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 $\text{PSL}(2,\mathbb{Z})$ a graph which we call its silhouette, which can be interpreted as a conjugacy class of free finite index subgroups of $\text{PSL}(2,\mathbb{Z})$.

31 pages. This is the second and last part of a thorough revision of arXiv:2011.09179. The first part of this revision is arXiv:2310.18923