By Yoichi Imayoshi, Masahiko Taniguchi

This publication deals a simple and compact entry to the speculation of Teichm?ller areas, ranging from the main ordinary points to the newest advancements, e.g. the position this idea performs in regards to thread idea. Teichm?ller areas provide parametrization of all of the complicated constructions on a given Riemann floor. This topic is said to many various parts of arithmetic together with complicated research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic varieties, advanced dynamics, and ergodic concept. lately, Teichm?ller areas have all started to play a massive position in string concept. Imayoshi and Taniguchi have tried to make the ebook as self-contained as attainable. They current quite a few examples and heuristic arguments so that it will aid the reader grab the tips of Teichm?ller idea. The e-book should be a great resource of data for graduate scholars and reserachers in advanced research and algebraic geometry in addition to for theoretical physicists operating in quantum concept.

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Y"(y)n0 f:. e o, we { j"(q") }J:"=l }f-, con:erges t' g for (J,it n. Smce ro7"(0) Since7rO followsthat In(U) 7, namely, namely,bn+d ("tny)-|,1n(0) n. It follows 0ln(U) = U. 0. ln (U) = U, 7"(U) = V, assertion (ii), we we conclude conclude that In+l By the assertion 0! is aa contradiction. 11 = In' 7,. This is = = = = = = = Exarnple 3. Here is is an a"nexample group which does 3. Here example of aa group Example does not act act properly discontindiscontinuously. Let a be be aa real real number not equal uously.

T) 9(st) for t E I. ,C(s)] fo: every s E I, we have a path Con R from [Io,po] to [C,p], which implies that R is connected. = = Now, we prove that R is simply connected. It is sufficient to see that every closed path G on R with base point [10 , Po] is homotopic to [Io,Po]. Put C = 'TroG. Then C is a closed path on R with base point Po, and the terminal point of G is represented by [C, Po]. Since G is a closed path, we conclude that [Io,Po] = [C, Po], which means that C is homotopic to 10 , Let F: I x I -+ R be a homotopy between them such that {F.

Now, by using paths we shall construct concretely a universal covering surface of a Riemann surface. Fix a base point Po on a given Riemann surface R. Let (C, p) be a pair of any point p on R and any path C on R from Po to p. These two pairs (C, p) and (C', p') are equivalent if p = p' and C is homotopic to C' on R. Denote by [C, p] the equivalence class of (C, p). Let R be the set of all these equivalence classes [C, p]. 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.

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