On computing finite index subgroups of PSL(2,Z)

TL;DR

This paper introduces a recursive method to compute finite index subgroups of PSL(2,Z), enabling the enumeration of all subgroups up to index 20 and analysis of their arithmetical properties.

Contribution

It develops a novel recursive approach for computing bivalent trees and their automorphisms, expanding the computational tools for subgroup analysis in PSL(2,Z).

Findings

Computed all subgroups of index up to 20

Generated tables of arithmetical properties

Provided a new database for further research

Abstract

We present a method to compute finite index subgroups of $PSL_2(\mathbb{Z})$. Our strategy follows Kulkarni's ideas, the main contribution being a recursive method to compute bivalent trees and their automorphism group. As a concrete application, we compute all subgroups of index up to 20. We then use this database to produce tables with several arithmetical properties.