Random subgroups of Thompson's group $F$

TL;DR

This paper studies the properties and distribution of random subgroups of Thompson's group F, revealing new phenomena such as persistent subgroups and the absence of a generic isomorphism class, with detailed combinatorial and asymptotic analysis.

Contribution

It introduces the concept of persistent subgroups in Thompson's group F and analyzes their density, providing the first examples of such subgroups for all large k, and explores the subgroup distribution under different stratifications.

Findings

Positive density of certain subgroup isomorphism classes varies with stratification

First examples of persistent subgroups in Thompson's group F for large k

Number of reduced pairs of trees is D-finite and not algebraic

Abstract

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results.