2023/11/14 by Frédérique Bassino, Bassino, Frédérique, Cyril Nicaud +3
Mathematics · Computer Science · #Advanced Topology and Set Theory #Topological and Geometric Data Analysis #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2311.08021
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 \textsfPSL(2,ℤ) a graph which we call its silhouette, which can be interpreted as a conjugacy class of free finite index subgroups of \textsfPSL(2,ℤ).