Let $\alpha \in \mathbb{R}$ and let$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\text{and}{B}_{\alpha }=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \alpha \hfill & \hfill 1\hfill \end{array}\right].$ The subgroup $G_{\alpha }$ of $\mathrm{S}{\mathrm{L}}_{\mathrm{2}}\mathrm{(}\mathbb{R}\mathrm{)}$ is a group generated by the matrices A and $B_{\alpha }$. In this paper, we investigate the property of the group $G_{\alpha }$. We construct a generalization of the Farey graph for the subgroup $G_{\alpha }$. This graph determines whether the group $G_{\alpha }$ is a free group of rank 2. More precisely, the group $G_{\alpha }$ is a free group of rank 2 if and only if the graph is tree. In particular, we show that if 1/2 is a vertex of the graph, then $G_{\alpha }$ is not a free group of rank 2. Using this, we construct a sequence of real numbers so that the sequence converges to 4 and each number has the corresponding group that is not a free group of rank 2. It turns out that the real numbers are algebraic integers.

让 $\alpha \in \mathbb{[R}$ 然后让$\mathrm{一种}=\left[\begin{array}{cc}\hfill \mathrm{1个}\hfill & \hfill \mathrm{1个}\hfill \\ \hfill 0\hfill & \hfill \mathrm{1个}\hfill \end{array}\right]\text{和}{乙}_{\alpha }=\left[\begin{array}{cc}\hfill \mathrm{1个}\hfill & \hfill 0\hfill \\ \hfill \alpha \hfill & \hfill \mathrm{1个}\hfill \end{array}\right]。$ 小组 $G_{\alpha }$ 的 $\mathrm{小号}{\mathrm{大号}}_{\mathrm{2个}}\mathrm{（}\mathbb{[R}\mathrm{）}$是由矩阵A和${乙}_{\alpha }$。在本文中，我们调查了该组的属性$G_{\alpha }$。我们为子组构造Farey图的推广$G_{\alpha }$。此图确定该组是否$G_{\alpha }$ 是排名2的自由组。更确切地说，该组 $G_{\alpha }$是且仅当图形为树时，才是第2级的自由组。特别地，我们表明如果1/2是图的顶点，则$G_{\alpha }$ 不是等级2的自由组。使用此方法，我们构造了一个实数序列，以使该序列收敛到4，并且每个数字都有对应的组，该组不是等级2的自由组。结果证明，实数是代数整数。