The University of Southampton
University of Southampton Institutional Repository

Four-punctured spheres

Four-punctured spheres
Four-punctured spheres
The four-punctured sphere can be viewed as a Riemann surface. It is then the quotient of the hyperbolic plane H by a Fuchsian group with signature [0; oo4]. In Chapter 1 we introduce Riemann surfaces and Fuchsian groups, in particular the subgroups of the modular group and Hecke groups. In Chapter 2 we introduce Teichmuller space via the Teichmuller space of the torus. We present the theory of Thompson [Th] and McNeilly [McNl] which gives a bijection between the Teichmuller space of groups of signature [0; oo4], denoted by T[0; oo4], and a domain V via a standard set of generators for such a group. We also describe the bijection between T[0; oo4] and T[l; oo] given by McNeilly In Chapter 3, the signatures of all Fuchsian groups Gi which contain a group G\ of signature [0; oo4] with finite index are found. For each such group inclusion G\ C G2 we find generators for the groups G\ and G2, and using the theory of Chapter 2 find the corresponding points in T[0; oo4]. The interesting case is when Gi is a triangle group in which case we say the corresponding G\ is exceptional. We also find the groups containing a subgroup of signature [1; 00] with finite index and the corresponding points in T[l; 00]. We introduce the Teichmuller modular group of a group of signature [0; oo4], and study its action on T[0; oo4]. We find that exceptional groups are congruence subgroups of the modular group and have rational points in V. We also give a short proof of a Theorem of Thompson [Th]. Finally, in Chapter 4 we introduce NEC groups and consider those NEC groups that have a group of signature [0; oo4] as their canonical Fuchsian group. Corresponding to these groups are some one-dimensional subspaces of T[0; oo4] which we calculate. We conclude with a brief discussion of further work related to onedimensional subspaces.
Kelk, Catherine Rosalind
722c21be-8490-49c7-9b61-644cc2dbe9c1
Kelk, Catherine Rosalind
722c21be-8490-49c7-9b61-644cc2dbe9c1

Kelk, Catherine Rosalind (1999) Four-punctured spheres. University of Southampton, Department of Mathematics, Doctoral Thesis, 99pp.

Record type: Thesis (Doctoral)

Abstract

The four-punctured sphere can be viewed as a Riemann surface. It is then the quotient of the hyperbolic plane H by a Fuchsian group with signature [0; oo4]. In Chapter 1 we introduce Riemann surfaces and Fuchsian groups, in particular the subgroups of the modular group and Hecke groups. In Chapter 2 we introduce Teichmuller space via the Teichmuller space of the torus. We present the theory of Thompson [Th] and McNeilly [McNl] which gives a bijection between the Teichmuller space of groups of signature [0; oo4], denoted by T[0; oo4], and a domain V via a standard set of generators for such a group. We also describe the bijection between T[0; oo4] and T[l; oo] given by McNeilly In Chapter 3, the signatures of all Fuchsian groups Gi which contain a group G\ of signature [0; oo4] with finite index are found. For each such group inclusion G\ C G2 we find generators for the groups G\ and G2, and using the theory of Chapter 2 find the corresponding points in T[0; oo4]. The interesting case is when Gi is a triangle group in which case we say the corresponding G\ is exceptional. We also find the groups containing a subgroup of signature [1; 00] with finite index and the corresponding points in T[l; 00]. We introduce the Teichmuller modular group of a group of signature [0; oo4], and study its action on T[0; oo4]. We find that exceptional groups are congruence subgroups of the modular group and have rational points in V. We also give a short proof of a Theorem of Thompson [Th]. Finally, in Chapter 4 we introduce NEC groups and consider those NEC groups that have a group of signature [0; oo4] as their canonical Fuchsian group. Corresponding to these groups are some one-dimensional subspaces of T[0; oo4] which we calculate. We conclude with a brief discussion of further work related to onedimensional subspaces.

Text
00125599.pdf - Other
Restricted to Repository staff only

More information

Published date: October 1999
Organisations: University of Southampton

Identifiers

Local EPrints ID: 50641
URI: http://eprints.soton.ac.uk/id/eprint/50641
PURE UUID: 96097216-6b62-4b5f-be03-fdcbe40ef292

Catalogue record

Date deposited: 06 Apr 2008
Last modified: 13 Mar 2019 20:50

Export record

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×