Silhouettes and generic properties of subgroups of the modular group

2020/11/18 by Bassino, Frédérique, Nicaud, Cyril, Weil, Pascal
#05A16 #05E15 #20E07 #20F69 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2011.09179

Abstract

We show how to count and randomly generate finitely generated subgroups of the modular group \textsfPSL(2,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 \textsfPSL(2,ℤ) a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of \textsfPSL(2,ℤ).

Related

• Silhouettes and generic properties of subgroups of the modular group
• Combinatorics of double cosets and fundamental domains for the subgroups of the modular group
• Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison
• Maximal subgroups of the modular and other groups
• Cone types and asymptotic invariants for the random walk on the modular group