On non-freeness of groups generated by two parabolic matrices with rational parameters: limit points and the orbit test

TL;DR

This paper explores the non-freeness of groups generated by two parabolic matrices with rational parameters, demonstrating counterexamples to the orbit test's converse and constructing sequences of rational numbers approaching 3.

Contribution

It provides explicit counterexamples to the converse of the orbit test and constructs rational sequences converging to 3 using the orbit test and a modified Pell's equation.

Findings

Counterexamples to the orbit test's converse are constructed.

Sequences of rational numbers approaching 3 are explicitly built.

The orbit test's limitations are demonstrated through these applications.

Abstract

For $\alpha \in \mathbb{R}$, let $$G_{\alpha}:= \left< \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ \alpha & 1 \end{bmatrix} \right> < \mathrm{SL}_2 (\mathbb{R}).$$ K. Kim and the first author established the orbit test, which provides a sufficient condition for $G_{\alpha}$ not to be a rank-$2$ free group. In this article, we present two main applications of the orbit test. First, using the corresponding modulo homomorphism, we show that the converse of the orbit test does not hold. In particular, we construct explicit counterexamples, all of which are rational. As another application, we construct sequences of non-free rational numbers converging to $3$. These sequences are given by $$ 3 + \frac{3}{2 (9 n - 1)} \quad \text{and} \quad 3 + \frac{9 n + 5}{3 (2 n + 1) (9 n + 4)},$$ and their construction relies on the orbit test together with a modified Pell's equation.