Miniatura

Fitxers

tfg_gerard_bargallo_gomez.pdf (1.71 MB)

Tipus de document

Treball de fi de grau

Data de publicació

2021-06-20

Llicència de publicació

cc-by-nc-nd (c) Gerard Bargalló Gómez, 2021
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/184272

Teichmüller spaces via Fenchel-Nielsen coordinates

Títol de la revista

Autors

Director/Tutor

ISSN de la revista

Títol del volum

Recurs relacionat

Resum

[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.

Descripció

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Ricardo García López

Matèries

Espais de Teichmüller, Treballs de fi de grau, Topologia de baixa dimensió, Geometria hiperbòlica, Varietats de Riemann

Matèries (anglès)

Teichmüller spaces, Bachelor's theses, Low-dimensional topology, Hyperbolic geometry, Riemannian manifolds

Citació

Col·leccions

Treballs Finals de Grau (TFG) - Matemàtiques

Citació

BARGALLÓ GÓMEZ, Gerard. Teichmüller spaces via Fenchel-Nielsen coordinates. [consulta: 13 de gener de 2026]. [Disponible a: https://hdl.handle.net/2445/184272]

Exportar metadades

JSON - METS

Compartir registre