Symmetries of Compact Riemann Surfaces (Lecture Notes in Mathematics, Vol. 2007) (Lecture Notes in Mathematics, 2007)

✍ Scribed by Emilio Bujalance, Francisco Javier Cirre, José Manuel Gamboa, Grzegorz Gromadzki

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 181
• Category: Library

✦ Synopsis

This monograph covers symmetries of compact Riemann surfaces. It examines the number of conjugacy classes of symmetries, the numbers of ovals of symmetries and the symmetry types of Riemann surfaces.

✦ Table of Contents

Symmetries of Compact Riemann Surfaces
Preface
Contents
Introduction
Chapter:1 Preliminaries
1.1 NEC Groups and Their Signatures
1.2 Normal Subgroups of NEC Groups
1.3 Centralizers of Reflections
1.4 Uniformization and Automorphism Groups of Riemann and Klein Surfaces
1.4.1 Maximal NEC Groups
1.4.2 Teichmüller Spaces
1.5 Symmetric Riemann Surfaces
1.5.1 Algebraic Conditions
Chapter:2 On the Number of Conjugacy Classes of Symmetries of Riemann Surfaces
2.1 Conjugacy Classes of Involutions in 2-Groups
2.2 Symmetries with Non-Empty Set of Fixed Points
2.3 Symmetries with Empty Set of Fixed Points
2.4 Symmetries of Surfaces Admitting a Fixed Point Free Symmetry
Chapter:3 Counting Ovals of Symmetries of Riemann Surfaces
3.1 Enumeration of Ovals of Symmetries at Large
3.2 Total Number of Ovals of Non-Conjugate Symmetries
3.3 Total Number of Ovals of all Symmetries of a Riemann Surface
3.4 Total Number of Ovals of a Couple of Symmetries
Chapter:4 Symmetry Types of Some Families of Riemann Surfaces
4.1 Symmetry Type of the Riemann Sphere
4.2 Symmetry Types of Tori
4.2.1 Symmetric Tori
4.3 Symmetry Types of Hyperelliptic Riemann Surfaces
4.3.1 A Geometric Method
4.3.2 An Example
Chapter:5 Symmetry Types of Riemann Surfaces with a Large Group of Automorphisms
5.1 Some General Results
5.2 Symmetry Types of Macbeath--Singerman Surfaces
5.3 Symmetry Types of Accola--Maclachlan and Kulkarni Surfaces
5.3.1 Number of Ovals of the Symmetries
5.3.2 Separating Character of the Symmetries
5.4 Algebraic Formulae for the Symmetries
Chapter:
6 Appendix
6.1 Compact Riemann Surfaces Without Symmetries
References
Index