Random generation of finitely generated subgroups of a free group

TL;DR

This paper presents an efficient algorithm for uniformly randomly generating finitely generated subgroups of a free group of a specified size, along with estimates of subgroup counts and properties.

Contribution

It introduces a novel algorithm for uniform random generation of subgroups and provides statistical estimates of subgroup characteristics.

Findings

Algorithm runs in average O(n) time in RAM model

Provides estimates for number and properties of subgroups

Analyzes the proportion of finite index subgroups

Abstract

We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $\O(n)$ in the RAM model and $\O(n^2\log^2n)$ in the bitcost model.