We show how to efficiently count and generate uniformly at random finitely generated subgroups of the modular group $\textsf{ PSL}(2,\mathbb{ Z})$ of a given isomorphism type. The method to achieve these results relies on a natural map of independent interest, which associates with any finitely generated subgroup of $\textsf{ PSL}(2,\mathbb{ Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $\textsf{ PSL}(2,\mathbb{ Z})$.
Categories: math.GR math.CO