A sequence of algebraic integer relation numbers which converges to 4

Abstract

Let $\alpha \in \mathbb{R}$ and let $$A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\text{and}{B}_{\alpha }=\left[\begin{array}{cc}1& 0\\ \alpha & 1\end{array}\right].$$ The subgroup $G_{\alpha }$ of ${\mathrm{SL}}_2(\mathbb{R})$ 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.