On the Moduli Space of Cyclic Trigonal Riemann Surfaces of Genus 4

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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