Random generation of subgroups of the modular group with a fixed isomorphism type

We show how to efficiently count and generate uniformly at random finitely generated subgroups of the modular group $\mathsf{\text{PSL}}(2,\mathbb{Z})$ 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 $\mathsf{\text{PSL}}(2,\mathbb{Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $\mathsf{\text{PSL}}(2,\mathbb{Z})$.