On non-freeness of groups generated by two parabolic matrices with rational parameters: limit points and the orbit test
Wonyong Jang, Dongryung Yi
TL;DR
The paper analyzes the non-freeness of groups $G_{\alpha}$ generated by two parabolic $\mathrm{SL}_2$ matrices, focusing on the orbit test and its interaction with modulo homomorphisms. It proves that the converse of the orbit test is false by constructing infinitely many rational counterexamples, and uses a Pell-type framework to generate infinite non-free rational sequences. It also establishes that $3$ is a limit point of non-free rationals, providing explicit parametric families and two concrete sequences converging to $3$, derived via the orbit test and Diophantine techniques. The results enhance understanding of non-free numbers in the real and rational setting and offer tools for analyzing congruence subgroups and stabilizers in associated geometric constructions.
Abstract
For $α\in \mathbb{R}$, let $$G_α:= \left< \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} , \begin{bmatrix} 1 & 0 \\ α& 1 \end{bmatrix} \right> < \mathrm{SL}_2 (\mathbb{R}).$$ K. Kim and the first author established the orbit test, which provides a sufficient condition for $G_α$ 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.