A sequence of algebraic integer relation numbers which converges to 4

Abstract

Let α∈R and letA=[1101]andBα=[10α1]. The subgroup Gα of SL2(R) is a group generated by the matrices A and Bα. In this paper, we investigate the property of the group Gα. We construct a generalization of the Farey graph for the subgroup Gα. This graph determines whether the group Gα is a free group of rank 2. More precisely, the group Gα 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 Gα 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.