Silhouettes and generic properties of subgroups of the modular group

We show how to count and randomly generate finitely generated subgroups of the modular group $\mathsf{\text{PSL}}(2,\mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for these subgroups. The combinatorial methods developed to achieve these results bring to light a natural map, 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 $\mathsf{\text{PSL}}(2,\mathbb{Z})$.