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

We present a method to compute finite index subgroups of $PS{L}_2(\mathbb{Z})$. 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.