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Congruence subgroups of groups commensurable with $\text{PSL}\left(2,ℤ\right)$ of genus 0 and 1. (English) Zbl 1099.11022

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

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].

##### MSC:
 11F06 Structure of modular groups and generalizations 11F03 Modular and automorphic functions 30F35 Fuchsian groups and automorphic functions 11F22 Relationship of automorphic forms to Lie algebras, etc. 20H05 Unimodular groups, congruence subgroups (matrix groups)
##### Keywords:
modular group; congruence subgroups; genus