tr'1srrg s8urddetr4l lBruroJrrocrsen$'6'7'1 ',t1t1eur.royuoc uorJ 3[ 3o uotletaap aql ernseauro1 pesn sl / Jo luerrlgaof, lruprllag aql pue 'g uo arnlrnrls xaldtuoc eql '(U)"f ul Jo uorleruroJepe sluasardar (y)"6 ul U'S] lurod e leql su€errrlr ef,uag {y :l 7eq1 Wl'lAl = [/'^g] ?eql s^roqssrq;'Surddeur crqdrotuoloqlq€ q,S * pue 'tusrqd.rouoeJrp Surl.rasard-uolleluelro ue sr /U * A :p? deu flrluepr eq1 'tl)D{ 's1as se (Io"*'("1)r-t) 1eq1 U = /U leql eloN } spooqroqq3raueleurp 'deilr slqt uI -rooc ;o uals,ts qlurr paddrnba IU ateJJns uu€uIeIU A\eu e aleq ea,t 'U uo arnlf,nrls xaldruoc " seugep v>a{(toDm'("1)vt) } spooqroqq3raueleu ',9 - g : ursrqdrouoeslp Sut,uasard-uolleluelro ue roJ -rprooc rt ;o ualsIs B puts S uo vl"{ ("*'"A) } spooqroqq3rau eleutp.roocJo ualsds e rog '.Lrop

'U uo / 1o Tuata$aocnaDr?Iegeql pallec$ q)nl^t

(e'r )

''P ,t - trl

zp

,tq ,{ldurrs (t't-) ed{1 ;o ruroJ l"rluereJlp slq} a}ouap e^r snq;, 'U el€ulprooc uo ad{1 (I'1-) Jo urroJ leltuareJlp € se)npul U Jo spool{ro,Q{31au - l{z areq^l uo /go sluerrlgeoc rur€rllagJo les eqt leql s.lroqsslql'rizors

(r r)

'(trut2)tz uo (#)

l@).(rzotrl)=

trl

e^tsqa,lr'Q * qnU ln '(qz'qn) pue rurerllag uaq,11',,(learlaadsar ol qll/'^ Jo sluarf,lgeoc (lz'fn) leadsar ./ eqt eq 'trl pup ld lr,1't1 1 (r2)l pue ln > (dI ?€qt qcns g p (tn'tn1 '(!m'11) spooq.roqq3reu eleurproocpue gr 1o (tz'qn) '(tr '.!2) spooqroqq3rau L1

a*dg

ralnurqf,ral pu? s8urddul4l purro;uorrsen$

'7'1

1. Teichmriller Space of Genus g

18

This shows that any infinitesimally small circle with center 0 is mapped by / to an ellipse whose ratio of the major axis to the minor axis is K(0).

L(z)

--t

cp- 0+aref,Q) a : ( 1 +l p ( o ) l ) r l l , ( o ) l b: (r-lp(0)l)rlf,(o)l

o:lareu$)

Fig. 1.11.

This statement holds at every point in D. Thus we also call the Beltrami coefficient , \ ft(r)

p t Q ) = f f i , z eD ,

the complen dilatationof / at z. As we saw before, Ft = 0 on D if and only if / is a biholomorphic mapping on D. We call f a quasiconfonnal mapping of D to Dt if f satisfies

Kr lrrl'J!. .". ' = supl* r-lpt?)l ,eb

Further, f is called a quasiconformal mapping with Beltrami coefficient Lrt.W" call K1 the maximal d,ilatation of f . In this chapter, we only consider smooth quasiconformal mappings. We shall study more general quasiconformal mappings in Chapter 4. tansformation formula (1.4) implies that the absolute value lprl(z)l of the Beltrami coefficient W = pJQ)dzldz of an orientation-preserving diffeomorphism I : R - .9 does not depend on local coordinates on l?. Thus lpy I is a continuous function and lpty| < 1 on ,t. Since r? is compact, we get

pl py( z) q > areq^4, 0 'srx€ z eql punor€ aueld-(z'f) "qt uo = zz * z(D - f) ala.rrcaql 3u-I^lo er zg fq paurelqo fl r{?rqr'r gll af,Bds ueaprTcng aql u! e?eJrns e aq W p1 'aldutoxg 'droeql pue drqsuorleler slql paztuEo uollcunJ ctrlatuoa3 aql papunoJ 'g 'ses?c -rer .raqSrq roJ enrl lou $ qtlq^\ 'sployrueur uu€uerg lsrs leuorsueurp ''e'l 'sp1o;rueur xalduroc I€uorsuaurp-1 .ro; ,tl,radord elqe4reur Iear l€uorsueurp-A -eJ e sr uorlrasse stq; 'Surdderu l"ruJoJuor € pells? sr Sutddeur cqdrouoloqtq e teql uos€er " $ srqJ '1ua1e,rrnbaare aln?cnrls IsruroJuo? Jo pu" aJnltnrls (teql s^\oqs uaroeql slqJ xaldtuoc ;o sldacuoc 'esea leuorsuaurp-oirrl aql q 'att1d.rouro1ot1lq tl fi1aopuo s! ([sp '.,1I)- Qsp'W):l s? *A lr lout^,t,otuoc S t! '([sp'N) splottuotu uvruuDutary uaql 'fi1aar7cedsar puo (zsp'W) louorsueurrp 'Z'I uraloaql -Z pa?ueuo frq pacnput,sacottns uuou?tg eq S puv A pI 'ureroeql 3ur,no11o; eql ol speal (61'1) uoll"lues -e.rda.raq1 go ssauenbrun eql esef, ar{l uI aas o1 fsee sl 'U uorsuaurlp ll 1eql Jo 'uaql uaa&rlaqSurddeur FluroJuol s slsrxa ereq? J! ernptury loru.totuoc euros eq1 eler1 ro Tuapatnba Qlou.t.r,oluoc a r e ( [ s p ' N ) p u e ( z s p ' W ) 1 e q 1{ e s a 1 l ' N u o ( z C ) t p u e ( t 5 l ; ; ' u e a a l l a q ' l s p fq pa.rnseaur 'a1Eue aql slenba W uo 7'C pue I, selrn? qloours fue uaeiu,leq 'f1arrr1rn1ul'W uo uorlcunJ rlloorus 'zsp fq, pernsperu 'e13ueaq1 leql srrcetu 1r penle^-lear v 4 6 araq,n '_;211 uo .sp(d)dxa o1 lenbe q ./ ,(q ltp to lceq 1nd e q 1y f u t t l d n u t p u t l o t u o c " s l N * W i t u s r q d r o u r o e g r p B u r , r r a s a r d - u o r l e l u a r r o ue '(|sp'N) poe (csp'A[) sploJrueurueruuetuel]r leuolsuetulp-U palualro rod 'zsp clrleru u€ruueruerg eql fq pecnpw ?rnpnr?s lout^to{uoceql palle) aq ,(eu U uo arnlcnrls xaldruoo eq; 'deu, slql q pautelqo aceJrns uueurelu eql g itq eloueq 'W uo ernlcnrls xalduroc s saugep r>!{(!n'h)} l€q} f;r.re,ro1 (,;ig;o qncJlp lou sr lI '!2 qcee uo lm elsurpJoo? I€r.uraqlofl ue slsrxe araql r)!{((n'!x)'12) eleurproocJo uralsdse ro; 'acua11'I > -llr/ll } spooq.roqqErau '7 .ra1deq3 ?eql paphord slsrxa s.ile,rale01 uorlnlos " qcns Jo A$ ul pe,rord st sy 'uo4onba ,urDrlleg aql pell€r sr uorlenba slql 'rn uorlnlos crqdrotuoaglp e seq

(elr) esp roJ /'1 al€urprooc l€uraqlo$

'

t

z

o

9a=9 m8

m8

uorlenba prluereJrp lerlred eql JI s?s1xe ue leql apnlcuoc erra'(0I'I) qlrrn Sur.reduroc t

, l z ni +

r

l

z P l z l z m=l"dl n n l d

sagsr??sesp roJ nl eleurProoc leruraqlosl u3 a?uIS sernlrnrls I"ruroluoC pu€ sarnlf,ulg xalduro3 'g'1

tz

22

l. Teichmriller Space of Genus g

\

a

Fig.1.12. , = ( o * 6 c o sg ) c o s 0 , y = (a * Dcos 1), set r = exp(-2r2/log,\) and A= {w € C l t < l.l < 1). Definer; H---+ Aby r(z) - exp(2trilogz/log.\), where logz denotes its principal branch. Then 11 becomes a universal covering surface of the annulus ,4. ( v ) t e t 4 b e a l a t t i c e g r o u p g e n e r a t e db y 1 a n d a p o i n t r € I / , a n d l e t r b e the projection of C onto the quotient space C f fr. Then C is a universal covering surface of the torus C/ lr. Any biholomorphic mapping '1, fr, - E wittr ro^l = z is-called a coaering transformalion of a covering (R,r,R). For a given covering (R,r, R), denote by l' the set of all its covering transformations. By the composition of mappings, l- forms a group, which is called the coaering transfonnation group of (,R.r,,R). In particular, we call I the uniaersal coaering lransformation group of (-R,r,.R) if ,R is a universal covering surface of r?.

(i)' (ii)' (iii)' (iu)' (")'

f = (rr) with 71(z)- z *2tri. f=(rr)with71(z)=z*1. 1= ('rt) with 71(z)- z exp(2riln),which is a finite groupof order n. r = (zr) with 71(z)= )2. 7 = (T,72) -- f,, where7{z) = z * 1 a,nd"fz(z)= z + r.

2.2.2. Construction

of tJniversal

Covering

Surfaces

First of all, we need several definitions. A path on a Riemann surface R means a continuous curve C: I - R, where.I is the interval [0, 1]. The points C(0) and C(1) are said to be the initial and lern'inal points of C, respectively. We also say that c is a path from c(0) to c(1). Throughout the book, if no confusion is possible, its image C(/) is also denoted by the same letter C.

fdolouroq € eleq am'f11eutg 'I ) n roy (n1)"g = (n)(c't)p fq g'uo (t't)p qled e ausep a,r'7 x / f (s'?) fue ro3'uaqJ,'1 3 s due .lo;'d = (I)"f, = (0)'J pue 'C = rI 'oI - og sagsrles tl'{ ("'.)uf = 'd } leql q?ns uaql uee^.rleq rldolouroq € eq U * 1 x I ii p1 'o7' o1 crdolouoq sI C leql su€etu q?Iqtl 'l"d'Cl,tq peluasarde.r 'q1ed pasoll " $ 'fod'Cl -C ecurs lod'ollleqt epnpuoa am sl C Jo lurod leurturel eql puie 'od lutod es€q IIII^a g uo qled Pasolee sI 5t uaql 'ei" '[od'olTol crdolouroq q C tnd [od''I] lutod eseq q]p\ g uo g qled Pesop ',r,r,o11 fierta 1eq1 aes ol luelrgns fl 1I 'Paleeuuos ,(ldturs q f€qt b,rord arr,r !f '[d'g] ol 'pelceuuoc q U leql satldtut qclqar lod'oll uro+ U uo I qled € a^eq al,r '7 ) s fle,ra iog [(s)g'"C] = (s)g 3ur11ag'I ) 1ro; (ls)j = (rt"C rq A uo'C Wed e eugep'1 I s q?ea ro,{'U uo qled e Aqlod'o1f ql.I^{ pal?auuoc sl lI f [d'g] lurod i(ra^a teql A{oqsol sjcgns }l 'U Jo ssauPa}ceuuoc aq1 a,rord ol1[t'0] - I ) ?,(ue ro; oa = (t)"1,(q paugap g-uo qled eq] aq oI p"I'too.t'4 'p?peuu@ fiylul'ts puD pep?uuoc st 'aaoqo pautoldxa sD peptulsuoc 'g acottns ?qJ 'T'Z Btutuarl 'ur uo ernlcnrls = 4z turl eql }eql ees err,r')Lodz xelduroc parrnba.raq1 splarf {(!z'!2)},tgurel -les 'gg ur ureurop pelcauuoc fldurrs e s\ dn l€qt qrns (or'on) pooqroqq3rau al€urprooc e e{sl 'U p [d'C] - gf lurod fue ro;'1ce; u1 'Surddeur ctqdrouroloq '1xe11 e setuof,aqA * U:1, l€qt os Ar uo alnltnrls xalduroc 3 euueP a,n 'deur e Eurra,roc Jo uorlrpuoc aql sagql"s pue U oluo Ur yo Eutddeur (uolllnrlsuof, eql ,cg 'a : (la'gl)r. rq eas o1 fsea q snonurluoc € sr l 1l 1eql ua,rr3 uorlceto.rd eql ee U * A:)L p1 'sceds 1ecr8o1odo1 JroPsneg € seuoteq g ueql 'A u\ 4 Jo spoor{roqq3rau pluaurepunJ Jo uralsfs e ouuep s^\ 'd2 aseql p;" dn uea/$leq aauapuodsauoc euo-ol-euo Isf,Iuouef, e a^€LI e,t 'uteutop ig,'h palcauuoc ,tldurrs e sr.d2 acurg 'D o1 d uroq d4 ur pauteluoc qled frerlrqre ue st d4 ';2 ul .{eirrr e qcns ul ur e sr b pw 1eq1 lutod 2I lb'bC.9] slurod II3 Jo les aqt dn pooqroqq3rau e 4p fq alouaq 'U ul ul€ruop pelceuuoc fldurrs-e $ qtul^r d lo '9, ol Peau an '1srrg e ecnPor?ul rog uo ,(3o1odol {ue e{el 'g lurod { Jo ld'Cl 's,lrolloJ se qqt leqt ees eA\ e satuoreq eceJrns Surra,roc U l"srelrun Jo B '[d'g] sasselcacuale,rrnba aseq?ileJo les eql eqU lef '(d'C) Jo ssplt acuele,unbaeqlld'CJ dq aloueq'g uo tC ol crdolouroq sr j pue d - d !\Tuapanba erc (d' ,g) pue (d'9) srted oarrl aseq; 'd o1 od uror; A uo C r{led fue pue U uo d lutod fue 3o rted e eg.(d'C) 'ecsJJns ulretuelg e 'U eosJrns uuetuell{ ualrE e uo od Jo lurod es€q 3 xld te1 'alotr1 ecsJrns Surre,rocl"srellun e i(lalarauoc lcnrlsuoc lleqs arrrsqled Sutsn fq 'd = (ilu tlA u\ il,"r;3:"9:r|"'; A uo CWed e sI Ar uo,, qled e o't11llv 61 pre6 sl q!. q d lurod y 'U ecsJrnsuueruelg e;o Sutrerroce eq (g')L'A) p"l '0),C lurod leururrel eql pue (0)C lutod prltut aq1 qf!^A U uo ,C . C rlled e 1aBaar 'rg;o lurod FIlluI eqt qq^a C;o lutod Ieunurel eq1 Eurlcauuoc ,(q '(g),9 = (t)C lsq? qcns A ao tC Pue C sqled oarrlrog s8urraaogl"sra^ru1'Z'Z

30

2. Fricke Space

F1r,";; I x I - E b"t*""n i andllo,polbysettingF1t,"; = [C1r,,;,r"(t)] for (r, s) € 1 x 1. Therefore, we conclude that E is simply connected.

B

On putting these observations together with the uniformization theorem, we obtain the following theorem. Theorem 2.2. Fo'reoerg Riemann surfaceR, there exisls a uniaersal coaering surfaceR of R, which is biholomorphic to one of the three Riemann surfacesA, C, or H. Throughout this section, 6 o universal covering surface E of a Riemann surface r? we always take the one constructed above. From the construction of such a universal covering, it is easy to get the following lemma by an argument similar to that used in the case of analytic continuation (Ahlfors [A-4], Chapter 8). Lemma 2.3. (Existence and uniqueness of a lift of a path) For any path C on R with initial point p, and for ang point F of R oaer p, there erists a unique lifl C of C wilh initial point fi. Ttreorem 2.4. (Litt of a mapping) For Riemann surfaces R and S, let (R,Tn,R) and (S,trs, S) be their uniaersal coaeringsconstrucled as etplained preaiously,respectiaely.Then giuen an arbitrary continuous mapping f : R- S, there etists a continuousmapping it fr,--- S with forp= osoi. Thit mapping - (1, where € fr. and I is uniquety tletermineduntler the condition that i@) fu 4r e S are such that rs(Q1) = f brn(Fr)) Morvooer, if f is differentiable or holomorphic, then f is also differentiable or holomorphic. This mappine i t Fl.- ^9 ir called a lift of f: .R *

S.

Proof of Theorem 2./. Setting fu = lCr,pr] and 4t = [Dt,/(pr)], we get a .f (C),/(p)l m a p p i n g d e f i n e db v f ( l c , p l ) _lDt.f (Ct)-t Jor all points [C,p] in R. Then it is obvious that /(f1) - {1 and fSrp - nsol. Since zrp a.nd n5 are locally biholomorphic and / is continuous, / must be continuous. It is also trivial that if / is differentiable or holomorphic, then so is /. the uniqueness assertion follows from Lemma 2.3. D Rernark. (Uniq:reness of universal covering) For any two universal coverings (R,r,R) and (r?1,11,R) of a Riemann surface r?, there exists a biholomorphic mapping g of R to r?q with Trog - n. See,for example, Ahlfors and Sario [A,-6], Theorem 18A of Chapter L

e Jo uorl-rugep ar{} ur uorlrpuo? aql sessrl€s qclq,$ u ur d 3o 2 pooq.roqqStau e esooqo 'ld'Cl = 4'@)" - d 1as pue 'g f gl tutod € a{€t '(rr) aes o5 " '(g),0 - i sagsp, 'l"C) -- L ', - o5l 3ur11n4 'A uo zg pue IC sqled euos rc1 t€rl? aas am IC . 7'C fd'z7l = p pue [d'rCj = 4 a,re{ er'ruaql 'd = (!)y = (4)a 1eq1esoddns '(r) a.,'o.rdoa '{oo.t4 'Q* X u ( y ) r p q l q c n s J ) L s l u a u a l a f i u o u t Q a T g u { I s o l l t?' o a . t ' oa " r ' a q y ' g sSaoi (rrr) lo y TasqnsTcoilutocfiuo.to!'st IDW:ry uo fipnonu4u@srp fr.1.r,ailo.ti1 'sTurodpat{ ou soy fiyquapt.ayy.tol Tilacxa - J ) L f r ^ r a a"ar o t Q = n U ( d L e ) D e ' " t o l n c q t o dq ' { p ! } 1 lo yueu,a1q ? D q l 1 l ? n sU u ! 4 ! o 2 p o o t l . r o q q \ n u a l q l p n s o f l e r e q l ' U ) g f . r a a a i o g ( n )

'@)L= D uo s?srseatayT'(p)tt.= (4)v q?!nU ) !'q fruo.tog o ltlln J a L \uau.ta1a

:sa4.tado.td6utno11otaq7 sa{stqos g acottns uuvur?tg eqJ 'g'Z BtutrroT 6ut.r,eaoc o Io (U':r-'A) to .1 dnotf uorTout.tolsuo"tT losJeaNutu 'a,rr1calrnssr acuepuodsa.rroc O srql ecueq pu€'.[C] - l, teql seqdrur7'Z ruaroeqJ'(d'A)ro Jo luetuele u€ = ["d'Cj = (l'd ''I])t reqr s^\oqs g'U €urureT snql sr [9] ecurg '(t'd'"tl).[C] 'flarrrlcadsar'g;o s1u1od pue', pue pue aql ar€ Ierlrur Ieurural ['d'oI] I'd'Cj C l€I{} Jo lJll € sl , acuaH'od lurod eseq qlu{ g. uo qled pesol? s sI Col, sarldurr y JLov uorleler aql ueql '(l"d'"tl)L ollod'oll uroq Ur rio qled e al Q 'a,rr1celrns slq] 1eq1 aaord o; sr ecuapuodsarJoc 3 ,L luaurela ,tue a1e1 'err,r1celur st eeuepuodseJJo?sltl] Ie.I'J

l€qt sA^olloJII'ed'a)rL Jo luatuale lrun eqt q [u ef,uaq Pu€'07 o1 ctdolouroq sr op 'snq;'[t'O] = 1 ) t fue to1 od - (l)'l reqr qrns U uo qled eq] sl o1 ereq!\ 'lod' o o " If = fod' Cf = (l"d' tl).|' Cl a^eq a^r ueql 'J Jo lueurele lrun eql q -[?] 1eq1asoddns 'arrrlcefutq ]l ]€q] e,rord oa '.7 o1 (od(g)tv;o rusrqdroruouor{e sr ecuapuodserrocsrq} }tsqt pI^Ir} sl rI'loord '(A'o'U) 0ut.taaoc losrearun n to 1 dnot'6 uotTotu.totsuo.r,T |uuaaoc losraarun?ql oluo A Io ("d'A)rv dnot6 pTuau,opunt aql uo spyaffi-['d - log)acuapuodsauocaaoqDeyJ'g'Z rraroaqJ to tustrlilrotuost. '(A'v'A) Jo uorleruJoJ -sue.r1Sur.rar'oce sr 1r 'sl leql '..,1o1 s3uolaq -[op] srql 'uo1]-ruuepeql fq 'fpea13

'ld'c.'c) = ([d'c1).1'c] ry> la'c) fq g uo -[op] uorlce eql eugep e^{'(od'U)rv>['C] ]ueuele fue rog 'f, 1o (od'g)rY eqt o1 crqd.rouroslsl J dnor3 Eurraaoclesrelrun slt l€q? aas dnorE leluaurepunJ 'p ace;rns uueuerg e II€qs alll Jo (g'.u'9,) ece;rns EutrarrocIesJeAIunuarrrEe rog sdno.rg uor+BrrrroJsuer;, EurreloC IT

lesrallun'e'Z'Z

s8urra,ro3l"sra^rufl'Z'Z

g2

2. FrickeSpace

covering map in $2.1, and denote by U the connected component of r-r(J) containing f. Actually, it is sufficient to take a simply connected domain U c o n t a i n i n ge . l t 1 ( 0 ) n 0 + { f o r s o m e1 € f , t h e n t h e r e a r e p o i n t s f u , f u e 0 with f1 = l(it). Since ro7 = n, we get T(fr1) = r(4), and hence Q1- fi1,for r is biholomorphic on U. Thus we have l(Ft) = id(Ft), where fd is the identity. By Theorem 2.4, we conclude lhat 1 - id. Finally, to verify (iii), assumethat there exists a sequence{ Z" }f,r consisting of mutually distinct elements of l- such that 7"(1() n I{ * / for all n. Then for each n, we can take two points [n,Fn € 1{ with fn = .ln([n). Since K is compact, taking a subsequenceif necessary,we may assume that { drl1T=r, {i" }Lr converge to Qo,io € /{, respectively, as n + oo. Since zro7, = ?r, we obtain r(4") = r(f") and o(4") = r(i,). Take a neighborhood [/ of r(,i'") in .R satisfyigg the condition of the definition of a covering map in $2.1, and denote by U and,I/ the connected components of zr- 1(U) containing f, and fo, respectively. Since { j"(q") }f-, convergesto fo, we hav,e.y"(y)n0 t' g for a sufficient-lylarge n. Since ro7"(0) = (J,it follows that 7"(U) = 7, namely, ("tny)-|,1n(0) = 0. By the assertion (ii), we conclude that 7,.11 - 7,. This is a contradiction. ! Exarnple 3. Here is a"nexample of a group which does not act properly discontinuously. Let a be a real number not equal to 2r multiplied by a rational number. Then the group generated by l(z) = edoz does not act properly discontinuously onC-{0}.

2.2.4. Representation

of Riemann

Surfaces as Quotient

Spaces

We shall explain a way to construct a Riemann surface Rlf fro a Riemann surface R and a subgroup l- of the biholomorphic automorphism group Aut(R), where f is assumed to satisfy the properties (ii) and (iii) in Lemma 2.6, that is, every element of f except for the unit element has no fixed points in E, and acts properly discontinuously on E. Two points F,C e Rare said tobe f -equiaalentor equiaalentuniler f if there exists an e-lement.f e f satisfying 4=t@). Denote by [f] the equivalence class of fi. Let R/f be the set of all these equivalence classesp], which is called the quolient spaceof r? by .i-. Define the projection r'. R * R/f by r(fi) = \fl. We introduce the quotient topology "n fr,/f . A subset U of R /f is said to be open if and only if the inverse image o-t (U) of [/ is an open subset of E. the p_rojectionr is readily seen to be a continuous mapping of E onto tr/f. Since ,R is connected, so is r?/f. Moreover, we see that Rl f is a Hausdorffspace, for l- acts properly discontinuously on .R. Now, we define a complex structure o" n/ f as follows: for any point f e E, take a neighborhood. Up of I satisfying the property (ii) in Lemma 2.6. We may assume that there exists a local coordinate zp on 0p. Then, putting p = r(fi), we see that r: 0O - tJ, is homeomorphic. Hence, setting zo = Uo =,tr(0), zpozr-r, we conclude that { (Up,zp)}rrrtl , definesa complex structure so ihut

G'z) ,

'29-r - (z)L ,'a , -; s, u?lprn oslo st slttJ 'T = "lql - "lrl ,ltp^ ) ) Q(o anqm

,p+z!_k)L

@'z)

9t zo fi.r'aag(nr) ru.ro!o soq (V)nV Tuaurap to '0*Dql?n C)q'oanym 'q+zo=(z)L

(s'z)

tuaag Q1) ut.rolo soq (g)wy to Tuaurala ' I = " q - p Dq w n ) c P ' c ' q ' oa n a y m ,P*zc _k\L

(z'z)

Q* zo

fr"raag(r) ur"roto soq (g)7ny lo Tuaua1a 8'Z stutuoT

erll e^sr{ Il pue 'V 'C'Q

:sturoJ 3utmo11o; sul"tuoP l€?Iuoue?;o surstqd.rouolne erqdrouroloqrg

srrrerrroq [BcruorrBC go sdno.rg rusrqdrourolny

'

crt1daoruoloqrg'1p'8'Z

1 g ) t n vp u e '( v ) t n v ' ( o ) t n v ' ( q ) n y

'puttu ul qql qllft sdnor3 ursrqdrourolne crqd.rouroloqlq aq1 fpnls sn 1a1 'g, uo flsnonulluo?$p fFadord eql roJ ldacxa g' ur slutod paxg E?" Pue'fllluepl '9'A €Luluel urord 'ur '19)tnv;o dnor8qns 3 sl Jo J 1noq1r,nslueruelaJo slsrsuo? 'i'Q sursrqdrourolne-clqdrouroloqtqlo dnorE aql (U)t"V,(q alouap eAyI/ to 'ureroeql uorlezrur.rd; seceJrns uuetuerg "".rq1 nq1 jo "uo o1 crqdrorioloqlq q U -run aql Jo anlnl fq 'era11 '.7 dno.r3 uolleuroJsuerl Sfrueaoc l€srollun 8ll ,(q Ur eceJrns Surra,roc lesralrun e p J/A eceJrns uusruarll luarlonb eqt fq Paluas lerdar sr U eceJlns uu"IuaIU ,(re,ra 16q1 ueas aAeq eal 'uollcas Surpeee.rdeql uI

suorleurroJsuBl,I,snlqgtrAtr'8'U

'@)".--,W ?Ul repun. A o7 Tuapanba filloatydtotuoto\?q q J nq V k I /U acuapuodsa.u,oc acottns uuotu?ty Tuat1onbeqt u?qJ 'tr dno.t6uotyont"totsuntT0uuaaoc losJearun 'Z'Z rrraroaq.1, qnn A acottns uuour?rg o to |uueaoc losrearun o eq (A')L'A) pI 'uorlresse Surrrrrollo;eq1 ,(lalerpeurul el"q a \ ueql 'J ,(q acopns uuouery tf 1o eW J /A eteJrns uueruelg sql ilec eM'J /A 3o Euirarroce st (tr f g';u'g) TuatTonb suorl"rurolsu?rl snlqol I 't'z

34

2. Fricke Space

where0eRandaeA. (iv) Eaery elemenlof Aut(H) has a form. t(z) =

az+b cd+d'

(2.6)

w h e r ea , b , c , de R w i t ha d -, b c= I . In (2.2), it is sufficient that complex numbers o,, b, c, and d satisfy the condition od - b" # 0. However, 7 does not change when a, b, c, and d are multiplied by a common constant. Hence,-we may normalize the expression of 7 by ad - bc - 1. Every element of Aut(e) is called a M1bius transformation or a linear fractional transformalion.In particula.r, an element of Aut(H) is called a real Miibius transformation or a real linear fractional transformalion. Proof^of Lemma 2.8. First of all, let us determine the form of an element 7 € Aut(C). If 7(oo) = oo, then in a neighborhood of oo, 7 has the Laurent expansion

tQ)=",+i

bnz-n,

where a I 0. Then tQ) - oz is holomorphic on e , and hence the maximum principle shows that lk) - oz must be a constant function, say 6. Thus we have l @ ) = a z l b w i t h o + 0 . I f z ( o o ) = z o # @ , t h e n s e t t i n gt { z ) - t l Q - z " ) , we see that both 11 and lpl are elements of Aut(e), and 71o7(m) = oo. Thus we havefolQ) = I/QQ)z o ) = a r z * b r , w h e r e0 r , 6 r € C w i t h o , * 0 . Therefore, 7 is expressedin the form (2.2). Next, every element t e Aut(C) is extended to an element of Aut(e) if we put 7(oo) - oo. By the above argument, it is obvious that 7 is represented in the form (2.3). Let 7 be an element in Aut(A). Set 7(0) - B. Then the Mcjbius transformation r(z) = (, - 0)/G - Bz) belongs to Aut(A). Hence .y2 - 1*.r also belongs to Aut(A) and 92(0) = 0. Schwarz' lemma implies that 72 is a rotation trQ) = eiqz,with real number d. Hence, T is expressedin the forrn (2.5). It is easy to see that 7 is written in the form (2.4). Finally, for any element 7 e Aut(H), taking a biholomorphic mappin gT(z) = (z-i)/(z+i) of fI onto .4, we have an elementlr = ToloT-r e AutlA). Thus 71 is a Mijbius transformation and is representedin the form (2.2). Since 7 sends ry' onto itself, we may assume that c, D, c, and d are real numbers, and ad- Dc ) 0. Therefore, this 7 is written in the form (2.5). tr For more on the fundamental properties of M6bius transformations, such as transformation of circles into circles, and the invariance of the crms ratio under them, we refer, for instance, to Ahlfors [A-4], $3 of Chapter 3; and Jones and Singerman [A-48], Chapter 2. Now, for every 7 e Aut(e) given by

'sl

ler{}1'L1o

'slueruel€?s3urno11o;eql e^€rl a,r,r'uor1e1nc1ec aldurs e ,(g 'oz - (oz)L Surr(;sr1es oz > C Jo tas aqt IIs 'r(1r1ujpreq} oz slurod pexgJo las aql eq (t)xrg 1a1 }ou sr qcrqn

, I = c q- p D , ) ) p , c , q , o

, ' . 1 1 , i= Q*zo

@)L

.{q ue,tr3 uor?euroJsuer} snrqory e aq I 1a1 suorlBurroJsuBrl

snlqgl trJo srrrroJ lBcruouBc

'z'e'z

'{1aar1aadsa.r '(1'1) atnTvufts to dno.r,6fil,opun lonads eq1 '1)29 pue (U'6)79 araqirr plle 7 aa.r6ap lo dno"r6nautl yonads lDa, eqt are (1

' { t+} lfi 'r)ns= G't)nsa = (v)wv

'{ r+

pue

}/(u'?,hs= (ll'z)tsa = (n)pv '{(e)t"v

a^eq a^{ 'flrelrurrg ) Ll 6oLor-.{ } = (ra)lny reqr pue

' = -, :; lz:ii';il li:1, | l

uorleuroJsuerl a.,rr1ce ford e o1 spuodser.roc(3)tnv Jo (p + zc)l&+ zD) - (z)1, lueurale ue 1eq1 ees e/tr uaq;,'rd Jo eleulProoc snoeueSouoqe sr [tz : 0z] araqrrr'rz/02 = (ftz : ozl)g * rd :J Surddeur crqd.louroloqrqe euuep 'paepul 'rd Jo uorleruroJsu€rl f,q ? a,rrlcefo.rde o1 spuodsauoc (3)lny Jo luetuela ue '1d aceds err,rlcalo.rd xalduroc Ieuorsuaurp-euo aql qlra pagrluapl q C a.raqdsuuetuerg eql ueq1\ llrDureq

'(c'dts ul yT sluauala o,rl!,1 ,tq peluasarder sI ,L 1eq1elop "L Jo uorlD?uesa.tdat rt.tTou e pelle)c$st(g)l"y 3 ,L luaruele ue rog (1,)r- W lo V lueuala wtr '4 aa.rfaplo dno.t6 = (C'Z)lSd toauq lotcads aar,Ttato.td eql tr lpc pue {1+}/6'dlS tas aIA

' { r + } / 6 ' z ) t s= ( q ) w v

ursrqdrourosrue secnpur (ruaroeq? tusrqdrouoruoqeql dq 'ecua11'xrr?Burlrun aql sI 1 ereq^r ltl '{ 'c'g 'o ereqrrl 'eloqe se y f + 1 sl W Jo loura{ aql ueqtr, Jo saulueeq?er€p pue '(C'Z)1S )>y .ro;(p+zc)/(q+zo) = (z)(y)W tq paugep(q)ny oluo (3'6)79 yo ;;4rusrqdroruoruoqe e^eqem'r(lasrarruo)'(C'dlS dnor3 reauqletcadsaq1;o lp c1 l'^ -l =V L q D) ,I =cq-pp

qll^{ j

> p'?'q'D

luauele u3 aAeq e^r D+z) (1__-i_ (z\L 9*zo suorl"urrolsu"rJ snrqol{'t'z

9t

2. Fricke Space

36

(i) The casewhere oo € Fix(7),i.e., c = 0. lf a/d = 1, that b, c - 6f = *1, then 7 has a sole fixed point oo and it is written in the form

1e)=z*b, where b is a non-zero complex number. On the other hand, if a/d f has another fixed point zo, and is represented as

1, then 7

u-zo=\(z-zo), where to =lQ),

and ) is a complex number equal neither to 0 nor to 1.

(ii) The case where oo f Fix(7), i.e., c f 0. If (a * d)2 = 4, then 7 has a sole fixed point zo and it is written as

w-%=;:1+o' where to = lQ), and a is a non-zerocomplexnumber.If (c* d)' # 4, then 7 has two fixed points z1 and 22, and it is representedin the form W - Z t

",- tr-

^, Z7 -- Z' l

where w = 1(z), and ,\ is a complex number equal neither to 0 nor to L. Now, two elements7r,jz e Aut(A) are said tobe Aut(X)'conjugate or conjagale in Aut(X) if there exists an element 6 e Aut(X) such that 1z = 6o"ho6-t , where X is one of C, C, H, or A. This leads us to the following lemma. Lemma 2,9. Eoery Mdbius tmnsformation 7(f id) has otueor two f,xed points on e , antl is Aut(e)-conjugate to the foltowing M|bius tmnsformation 7o: ( i ) I f t h a sa s o l ef i x e i l p o i n t t, h e n T . Q ) = z + d f o r s o m ea € C , a * 0 . ( i i ) I f 7 h a st w o f i t e d p o i n t s , t h e n T o ( z ) = ) , 2 f o r s o m e \ € C , ) + 0 , 1 . We call this 7o a canonical form of 7. Matrix representations of canonical forms in (i) and (ii) correspond to

[;?], tf ,f.^), respectively. A real Mobius transformation 7(l fd) whose fixed points a.rein fl = RU{ m } is Aut(H)-conjugate to a canonical form 7o such that the entry a or ) of a matrix representation.To is a real number.

'peuueplla,lr '1, sr sll a.renbs (1,)rr1 Tnq fq flanbrun peurrurelap olw peretlesr ec€r1slr ueql'V- f,q pacelda.r sr y 1ousr (l).r1 snql'(p+D)'.{. uorleurroJsusrlsnlqgl4le 'p ;1 Jo nD4 " pelpc q qcg/'r * o (l)r1 1nd a14

' T= ? q,C , l \ ' ^ l= , PD ) p,c,q,o L 9 DJ :,L;o uorleluasa,rdarxrrleru " Jo ecerl eql np+p tsql atoN 'Z+y/I+y= e , ( pa r ) u o r l e n b ea q l s e g s l l e sy r a q d r l p t u s 1 1 'ol,;o lurod pexg e^r1?€r11ea{} o1 Eurpuodsarroc,L;o lurod pexg sJo alloq? aql uo spuadep yy/I pue aqt roJ sarroqr o,lrl e eq a64'l;o 1''a'r'l,yo.rar1dr11ruu .taqd41nut, e pallsc sl y slql'0'O* y'C ) y) zy - Q)'L uroJ Iscruouer € ot apSnluoc-(g)tnv o. cqoqered lou sr rl?rqar ,L uorleurroysrrerl snrqotrtre 'alo1q 'g uo sTurodpae{ omy soq L fipo puo ctloqtedfitl s! ,t (H) ta tt, .zz = rz puV , *H 'H 'rz slutod par{ on\ sorl L ) zz ) rz pUl qtns zz tt fi1uopuo tt, ctTdqla st' L 'g uo sr L Tutod pat{ alos o soy L lr fiyuo puo lt cqoqn.r,od

(r) (r)

:p7or16utmo71olaq7 uayl 'fr,7t7uapt ?ql pu s, q)rqn uotgou.t.r,olsuorl sntqory loar o eq L pT 'Ot'Z eurtuarl 's11nsaresaql Surulqurop 'uor?ross€ aqt urelqo ea,r 3ura,ro11o; /tls"a 'cqoqradr(q ro 'cr1dt11a ro 'cqoqered reqlla sI ','[1t1uapt ro (11)7ny Jo luetuale fra^a 1€ql ^\oqs ol fsea st 11 'fla,rrlredsar '1,

eql lou sr q?rq^r'(V)lnV

go paxg Suqledar eq} !o pue r; ,(q alouaq pexg a^rlrer11e a{l pue lurod lurod 'flarrleadser 'L elql pve Turotl pax{ 0ur11ada,eq} pell€, 1o yut,odpar{ aatyco.tqqo elre ez pue rz uaqJ, 'I < lVl qll^ y euos to! zY - Q)'L ruroJ l"tluouec e Jo qurod pexg eql o1 puodsarroa 'fla,rrlcadser'zz pu€ Iz 1eq1 asoddng oo pu" 6 'l, uorleur.ro;suer? snrqoq cnuorpoxol e;o slurod pexg aql aQ zz pue rz lo.l 'seuo ?rloqredr(qapnlcur suorleurroJsusrl c[uoJpoxol eql airr '4ooq slql uI 'crldqe rou ttloqersd .raqltau sl ?l JI cruoJporol l€ql eurrrnss€ eq ol pres sr flrluepr eql lou s! qrlqa uorleurroJsuerl snrqory y'((oo'O] / V zV : ue,rrEsr aldurexauy 'cqoq.redfq rou 'cr1dq1a pu€ > I f) I lVl'C Q)L.{q 'cqoqered raq?reu are qf,rqra suorl€ruroJsu"rl snlqol are eraq+ teql aloN I 'I+y '0 = c t l o y , a d f i qq f ( U ) y e r u o s r o J z y u o r t e l l p e o l a l e 3 n f u o c s r l I ; r ( z ) o L < '(7 3 u) ttuT - (r)'L uorlelor e o1 ele3nluoc sr ycr,Ttlqla q t (I) ]I * 0'1g )f g auos tol zsp

'o+n'c>p

eurrros roJ a + z -

(z)ot uorlelsuerl e o1 ele3ntuoa s! 1l y cqoqotod q f (t)

'f1r1uapr eql tou $ qcrrlAr uorl€ruJoJ -suerl snrqontr e eq ,L 1a1 'sadf1 earql otq suorl"ruroJsusrl snlqotr tr fSsselc all

snlqg,trtrJo uorlBcgrssBlc '8,'8'z suorlBrrrroJsrrB.LT. suorl"urolsuerl snlqgl I 't'z

LT

38

2. Fricke Space

By a simple calculation, we see that Mobius transformations are classifiedby trace. Lernma 2.LL. Let 7 be a M6bius transfonnation which is not the idenlity. Then the following hold: (i) 7 is parabolic if and only if tf (7) = a. (ii) 7 is elliptic if and only if 0 f tf (1) < a. (iii) r is hyperbolic if and only if tf Q) > . (iu) r is lorodromic if and only if tf (1) e C - [0,4]. Finally, we define the axis of a hyperbolic real Mobius transformation 7. Suppose that 7 is conjugate to a canonical form U@) - )z with ,\ > 1, by a real Mcibius transformation 6. Namely, suppose that 7 = 6oloo6-1. The half-line L = {iy | 0 < y < oo} in the upper half-plane .Il is the geodesic,joining 0 and oo, with respect to the Poincar6 metric ldzl2I [m z)2 on I1 (see $3 of Chapter 3) . The image 6(L) of .L under 6 is called the oris of 7 and is denoted by Ar. Then ,4', is the geodesicjoining the fixed points r,, and a., of 1, which is characterized as a semi-circle which joins r., and o., and is orthogonal to the real axis. Similarly, we define the axis A, of a hyperbolic transformation 7 in Aut(A).

2.4. Fuchsian Models First, we show that a Riemann surface whose universal covering surface is not biholomorphic to the upper halfplane f/ is biholomorphic to one of e , C, C { 0 }, or tori. Next, we study some fundamental properties of discrete subgroups of Aut(H), i.e., Fuchsiangroups. 2.4.L. Riernann

Surfaces of Exceptional

Type

Let us determine Riemann surfaceswhose universal covering surfacesare biholomorphic to either 0 or C. Theorem 2.L2. A Riemann surface^R has a uniuersal coaering surface fr. biholomorphic to lhe Riemann sphereC if and only if R itself is biholomorphicto C. Prool. Assume that .E = e . Sitt"" every element 7 of its covering transformation group f is a M o g ) o q + z - ( z ) o L r y q 1 e r u n s s ef e u r e a ru a q l , c q o q e r e ds o , L ; 1 'cqoqrad{q @*'q'U

'H uo slurod pexg ou seq ol, acurg .ro crloqered sr o,L saqdurr etuure1 1eq1 0I'Z 'p! oL qtl,lr '{pgl1 o,L .too.t4 luaurale ue a{€tr * J 3 * J leqt arunsss,(eu e11 'ct1cfics, u?Vl 'uoqaqos? 'H uo snonut?uocsrp lN J II fr4.rado.rd s! J to uorl?o eW llUI q?ns puv g uo sTutoilpac{ ou sly {p?} - J .VTZ BururaT lo Tuaua1e fi.raaa Toqgqcns (g)nv {o dnolfiqns D eq J pI 'adfi7 TouorTdnr? lo eq ol pres sr rrol ro '{ O} - C 'C 'C Jo euo o1 erqd.rouroloqlqq qcrq^\ ac€Jrnsuueuer}I V ' .t / C ot cryd.toutoloqrq s! A pW q?ns J dno.tf ac47ol D slstr,? a.r,aq1'g sn"to7fr.tana"lo3r .z(.re11o.ro3 '3 o1 crqd.rouroloqlq sr snrol € Jo aceJJns D 'dno.r3erlc{c e aq pFoqs 3ur.ra.,loc I€srelrun e e)ueH J e qcns leq} slras$ qcrrl^r (p1'6 eurtual) eurural 3ur,rlo11og eqt s1?rp€rluoc srqJ .g. ;o dno.r3 FluauepunJ eq? o1 crqdrourosr sl J roJ (6 4uer;o dno.r3 uerlaqe aarJ € eq lsnur J uar{} ,.Fl o1 ctqdrouroloqlqq U JI'snrol e q g 1eq1asoddns'fleurg'C = U leqt ^rou{ a , $ ' I ' Z $ y o 1 e l d u r e f g u r u a e ss e A rs V . { O } - C = A 1 a 1 , } x a N. C = A 1 a Ba m 'palcauuoc 'C fldurrs =A fl ?eql aurnsselsrg,asra,ruoc aq1 lroils o1 C ecws 'flalrlcadsa.r 'snro1 e pue '{ '9 o1 crqdrouroloq 0} - C '(II) pue '(ll) ,(r) sasecur ,e.ro;araqa U ac€Jrnsuueuerg eql -lq sl J/C

-

'U re^o luapuedepur fpeauq er€ qcrq^r C f r g ' o g a u t o sr o y t g * z = ( z ) r L p r r s o g* z = ( z ) o l a r a q a r ' ( r t ' . t ) = J ( l l l )

'{o}-c)oq

oruos roJ oq*z

(sr - (z)ol uorlelsuerl e {q pele.reueErl ,('tl leq} J

--,t

(tr)

{ p t l = . t (r)

:(1'deq3 Jo Z$ '[t-V] sroJlqy ';c) sesecaerql 3ur,rao11og aql rncco ereql l"ql elo.rd uec e^{ ueql '(9'6 eruuel ;c) g uo flsnonurluocsrp '.ra,roaro1,1i 'g fpadord slc€ em uo slurod pexg ou seq {pl } -,f > f teql J [eeer fue asnereq'l = D ' r a q l r n g ' ( O " ' C g ' o ) q + z o - ( z ) L t u r o Je q l u r f f uellrr^,l\sl J ;l,L fre,re g'A"uure.I fq,(g)lnV;o dnorEqnse sl J acurg.dno.rE uol+€ruroJsue.rl Sur.ra,rocl€sralrun st! eq J lel 'C = U leqt aunssy '{oot4 sIePoI{ u"rsrlf,nJ 't'z

6t

2. Fricke Space

2.4.2. Fuchsian

Models

and f\rndamental

Dornains

The following is an immediate consequenceof Theorems 2.I2 and 2.13. Theorem 2.L5. A Riemann surface R has a unioersal couering surface fr, biholornorphic to H if and only if R is not ^of exceptional lype; that is, if and only if R is not biholomorphicloany one of C, C, C - {0}, ortori. If a universal covering surface E of a Riemann surface ,R is the upper halfplane fI, we call its universal covering transformation group I a Fuchsian rnodel of .R. In this case, f is asubgroup of Aut(H). However, identifying I/ with 4, we sometimes consider a F\chsian model f as a subgroup of Aut(A). Remark -1. By an argument similar to that in the proofs of Theorem 2.13 and Lemma 2.I4, we see that the fundamental group of a Riemann surface R is commutative if and only if .R is biholomorphic to one of C, C, C - { 0 }, tori, the unit disk .4, 4- {0}, or annuli {z e C | 1 < lzl < r}. In order to obtain a geometric image of correspondencebetween a Riemann surface R and its Fuchsian model f, we use a fundamental domain for f. An open set F of the upper half-plane 11 is a fundamenlal domain for f if F satisfies the following three conditions: (i) z({) oF = / for every 7 e f with 1 { id. (ii) If .F is the closure of .F in 11, then

a = [J ,r(F). 7el

(iii) The relative boundary 0F of F in H has measure zero with respect to the twodimensional Lebesgue measure. These conditions tell us that the Riemann surface R = H /f F with points on dF identified under the covering group l..

is considered as

Emmple y'. For each covering group l- in Example 2 in $2.1,we define similarly its fundamental domain. The following (i)", ..., (t)" give examples of fundamental domains for covering groups of (i)', . . . , (t)' in Example 2, respectively'

( i ) " r - { , e C | 0 < I m z< 2 r } . ( i i ) " r - { , e H | 0 < R e z< 1 } . ( i i i ) "F - { , e c - { 0 } | 0 < a r s z< 2 r / n } . ( i v ) " . F - { z e H l I < l r l< f } . ( u ) "F - { , e C l r - a } b r , 0 < a < 1 , 0 < D < 1 } . There is a simple way to construct canonically a fundamental domain for a Fuchsian model of a Riemann surface r?. First, cut -Ralong suitable smooth paths on ,R to get a simply connected domain Ro. Let .F be a connected component of

aa.rq13uo1e U ln? pue U 3 od lurod e e{e; 'tO pue 'zO 're self,rrc ea.rq1fq pepunoq 'aue1dxalduroe eql ut Ar ureuop e 'g'Z '3ld ur pal€rlsnll se 'raprsuo3

'8'Z'8tJ

'V I

'g alilu,oag

'z'z'Btr

€- )t

'6, u1 $ qcrqrr ureluop Fluarrr€punJ e ser{ J l"ql leeduroc f1a,rr1e1a.r smolloJ 'rslncur€d ul'?,'7,'3U ul patertsnll sB J roJ ureuop le]uetuepunJ e l.I sl ('U)r-l;o g luauoduoc pelf,euuoc V'oU ur€ruop pelrauuoe rtldurrse 1e3 aa,r 'lg pue fy 1yeEuop g,3ur11ng'od lurod aseq qlr&t salrn? pesolc alduns qloorus il€ arp {g pue fy trqt qans (6 l) d 6nua3 Jo Ur eceJrns uueruarg pasop " roJ (od,A)to Jo srol"reuaS;o ualsfs l"cruouec " nq,=f{ lg'lV } tet 'g a\duorg 'saldurexa 'fe,u, slql uI aa,rq1 eql u-r"lqo aal Eur.nolloJ 'J IoJ uleuroP IelueuepunJ e sI d slql 1eq1flrsea aeseAr'Z'Z$ul (A'y'A) Eurraloc lesJe^IuneJo uorl?nrlsuoc eql ,tg'.u detu Sur.ranoceql rapun tA p (;A)t-! a3eurr esra^ur eql sIePoII u"rsqf,nJ't'z

TV

2. Fricke Space

42

smooth curves Ct, Cz, and C3. By the same argument as that in example 6, we have a fundamental domain tr'for .R as is shown in Fig. 2.3. The elements Ir,''lz € l- corresponding to the elements [At],[Ar] E q(R,po), respectively,give a ca.nonicalsystem of generators of f. In $1.5 of Chapter 3, we shall describe another way of cutting .R to get a fundamental domain for this group.

Example 7. As a limiting case of Example 6, let each circle D; degenerate to a single point pi to obtain a Riemann surface,R biholomorphic to C-{n,pz,ps}. A Fuchsian model of .R is conjugate to the principal congraencesubgroup f(2) of leoel 2, which consists of all elements 7(z) - (o, + b)/(cz * d) such that a , b , c , d € Z , a d - b c = l , a n d a = d = 1 , 6 : c = 0 m o d 2 . A s a s y s t e mo f generatorsof f(2), we have fQ) = z *2 and fQ) = z/(22 * 1). The picture on the left hand side of Fig.2.4 shows an example of a fundamental domain for f(2). The picture on the right hand side of this figure illustrates a fundamental domain for a subgroup of Aut(A) conjugate to l-(2). For details, see Ahlfols [A-4], $2 of Chapter 7; and Jones and Singerman [A-48], Chpater 6.

----)

I

z-t

R

a fundamental domain in .F/ for

rQ)

a fundamental domain in 4 for f(2)

Fig.2.4.

Remark 2. As canonical fundamental domains for a subgroup of Aut(H) acting properly discontinuously on ff , we have Dirichlet regions and Ford regions. For details, we refer to standard text books such as Beardon [A-11], Ford [A-31], Jones and Singerman [A-48], Lehner [A-66], [A-67], and Maskit [A-71].

on y > g y'acua11 o1 sa3ra.iluoc

o'l = ' I 'I | . ( , r ,- I ) e D; i arrrlrsod{ue.ro; u-VrBuVB 1eBea,r'u.ra3a1ur

, o* q e , u ) 9 , o

- ug 3ur11as'arrolq'flalrlcadsar

,

g] -

[,;,

, r + y, o < y , l ' ; .

i]

g

_V

dq uerrr3an g'L lo g'V suorle?uas '{ -e.rda.r xr.rleur ueql oo } = (g)*1.{ U (l)xrg pue { m'O } = (l)*tg t€rll eunsse feur aar 'uorle3nfuoc-(Hhnv fg 'sp1oq (rr) rou (r) raqlrau l€ql eunssv 'loo.r,4

.0 = (g)xr.{ u (r)4.{ (r) :sp1oy furmoilo! eyrlo euot"rl, 'plTg"r;r'::r:;"::, 'OZ'Z

s? L lI 'tr dno.r0uorsq?nf,o to sTuaualeonl ?q g puo L pI

BruuraT

'ralel pesn ar€ rl?rq,a,r, 'sdnor3 u"rsqcr\{ sarlradord auros luesard all ;o sdno.rg uBrsqcr\{ ;o sarl.radord Jaq}JqlI'V'?'Z sIePoI{ u"rsqf,nJ't'z

9t

2. Fricke Space

46

6".l "o" -- l o n d nJ ' l"n where o, = I - an-rcn-r, Thus

it follows

that

cr

bn = (an-!)2 , cn = -(cn-1)2, and d,. = I * a n - ( n - t . * = -"'n-t oo. Next, setting M = 0 as n *

max{ lol,l/(L - lcl) }, we obtain inductivelylo"l S M for any n. Thus each ant bn, and d, converges to 1 as n + contradicts the discreteness of l-.

oo. Hence, .Ar converges to Ao, which

tr

Theorem 2.22. Eaery element of a Fuchsian model of a closedRiemann surface of genusS (|=2) consislsonly of the identity and hyperbolicelements. Proof. Since every element 7 e f - { id } has no fixed points on I/, it is parabolic or hyperbolic. Assume that f contains a pa"rabolic element lo.By Aut(H)conjugation, we may suppose lhatT"Q) - z*1. From Lemma2.20, any element .y (+ id) of .i- with f(m) = oo is parabolic, which is written in the form = oo} is a ilz) = z *b for some real number 6. Hence, ]-- = { j e f I r(*) cyclic group. Replacing 7, with another element, if necessary'we may assume ad-bc= L, t h a t 7 o i s a g e n e r a t o rf o r f t . S i n c ee v e r y 7 ( z ) - ( a z * b ) l @ z * d ) , belonging to l- - ,i-- satisfies c + 0, Lemma 2.21 shows that lcl 2 1. Thus we obtain

rmTQ)S

1

1r-;pp

51

for all z with Imz ) 1. Set Uo - {z e H llmz > 2}. Then any two distinct points on [/o are not equivalent under any element of f - l--. Thus the quotient space Do = Uo/f* is biholomorphic to a domain Ro in r?. since 7o corresponds to a non-trivial element of the fundamental group of .r?,the closure E of R, in R is not simply connected. Since Do is biholomorphic to the punctured disk { z e C | 0 < l " l ( 1 } , w e i n f e r t h a t 4 m u s t b e h o m e o m o r p h i ct o { z € C | 0 < tr ltl S | ). This contradicts that R is compact. Remark. This theorem is also obtained by using the hyperbolic geometry discussedin $1 of Chapter 3. We present its outline. Let dsz = ldzl2l(Imz)2 be - H/f . the Poincar6 metric on 11, which induces the hyperbolic metric on R z * 1. For any positive number o, Assume that f has a translation 7o(z) denote by C" u closed path on ,R which is the image of the segment tro joining fo and 1o(ia) by the projection r: H -.R. Let l(C")be the hyperbolic length of Co,i.e., the length of .Lo with respect to the Poincar6metric. Then we see that t(C") - 0 as n + oo. On the other hand, .R being compact, we have a sequence + oo and r(l'o") + po as r, + oor { o" }L[r of positive numbers such that dn where po is a point on rt. Hence, if we take a simply connected domain u which contains po, then the closed path C,. is included in [/ for sufficiently large n. This implies lo - id, a contradiction.

,Hor{ro ,,u",u^ -,;,j"ii 1.j.,,i,",,; i";:;"Jn

'/G ol tuale^rnba sr (, lrl I C ) zI = 7 }tslP }Iun eqtr crrlatr l ?rBcurod 'T'I'8

l(llaruoag

rrloqrod/tH

pue rlrlatr tr ?rerutod

'I'g

'uolsrnql 'A\ fq pasodorddlluacars€^rqcrq/!\'acedsrellnurq)Ial aqt uotlecyrlceduroe Jo q)le{s e e,rr3 aaa.'p uorlcag ut 'fleutg elq€1oue uorlf,nrlsuot eql Jo Jo 'sq1Eua1 'urely pup a{?tJd suorle3tlsa,rul Jo Ieclsselcut ur3trosl! seq qctq,lr crsepoe3go sueeru,(q acedsueaprlcngue olur acedsrallntuqclel eql JoSutppequa ue ssncsrpa,n 'g uorlces uI 'e?€Jrnsuu€ureru pesoll e 3o ecedsrallnuq?Ial 'saleurprooc uralsdse auuaPe1'r aql uo 'saleurp.roolueslerNleq?uad ;o Pallec 'd.rlauroaE esoql fletcadsa crloqradfq 3urs11'scrsapoe3 Sutu.recuof, 6 uorlces ur 'serlredordcrseq{pn1spue elrlau ereculodaql eugepaiu.'1uorlcagut'1s.rtg '{srp uuetuelg }run eql uo crr}eruersculodaql dq pecnpulfl tlclq^rseceJrns uo frleuroa3 cqoqredfq aq1;o slcedseaurosssnc$pII€qsa^\ 'reldeqc sql uI

salBurProoc uoslarN-Iaqruad puB rt.rlauroa.D rrloq.radfll t raldBrlc

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

52

lf'.(:)l

=. - l " r I - l f ( r ) l ' = 1+,

z € a.

Moroaer, if the equality holds at one point in A, then f is a biholomorphic aulornorphism of A, and the equalitg holds at any point of A. Proof. Fix a point z in A arbitrarily, and set w*z .tt\w) =TlZw,

/ \ w-f(z) lztwt=, -fz1w' Then 71 and 72 belong to Aut(A), and ,F(to) = J2 o f " lr(w) is a holomorphic mapping of 4 into 4. Since .F(0) = g attd 'l - l,l2

F,(o)= ffif'(r), tr

we have the assertion by Schwarz'lemma.

When we denote by /-(ds2) the pull-back of the Poincard metric ds2 = aldzl2/(l - lrl')' by /, Proposition 3.1 implies that f* (ds2) < ds2 and that I.(dt2) Corollary.

- dsz if and only if belongs ro Aut(A). /

Eaery holomorphic mapping f : A ------A satisfies p(f (zt), f (zzD 1 p(21, z2),

21,22 € A.

Remark. In general, the Gaussian curaalure /{(h) h ( z ) 2 l d z l z( h ( r ) > o ) i s g i v e n b y

of a Riemannian metric

4 fl2logh

,h(h)=-Fd

A simple computation shows that the Gaussian curvature of the Poincar6 metric is identically equal to -1 on 4. Moreover, we can see that, when a metric h(z)2ldzl2 is invariant under the action by Aut(A), it is coincideni with the Poincar6 metric, up to a constant factor.

'y otuo H lo (! + z)/(? - ,) = (z)1,uorleurroJsuert snlqgl i eqt ,,(qv uo zspcrrleru ?r"curod eql Jo {req11nd aql 1nq3mq1ouq qclq,tr

, z(lu'l) = ,rrp

"l'Pl

3ur11as,(qpaugap sr g aueld-geq .raddnaql uo {sp crrlaur ar"curod eq&'tlrout?[ ''V LV uo slurod oall i(ue Eurlcauuoc crsapoe3fre,ra'6'9 Jo rr"qns e q uorlrsodor4 ,tq 'teql a?ou eJaH ',0 fq uorlce eql rapun luerrslur sr f,y uxe aq; 'L p'V sDr€eql palpc sr Ve oI 1euo3oq1.ro sr pue slurod asaql q3no.rq1sassed qcrqa,llluaur3as auq eql ro alcrr? eql Jo y u1 lred aql l"ql Ip?eU 'Vg uo Lo prte L.r, slurod paxg Irurlsrp otrl seq l, 'crloqredfq 4 (VhnV 3 ,L uaqal '1eq1 lpcag

'lzz'o)

tr -

7 luaur3es euq eq? qlr^r lueprcurof, sr Cyr fluo pueJI (rh .2, I xp7,

= (zz'1)d acuag

,W"[ ",ol

eleq era 'zz pue g Surlcauuoo trts pesolc frala ro;'r".II C 'U f d elq€1lns qYal. (V)7nY P zlz -t

#eP

= Q) L

'0 1 zz pue = rz luauela ue fq tuaql Sururrogsuert /tq 1eq1 etunsse deur 0 en'(V)1ny fq uorlce ar{l repun lu€rrelur sr crrleru ar€curod eq1 acurg /oo.l4,

'v lo

sl puD zz puo rz q0norqt sassodt1cn1m7uau,6as Vg fi.topunoqeql oI 1ouo0or17.to euq eql ro el?Jr?eyyto ctoqns o s, puD anbtun st 7r |teaoanory'V ul zz puo rz |utTcauuoc crsapoa0o slsNaeere1l 'V ) zz'rz fi^to4tq.toro,I 'Z'g uorlrsodor6 '(C)l = (zz'rr)d eleq e^r JI '9z ul zz pve Iz $ullcauuoc (cr.r1eru gr€oulod aq1 o1 lcadsar q1ra,r)ctsapoe0e 'V ul zz pue rz Surlcauuot'g cre pasolc elq€Urlcar€ IIef, e \'V ) zr 'rz s?ulod orrr1fue rod'(r)/ fq 1r elouap pu€'C p y76ua1cqoqtadfr,tlaql sp "[ lV, ell.'V ur , ]re pasolc alqegrlcer fre,ra rog

scrsaPoaD 'z'T't ,{r1auroag cuoqradifll pu" f,rrlel{ gr"f,u-Iod 'I't

t9

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

54 3.L.3. Hyperbolic

Metric

on a Riemann

Surface

Let it be a Riemann surface whose universal covering surface is biholomorphically equivalent to 4. Consider a Fhchsian model I of ,R acting on /. Let r: A r? be the projection of 4 onto R = A/f. Since the Poincar6 metric ds2 is invariant under the action by .l-, we obtain a Riemannian metric ds2pon R which satisfies r- (dszp)= d,s2. We call this dsft the Poincar6 metric, or the hyperbolic metric on R. Now, every 1 e f corresponds to an element [Cr] of the fundamental group r{R,po) of ,R (Theorem 2.5). In particular, 7 determines the free homotopy class of C7, where C,, is a representative of the class [Cr]. We say that 7 coaers the closedcurue Cr. When j € f is hyperbolic, it is seen that the closed curve -t, - A-, I 1l ), the image on .R of the axis A, by zr, is the unique geodesic(with respect to the hyperbolic metric on ,R ) belonging to the free homotopy class of Ct. We call L-, the closed geodesiccorresponding to 7, or to C.,. Proposition 3.3. Lel R be a Riemann surface with uniaersal coueringsurface H , and 11 be a Fuchsian model of R acting on H. Let

tk) =

az*b cz*d'

a , b , c , de P - ,

a d - b c= 7 ,

on R corresponding elemenlof 11, and L, bethe closedgeodesic be a hyperbolic to 7. Then the hyperboliclenglhI(Lr) of L, satisfies

t.'(r) - @+d)2= 4cosh2 e) Prool. Since t(L-r) and tr2(7) are invariant under the conjugation of 7 by an element ol Aut(H), we may assume that 7(z) - )z (.\ > 1). We may also this case,we have assumethat o - t5, b = c =0, and d = I/\5.In

((L.t) = Ir^ + = log) = 2log a. Hence we have the assertion.

!

3.1.4, Pants Consider cutting a Riemann surface r? which admits the hyperbolic metric by a family of mutually disjoint simple closed geodesicson R. Let P be a relatively compact connected component of the resulting union of subsurfaces.If P contains no more simple closed geodesic of .r?,then P should be triply connected, i.e., homeomorphic to a planar region, say

'd louoNeuelseueslerNaql Pellet sr d Pu€'d Jolaweq ueslerN erll Pall€l 'd st 2r 'f11en1rq€H'-dJ fq paurunalap flanbiun ,, j '.pto^ reqto uI Jo slued ',,(pee13 '2' 3o aee;rnsqns € s'e pereplsuoc sl d ;o rred anbrun eql s! plq^{ 'pelcauuoc ,{1dr.r1 '(t'g '31.{ eas) ; Jo lapour uelsqcqil e q d.7 pue 'acue11'lueuodtu6r ,t.repunoq qcee Suop uorSar palceuuof, .{1qnop ure3e sr 2' paul€tqo areJrns€ sI uaql 'a,t/V - d las elq€lrns e Surqcelp fq dr 4'uro.r; ql/d d pue'suotleurrogsuerl crloq.rad,tqom1fq pele.reue3dno.r3aa.ge s1 d.7 uaql 'd = ({)t }€tI} q?ns J Jo L dnor3qns eqt dJ fq alouaq '(d) r -! Jo lueuoduroc sluetuele yo Surlsrsuoc J Jo 1e = U aq1 aq pelceuuoce aQ V i )L pve'7 uo 3ur1ce JIV d 1e1'uorlcelord 'fpre.r1rqr" g 3o 7 slued ;o .rted e xtg U Jo lapour uersqcqE e aq J 1e1

'r'8'tIJ

4J/V:d

'U uo rlsapoe3 pasolc elduns € sl 2I ul d Jo frepunoq e^rleler eqt Jo luauoduroc palrauuoc frarra 3r Pue Palceuuoc i(1dt.r1q d JI g' 'g e IIef, e^\ '.re1;eara11 1o sTuod;o .rted e U Jo d eD€Jrnsqnslcedtuoc f1errr1e1e.r Surppnqa.r.ro; secerdlseilerus eql Jo auo s€ pereplsuoc aq ue? d et"Jrnsqns e qcns

({i t rr- ",}^{i >rr+,r})- { z > l z l } = 0 4 frlauroa.g rqoqraddll

cc

'I't Pu€ f,rrlel{ grsf,urod

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

3.1.5. Existence

and Uniqueness of P'-ts

We shall discuss the relationship between the complex structure of a triply connected domain J? and the hyperbolic structure of P, the unique pair of pants of O, induced by the hyperbolic metric on O. Let L1,L2, and..L3 be the boundary components, which a.re simple closed geodesics, of the pair of pants P. Let J-e be a Fuchsian model of the domain O acting on A. Then'i-s is a free group generated by two hyperbolic transformations, say, 7r and. 72. We may assume that 71 and 72 cover .L1 and L2, respectively. Theorem 3.4. For an arbitrarily giaen triple (ayaz,as) of positiae numbers, lhere erists a triply connected planar Riemann surfoce Q such that t(Li)=a1,

i-1,2,3.

Proof. We prove it by constructing O explicitly. Let Cr be the part of the imaginary axis in A. Fix another geodesic, say C2, on 4 such that the Poincar6 distance between C1 and C2 is equal to a1f 2. On the other hand, geodesics on 4 from which the Poincar6 distance to C1 are equal to oBf2 form a real one-parameter family (i.e., the family of circular arcs Cl tangent to the broken circular arc in Fig. 3.2-i)). Hence there exists a geodesic, say Cs, in this family such that the Poincard distance between Cz and Cg is equal to a2f2.

Fig.3.2.

Next, let 21 arrd z2 be the points in 4 uniquely determined by the condition

p(rr,r) =

?,

zr e Ct, zze Cz.

Let L\ be the geodesic connecting 21 and 22. Similarly, let {z3,za} and {rs,re]1 be the pairs of points uniquely determined by the conditions

feur e,r. 'r(.ressacau.l uorleEntuot-(g)7ny ue 3ur:1et 'asodrnd srql lsrll eutrrnssp rog'r={{{o},(q peururralap,{lanbrun are zL pue I,L }eql ^\oqs o} seclsns U 'e? 'I zL) eL sraloc o teql pue (e = 1) 17 s.rairoc{1, ?sqt arunsss feur aa,r ,_(tf 'ara11'0.7 srolereuaS;o rualsfs e aq 'I.L} 'g eueld-y1eqraddn eq} uo leT Jo {zt 3ur1ce Jo lepour uersqr\{ e eq 0J pu€ 'd Jo uorsuelxe ueslarN eql eq d ?c,-I'd d Jo (g'Z'l = f) f7 lueuoduroc frepunoq aq1 ;o q1Eua1crloqlad,(q eql eq lp p"I 'fy.rer1rq.reuaarEsr sluedgo ned s 'too.r4 teql esoddng ('IIt'[Ott] uaay'93) 2, '4 perepro aq1lo st176ua7 cqoqtedfrq aqy fiq lo sTuauoduoefi".topunoq p?aunrepp fryanbrunsa 4 syuodto .ttorl o to ernlrtuls aelilutoc ?ttJ .g.g ruoJoaql '(g'g '81.{ eas) ace;rns peilsap " sr snqtr '0J ,tq uorlce eql rapun (O)tbnO tr {Ji go ,trepunoq eq1 Surf;rluapl fq paurclqo ?as eql Jo rorrelur eql sl (J ;o 2, slued yo rpd anbrun arl? lstll pue 'pelceuuoc f1du1 sr.oJ /V = U leq+ realc s-rlt

'8'8'ttJ

pele.rauaEdno.rEaq1 aq o.ir1e1 '(V)l"V

.z,L pue rl, aseqt fq t" sluauele arloqrad,tu are zl, pue rl, ueq,L

'Vtoeb-zL

tebsV)-rl-

'eslrlurod fg Eur,rraserd les 3 go ursrqdrour ''e'l'f^2 -o1ne crqd,rouroloq-rlu" aqt qlr^r o1 uorl?auer eql;q (g'Z't = f) laadsar 't=[{!,1'!c} lh p1 pepunoq uo3exaq cqoqrad,(q pesolt eq? eq o p"r fq ('(fa'g 'ft9 eag) 'ez Pue ez Pue'?z pue 8z Eurlceuuoc scrsapoaSeq1 'f1a,rr1cadse.r'!7 pve 1I ,{q alouaq 'flelrlcedser

'rC)sz'eC)sz

, 7 , = (sz(s2)i tp

' 8 8 8 8 8 C ) v z ( z C ) ,E zz - ( v z ' e s ) 6 Z9

L9

i(rlauroag ruoqradi(g

pu" rrrlel l ?r"3ulod 'I't

58

3. Hyperbolic Geometry and Penchel-Nielsen Coordinates It(z)=\22, 0