Type of Document Dissertation Author Caranica, Constantin Cristian Author's Email Address ccaran1@lsu.edu, ccaranica@yahoo.com URN etd-07092009-200839 Title Algorithms Related to Subgroups of the Modular Group Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title Helena Verrill Committee Chair Mark Davidson Committee Member Patrick Gilmer Committee Member Robert Perlis Committee Member William Adkins Committee Member Warren Waggenspack Dean's Representative Keywords

- cusp width
- fundamental domain
- trivalent diagram
- farey symbol
- noncongruence subgroup
Date of Defense 2009-07-07 Availability unrestricted AbstractClassifying 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.

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