Algorithms related to subgroups of the modular group

Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose of this thesis is to find and characterize small-index noncongruence subgroups of the modular group PSL_2{Z}. We use the concept of Farey symbol to describe the subgroups of PSL_2{Z}. The first part contains results concerning the geometry of subgroups of PSL_2{Z}. The second part describes a graph-theoretical approach to finding all subgroups of a given index. In the third part we describe two algorithms for testing the membership of a matrix to a subgroup given by a Farey symbol. As an application we find the noncongruence subgroups of PSL_2{Z} of index less than 10.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2009

Author dc:creator

• Caranica, Constantin Cristian

• cusp width
• fundamental domain
• trivalent diagram
• farey symbol
• noncongruence subgroup

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-07092009-200839
https://repository.lsu.edu/gradschool_dissertations/854

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-1853