Algorithms Related to Triangle Groups

Abstract

Given a finite index subgroup of $\PSL_2(\Z)$, one can talk about the different properties of this subgroup. These properties have been studied extensively in an attempt to classify these subgroups. Tim Hsu created an algorithm to determine whether a subgroup is a congruence subgroup by using permutations \cite{hsu}. Lang, Lim, and Tan also created an algorithm to determine if a subgroup is a congruence subgroup by using Farey Symbols \cite{llt}. Sebbar classified torsion-free congruence subgroups of genus 0 \cite{sebbar}. Pauli and Cummins computed and tabulated all congruence subgroups of genus less than 24 \cite{ps}. However, there are still some problems left to be solved. In the first part of this thesis, we will use the concept of Farey Symbols and bipartite cuboid graphs to determine when two subgroups of $\PSL_2(\Z)$ are in the same conjugacy class in $\PSL_2(\Z)$. We implemented this algorithm, and other related algorithms, with SageMath \cite{baowebsite}. In the second part of the thesis, we will extend these ideas to general triangle groups. Specifically, we will classify some small index conjugacy classes of subgroups of the triangle group $\overline{\triangle}(2,4,6)$.