Statistics of subgroups of the modular group

We count the finitely generated subgroups of the modular group $\mathsf{\text{PSL}}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $\mathrm{\Gamma }(H)$, we consider the number of vertices of $\mathrm{\Gamma }(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $\mathsf{\text{PSL}}(2,\mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $\mathsf{\text{PSL}}(2,\mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviations statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (resp. finite index subgroup, free subgroup) of $\mathsf{\text{PSL}}(2,\mathbb{Z})$.