Classification of torsion-free genus zero congruence groups (Q2716092)

The purpose of the article is to classify all torsion-free genus zero congruence subgroups of the modular group \(\text{PSL}_2(\mathbb{Z})\). It had been shown by the author in a previous paper [in Duke Math. J. (to appear)] that a torsion free congruence subgroup of genus zero of \(\text{PSL}_2(\mathbb{R})\) is necessarily conjugate to a congruence subgroup of \(\text{PSL}_2(\mathbb{Z})\). The classification in \(\text{PSL}_2(\mathbb{Z})\) would thus lead to a classification in \(\text{PSL}_2(\mathbb{R})\). The two main technical results in this paper are firstly, that any torsion free genus zero congruence subgroup of \(\text{PSL}_2(\mathbb{Z})\) is conjugate by an element of \(\text{PSL}_2(\mathbb{Z})\) to a Larcher congruence subgroup (Proposition 4.1) and secondly, that the possible list of torsion-free genus zero Larcher congruence subgroups is finite (Proposition 6.1) both of which are fairly straightforward. From these, a careful study of the possible candidates provides the main result that there are 33 conjugacy classes in \(\text{PSL}_2(\mathbb{Z})\) which are partitioned into 15 \(\text{PSL}_2(\mathbb{R})\) conjugacy classes. A table of the classification is also given.