Title page for ETD etd-07092009-200839


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
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.

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