Congruence Subgroups Associated to the Monster

Abstract

Let {\small $\Delta = \{ G: g(G) =0, \Gamma_0(m) \le G \le N(\Gamma_0(m))$ $\mbox{ for some }m\},$} where {\small $N(\Gamma_0(m))$} is the normaliser of {\small $\Gamma_0(m)$} in {\small $PSL_2(\Bbb R)$} and {\small $g(G)$} is the genus of {\small $\Bbb H^*/G$}. In this article, we determine all the {\small $m$}. Further, for each {\small $m$}, we list all the intermediate groups {\small $G$} of {\small $\Gamma_0(m) \le N(\Gamma_0(m))$} such that {\small $ g(G) =0$}. All the intermediate groups of width 1 at {\small $\infty$} are also listed in a separate table (see www.math.nus.edu.sg/$\sim$matlml/).