By Yoichi Imayoshi, Masahiko Taniguchi

This e-book deals a simple and compact entry to the idea of Teichm?ller areas, ranging from the main hassle-free elements to the newest advancements, e.g. the function this concept performs with reference to thread idea. Teichm?ller areas supply parametrization of all of the advanced buildings on a given Riemann floor. This topic is said to many various components of arithmetic together with advanced research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic kinds, advanced dynamics, and ergodic conception. lately, Teichm?ller areas have all started to play an incredible position in string conception. Imayoshi and Taniguchi have tried to make the booklet as self-contained as attainable. They current various examples and heuristic arguments on the way to aid the reader seize the guidelines of Teichm?ller concept. The ebook may be an exceptional resource of data for graduate scholars and reserachers in advanced research and algebraic geometry in addition to for theoretical physicists operating in quantum thought.

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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. Construction of tJniversal Covering Surfaces First of all, we need several definitions. 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).

IX) ' H J o ' J ' 3 s e c o t t n su u D u r e l y st acoltns-uuouaty peloauuo? repunoq euo u"q? ta ol Pa)nPer sI rueroaq? oeJalqe{r€IueJ e }sq} u^lou{ 11ams! uot a,rl 'dem srql uI 'suolleuroJsueJl snlqgl 'dnor3 'relnarlred uersqrnd e uI tr J II€r ell'' 11 = U uerl^r 'Il ro'C '? raldeqc luesard eql Jo esodrnd aqa ar€ds a{rlqjt u raldBrlc 2. FrickeSpace 26 Remark. These Riemann surfaces 0, C, and /y' are not mutually biholomorphically equivalent. The Mobius transformation tr = (z -i)lQ * f) maps biholomorphically 11 onto the unit disk 4, and hence we often use the unit disk 4 instead of the upper half-plane If .

Hence,-we may normalize the expression of 7 by ad - bc - 1. r, an element of Aut(H) is called a real Miibius transformation or a real linear fractional transformalion. 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 .