Silhouettes and generic properties of subgroups of the modular group

Abstract

We show that the probability for a finitely generated subgroup of the modular group, of size $n$, to be almost malnormal or non-parabolic, tends to 0 as $n$ tends to infinity – where the notion of the size of a subgroup is based on a natural graph-theoretic representation of the subgroup. The proofs of these results rely on the combinatorial and asymptotic study of a natural map, which associates with any finitely generated subgroup of ${\mathsf{PSL}}_2(\mathbb{Z})$ a graph which we call its silhouette, which can be interpreted as a conjugacy class of free finite index subgroups of ${\mathsf{PSL}}_2(\mathbb{Z})$.