Statistics of Subgroups of the Modular Group

We count the finitely generated subgroups of the modular group PSL2(Z). More precisely, each such subgroup H can be represented by its Stallings graph Gamma(H), we consider the number of vertices of 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 PSL2(Z). We give asymptotic equivalents for the number of finitely generated subgroups of PSL2(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 deviation 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 (respectively, finite index subgroup, free subgroup) of PSL2(Z).