This paper deals with the problem of classifying the set of discrete, torsion-free, genus zero congruence subgroups of \(\text{PSL}_2({\mathbb R})\), up to conjugacy. Such groups are of interest because of their relation to Moonshine, and previously, only finiteness results were known by the work of J. Thompson.
The main result of the paper is a complete classification of this set. The author shows that there are exactly 15 conjugacy classes of such subgroups of PSL\(_2({\mathbb R})\) (theorem 1), and he gives a list for the representatives of each conjugacy class. The proof of the main result is obtained by a series of steps. First, the author shows that every torsion-free genus zero discrete subgroup of \(\text{PSL}_2({\mathbb R})\) commensurable to the modular group is conjugate to a subgroup of \(\text{PSL}_2({\mathbb Z})\) (theorem 2), thereby transferring the problem to one of classifying torsion-free genus zero congruence subgroups of \(\text{PSL}_2({\mathbb Z})\).
Next, he shows that any such subgroup is conjugate to a Larcher congruence subgroup (proposition 6.1), a special class of congruence subgroups introduced by Larcher. It is in the class of Larcher congruence subgroups that the classification is affected, since there are only a fairly small number of possibilities to be considered here. Finally, the author gives an application to modular curves by exhibiting the finite set of values of \(z\) for which the curve \({\mathbb P}^1\setminus \{0,1,\infty, z\}\) is a modular curve (theorem 9.2).