By Yoichi Imayoshi, Masahiko Taniguchi

This booklet bargains a simple and compact entry to the idea of Teichm?ller areas, ranging from the main ordinary points to the newest advancements, e.g. the function this idea performs with reference to thread thought. Teichm?ller areas supply parametrization of the entire advanced constructions on a given Riemann floor. This topic is expounded to many alternative parts of arithmetic together with complicated research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic types, advanced dynamics, and ergodic conception. lately, Teichm?ller areas have began to play an enormous position in string conception. Imayoshi and Taniguchi have tried to make the ebook as self-contained as attainable. They current a number of examples and heuristic arguments on the way to support the reader snatch the guidelines of Teichm?ller concept. The publication may be an outstanding 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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With the common pole p. Then, in the same way as before, we can construct a biholomorphic mapping fn: Rn - Alor every n. rns|uuaaoc e g iue 'g 1o |uttaaoc e (A'v'U) IIec aaa'r1ooq srql uI 'crqdlouroloqlq q /? roqqSreue s€q d fra,re y dout, Fuueaoc e eq ol pres sr lurod , A Jo 'sac€JJnsuueuaru aq pue Ur Ar ]erl A - U : ! unJuaarg erlt Jo qder3 I I +--I sBurra,rog lesrel-ru1'Z'Z LZ 2. Fricke SPace 28 "highest" covering surface of all coverings of covering of R mea,nsthat it is the r? (cf. 2). Example I.

5). 4). 4, we have an elementlr = ToloT-r e AutlA). 2). Since 7 sends ry' onto itself, we may assume that c, D, c, and d are real numbers, and ad- Dc ) 0. 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. 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 , ' .

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. R. Lo with respect to the Poincar6metric. Then we see that t(C") - 0 as n + oo.