Computations with finite index subgroups of $PSL_2(\mathbb Z)$ using Farey Symbols

TL;DR

This paper discusses Farey symbols as a compact and efficient tool for performing calculations with finite index subgroups of the modular group, including generators, cosets, and group properties.

Contribution

It provides explicit algorithms and an expository overview of using Farey symbols to analyze finite index subgroups of the modular group.

Findings

Algorithms for computing generators and coset representatives

Methods for identifying elliptic points and genus

Procedures for testing congruence of groups

Abstract

Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group $\Gamma$, a Farey symbol for $\Gamma$ is a certain finite sequence of rational numbers (representing vertices of a fundamental domain of $\Gamma$) together with pairing information for the edges between the vertices. They are a compact way of encoding the information about the group and they provide a simple way to do calculations with the group. For example: calculating an independent set of generators and decomposing group elements into a word in these generators, finding coset representatives, elliptic points, and genus of the group, testing if the group is congruence, etc. In this expository article, we will discuss Farey Symbols and explicit algorithms for working with them.