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Let $\alpha \in \mathbb{R}$ and let $$A=\begin{bmatrix} 1 & 1 \\ 0 &1\end{bmatrix} \ \text{and} \ B_{\alpha} = \begin{bmatrix} 1 & 0 \\ \alpha &1\end{bmatrix}.$$ The subgroup $G_\alpha$ of $\mathrm{SL}_2(\mathbb{R})$ is agroup generated by the matrices $A$ and $B_\alpha$. In this paper, weinvestigate the property of the group $G_\alpha.$ We construct a generalizationof the Farey graph for the subgroup $G_\alpha.$ This graph determines whetherthe 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. Inparticular, we show that if $1/2$ is a vertex of the graph, then $G_\alpha$ isnot a free group of rank $2$. Using this, we construct a sequence of realnumbers so that the sequence converges to $4$ and each number has thecorresponding group that is not a free group of rank $2$. It turns out that thereal numbers are algebraic integers.

A sequence of algebraic integer relation numbers which converges to 4