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

Wonyong Jang, Dongryung Yi

For $\alpha \in \mathbb{R}$, let $$G_{\alpha }:=⟨\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ \alpha & 1\end{array}\right]⟩<{\mathrm{SL}}_2(\mathbb{R}).$$ K. Kim…