Teichmüller spaces via Fenchel-Nielsen coordinates

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Ricardo García López[en] The main goal of this work is to study Teichmüller spaces of Riemann surfaces and hyperbolic surfaces via Fenchel-Nielsen coordinates. To do this, we first determine the universal holomorphic covering of every Riemann surface and briefly study Fuchsian groups. Then, we introduce moduli and Teichmüller spaces for Riemann surfaces and use the previous characterization to imbue the Teichmüller space with the so-called algebraic topology. We also compute said spaces for the torus and give a short remark on the mapping class group. Afterwards, we introduce hyperbolic geometry as the natural geometry compatible with the complex structure of the complex unit disc and use this along our previous effort to geometrize Riemann surfaces. We go on to study hyperbolic surfaces with and without boundary with a special emphasis on building the necessary machinery to prove the existence and uniqueness of closed geodesics in a given homotopy class via the axis of hyperbolic transformations. Finally, we undergo a thorough study of pairs of pants and X-pieces in order to demonstrate the main theorem about the Fenchel-Nielsen coordinates. Also, this study provides the necessary background from hyperbolic geometry for the short paper that has grown from this project. We conclude with some applications of these efforts, like the collar lemma