Random Generation of Thompson's group $F$

TL;DR

This paper demonstrates that under certain probabilistic models, the likelihood of randomly generating the entire Thompson's group F or specific subgroups is positive, revealing new probabilistic properties of F.

Contribution

It establishes the positivity of probabilities for random generation of Thompson's group F and its subgroups under natural probabilistic models, a novel insight in group theory.

Findings

Positive probability of generating F with two elements

Positive probability of forming specific subgroups with additional generators

New probabilistic properties of Thompson's group F

Abstract

We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability that a random pair of elements of Thompson's group $F$ generate the entire group is positive. We also prove that for any $k$-generated subgroup $H$ of $F$ which contains a "natural" copy of $F$, the probability of a random $(k+2)$-generated subgroup of $F$ coinciding with $H$ is positive.