*(English)*Zbl 1099.11022

Thompson has proved that, up to conjugation, there are only finitely subgroups of ${\Lambda}={\text{PSL}}_{2}\left(\mathbb{R}\right)$ containing a (non-trivial) principal congruence subgroup of the modular group ${\text{PSL}}_{2}\left(\mathbb{Z}\right)$ of a fixed genus. (The *genus* of a (discrete) subgroup $G$ of ${\Lambda}$ is the genus of the Riemann surface ${\mathbb{H}}^{2}/G$, where ${\mathbb{H}}^{2}$ is hyperbolic 2-space.)

This paper provides a detailed version of Thompson’s theorem for the two lowest genus values. They compile a complete list of all congruence subgroups of ${\Lambda}$ of genus 0 and 1. The proofs involve computations using

The results answer a question of Conway and Norton by listing all the so-called “moonshine groups”, i.e. the genus zero subgroups $G$ of ${\Lambda}$ which contain (i) a subgroup of the type ${{\Gamma}}_{0}\left(N\right)$ and (ii) translations *only* defined by rational integers.

The authors discuss the overlap between their results and those of *K. S. Chua* and *M. L. Lang* in “Congruence subgroups associated to the Monster” [Exp. Math. 13, No. 3, 343–360 (2004), see Zbl 1099.11018 above].