Statistics of subgroups of the modular group

We count the finitely generated subgroups of modular group [Formula: see text]. More precisely, each such subgroup text] can be represented by its Stallings graph text], we consider number vertices to size and Since an index has our results generalize known on enumeration finite give asymptotic equivalents for as well subgroups, free subgroups. also expected value isomorphism type a prove large deviation statement concerning this value. Similar are proved Finally, show how efficiently generate uniformly at random (respectively, subgroup, subgroup)