## Fake Congruence Subgroups and the Hurwitz Monodromy Group

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 3, Page 559--574.

Berger, Gabriel
Fake Congruence Subgroups and the Hurwitz Monodromy Group
Suppose $G$ is a finite group, embedded as a transitive subgroup of $S_n$ for some $n$. Suppose in addition that $(\mathcal{C}_1, \dots ,\mathcal{C}_4)$ is a quadruple of conjugacy classes of $G$. In earlier papers ([F], [D-F], [B-F]), it was shown that to these data one can canonically associate a finite index subgroup of $PSL_2(\mathbb{Z})$. For example, when $N$ is an odd integer, $G$ is the dihedral group $D_N$ and the conjugacy classes all consist of involutions, the associated subgroup is $Î_0(N).$ In this paper we investigate the case in which $G$ is the semidirect product of the abelian group $\mathbb{Z}[Î¶_d]/\mathcal{N}$ (where $Î¶_d$ is a primitive $d$'th root of unity and $\mathcal{N}$ is an ideal of $\mathbb{Z}[Î¶_d]$ relatively prime to $d$) and the cyclic group $\langle Î¶_d \rangle$. We relate the corresponding subgroup of $PSL_2(\mathbb{Z})$ to the "fake congruence subgroups" described in \cite{B2}. Specifically, if we let $\mathcal{C}$ denote the conjugacy class of $Î¶_d$ in the multiplicative subgroup $\langle Î¶_d \rangle$ and choose our conjugacy classes to be $(\mathcal{C}, \mathcal{C}, \mathcal{C}, \mathcal{C}^{-3}) ,$ then the subgroup is in fact $Î_0(\mathcal{N})$ (defined originally in \cite{B2}; see section 2).