Evrensel gönderim sınıf grubu elemanlarının yarı-konformal temsilleri

The universal Teichmüller space T , which is the space of normalized quasi- symmetric homeomorphisms of the unit circle S1 fixing three points, is defined by L.Bers [1] in 1965 and it contains classical Teichmüller spaces as complex submanifolds. In this thesis, we first focus on quasi-conformal homoemorphisms and the universal Teichmüller space. Since quasi-conformal maps have a fundamental role in Teichmüller theory, we study them in depth by presenting their various definitions and examples. Moreover, we introduce the Douady-Earle extension in order to find a quasi-conformal extension of a given quasi-symmetric map which is a homeomorphism satisfying the following inequality condition 1/k ⩽ g(x + t) − g(x)/g(x) − g(x − t) ⩽ k for all x ∈ R and t > 0. After constructing the bijection between the collection of all tessellations, Tess′, with a distinguished oriented edge of the unit disc D and the group of orientation-preserving homeomorphisms of S1, Homeo+(S1), we obtain a homeomorhism between Tess′/PSL2(R) and Homeo+(S1)/PSL2(R). Moreover, we consider the tessellation produced from the flip along the distinguished oriented edge of the Farey tessellation. Since the characteristic mapping of the tessellation produced is quasi-symmetric, we obtain the quasi-conformal map of this characteristic map by the Douady-Earle extension.