Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) finitely presented? If so, is there a proof of this? Assuming they are finitely presented, are the presentations of e.g. the principal congruence subgroups $\Gamma\left(n\right)$ (for small $n$) documented anywhere? I know they could be computed using the Reidemeister-Schreier process, but it would be nice to have some independent confirmation of the presentations.
Many thanks!
Best Answer
A subgroup of finite index in a finitely presented group is finitely presented (see Exercise 6.1.6 in Robinson: A course in the theory of groups), so all congruence subgroups of the modular group are finitely presented. I cannot answer your second question.