A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature): Let $f$ be a holomorphic fu …

By the modular group I either mean $SL(2,\mathbb{Z})$, or $PSL(2,\mathbb{Z})$. Where can I find examples of these? Another question - is there a good (ideally analytical, but pos …

Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does n …

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\r …

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. Fo …

Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to ide …

This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup $$ \pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix …

I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can e …

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topologic …

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. Th …

This question is something of a follow-up to http://mathoverflow.net/questions/19400/ . How does one recognise whether a subgroup of the modular group $\Gamma=\mathrm{SL}_2(\mathb …