# Non-freeness of groups generated by two parabolic elements with small rational parameters

## TL;DR

This paper investigates when groups generated by two specific parabolic matrices with rational parameters are non-free, providing computational criteria and density results, supported by computer-assisted proofs and Mathematica code.

## Contribution

It introduces a robust computational criterion for non-freeness of these groups and establishes density and sequence results for various rational parameters.

## Key findings

- Groups are non-free for all rational parameters with numerator |s| ≤ 27, except possibly s=24.
- For s=24, the set of denominators r making the group non-free has natural density 1.
- For fixed s, there exist arbitrarily long sequences of denominators r where the group is non-free.

## Abstract

Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let $G_q<\mathrm{SL}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In this paper, we study the problem of determining when the group $G_q$ is not free for $|q|<4$ rational. We give a robust computational criterion which allows us to prove that if $q=s/r$ for $|s|\leq 27$ then $G_q$ is non-free, with the possible exception of $s=24$. In this latter case, we prove that the set of denominators $r\in\mathbb{N}$ for which $G_{24/r}$ is non-free has natural density $1$. For a general numerator $s>27$, we prove that the lower density of denominators $r\in \mathbb{N}$ for which $G_{s/r}$ is non-free has a lower bound \[ 1- \left(1-\frac{11}{s}\right) \prod_{n=1}^\infty \left(1-\frac{4}{s^{2^n-1}}\right). \] Finally, we show that for a fixed $s$, there are arbitrarily long sequences of consecutive denominators $r$ such that $G_{s/r}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.