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admits a quasiconformal Ahlfors algebraic curve analytic function Automorphic forms Banach space Beltrami equation Bers boundary curves canonical closed Riemann surfaces complete metric space computation conformal map continuous and holomorphic converges defined definition denote depends holomorphically differentials dim T(G domain of holomorphy elliptic elements equivalence class exists finite fixed follows Fuchsian group G o w genus g hence holomorphic and strongly holomorphic functions holomorphic homeomorphism integral Jordan curve Jordan domain Lemma Let G linear mapping Main Theorem mapping f marked surfaces Mbbius transformation modular group norm normalized self-mapping open Riemann surface parameters Poincare metric proof properly discontinuous Proposition quasiconformal map quasiconformal reflection R(So real axis reference surface References for Lecture Remark schlicht function Schwarzian derivative statement strongly continuous T(So T(SQ Teichmuller metric Teichmuller spaces topological map topological rectangle topological self-map uniformly upper half plane whole plane wy(z