Teichmüller Spaces and Grothendieck’s Dessins

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Ghorbani, Hanie | 2022

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 55048 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Safdari, Mohammad; Kamalinejad, Ali
Abstract:
The main goal of this thesis is to study the Teichmüller space and its geometric properties. The Teichmüller space of a surface can be studied from complex geometric or hyperbolic geometric points of view, and in this thesis we refer to both of these viewpoints where needed. Another goal of this thesis is to review a method of compactification of the Teichmüller space. Therefore, we introduce the compactification method proposed by Thurston. To this end, we first briefly review hyperbolic geometry. Then we extend some tools, and use some techniques in hyperbolic geometry, to describe Thurston’s construction and pave the way for it. In the second phase of this study, we introduce Grothendieck’s Dessins and peruse its relation with Belyi functions
Keywords:
Tichmuler Space ; Belyi Functions ; Measured Foliations ; Teichmuller Space Compactification ; Grothendieck’s Dessins ; Measured Laminations