A sequence of algebraic integer relation numbers which converges to 4

TL;DR

This paper explores the properties of certain subgroups of SL(2,R) generated by specific matrices, constructs a generalized Farey graph to determine their freeness, and creates a sequence of algebraic integers converging to 4 with particular group properties.

Contribution

It introduces a generalized Farey graph for subgroups G_α, linking graph structure to group freeness, and constructs a sequence of algebraic integers approaching 4 with non-free group properties.

Findings

G_α is free iff the associated graph is a tree.

If 1/2 is a vertex, G_α is not free.

Constructed sequence of algebraic integers converging to 4 with non-free groups.

Abstract

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 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.