Identifier
etd-07092009-200839
Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
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.
Date
2009
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Recommended Citation
Caranica, Constantin Cristian, "Algorithms related to subgroups of the modular group" (2009). LSU Doctoral Dissertations. 854.
https://repository.lsu.edu/gradschool_dissertations/854
Committee Chair
Helena Verrill
DOI
10.31390/gradschool_dissertations.854