A closed **Riemann surface** which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis characterizes the cyclic trigonal Riemann surfaces of genus 4 with non-unique trigonal morphism using the automorphism groups of the surfaces. The thesis shows that **Accola’s bound** is sharp with the existence of a uniparametric family of cyclic trigonal Riemann surfaces of genus 4…

*Contents*

Introduction

1 Preliminaries

1.1 Hyperbolic geometry

Classifying isometries of H

1.2 **Riemann surfaces**

Holomorphic functions on a Riemann surface

Meromorphic functions on a Riemann surface

Holomorphic maps between Riemann surfaces

Properties of holomorphic maps

The Euler characteristic and the Riemann Hurwitz formula

1.3 Uniformization

Group actions on Riemann surfaces

Monodromy

Uniformization

Fuchsian groups

Fuchsian subgroups

The quotient space H/Γ

Automorphism groups of compact Riemann surfaces

1.4 Teichm¨uller theory

Quasiconformal mappings

The Teichm¨uller space of Riemann surfaces

The modular group

Action of the modular group

Maximal Fuchsian groups

1.5 Equisymmetric Riemann surfaces and actions of groups

Equisymmetric Riemann surfaces

Finite group actions on Riemann surfaces

Algebraic characterization of B

November 8, 2006 (9:45)2 Trigonal Riemann surfaces of genus 4

2.1 Trigonal Riemann surfaces

2.2 Existence of cyclic trigonal Riemann surfaces of genus 4

Producing cyclic trigonal Riemann surfaces of genus 4

2.3 Cyclic trigonal Riemann surfaces of genus 4 with non-unique morphisms

2.4 The equisymmetric strata of trigonal Riemann surfaces of genus 4

Equisymmetric Riemann surfaces …

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Author: Ying, Daniel

Source: Linköping University

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