By Yoichi Imayoshi, Masahiko Taniguchi

This booklet bargains a simple and compact entry to the idea of Teichm?ller areas, ranging from the main straightforward features to the latest advancements, e.g. the position this thought performs in regards to thread conception. Teichm?ller areas supply parametrization of the entire advanced constructions on a given Riemann floor. This topic is said to many various components of arithmetic together with complicated research, algebraic geometry, differential geometry, topology in and 3 dimensions, Kleinian and Fuchsian teams, automorphic kinds, complicated dynamics, and ergodic thought. lately, Teichm?ller areas have started to play an incredible function in string conception. Imayoshi and Taniguchi have tried to make the booklet as self-contained as attainable. They current a number of examples and heuristic arguments which will aid the reader clutch the tips of Teichm?ller conception. The e-book should be an exceptional resource of knowledge for graduate scholars and reserachers in advanced research and algebraic geometry in addition to for theoretical physicists operating in quantum conception.

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**Example text**

H " t t . R = R/f = C. Conversely, suppose that ,R = e . 2 it follows that E = e . L3. A Riemann surfaceR has a uniaersal coueringsurface biholomorphic to lhe complexplane C if and only if R is biholonrorphiclo one of C, C-{0}, ortori. ro crloqered sr o,L saqdurr etuure1 1eq1 0I'Z 'p! Vl 'uoqaqos? rd s! raaa Toqgqcns (g)nv {o dnolfiqns D eq J pI 'adfi7 TouorTdnr? toutoloqrq s! tf ac47ol D slstr,? g. r3 FluauepunJ eq? 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 .

R,6uvrsycnl e Jo palp? q (H)InV;o dnorEqns eterf,srp O '19)tnV Jo leql uorJ pernpu! t J ! 7 dnol3qns y 'oo ol sPuel u se 'dle,rrlcedsar'p pr. '(11)WV uo ,(Eo1odo1Iernleu e Surugap q1ral ur3aq e11 (n)pV go sdnorEqns alarcsrq 'g'V'Z sIaPoII u"rsqsnd 't'z t 2. Fricke Space 44 as n --+ oo. Since { r" }Lr is a normal family, taking a subsequence,if necessary, we may assume that { 7, }p, converges uniformly on compact subsets of f/ to a holomorphic function 7 defined in 11 . 18), this 7 must be an element of Aut(H).

Next, every element t e Aut(C) is extended to an element of Aut(e) if we put 7(oo) - oo. 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). r also belongs to Aut(A) and 92(0) = 0. Schwarz' lemma implies that 72 is a rotation trQ) = eiqz,with real number d. 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.