Silhouettes and generic properties of subgroups of the modular group

TL;DR

This paper introduces combinatorial methods to count and generate subgroups of the modular group, revealing that certain properties are rare and defining a new graphical representation called the silhouette.

Contribution

It develops a novel approach to classify and analyze subgroups of the modular group using silhouette graphs, including counting, generation, and property analysis.

Findings

Almost malnormality and non-parabolicity are negligible properties.

Introduces the silhouette graph as a natural representation of subgroups.

Provides methods for counting and randomly generating subgroups of a given isomorphism type.

Abstract

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