A sequence of algebraic integer relation numbers which converges to 4

摘要

Let alpha is an element of R and letA =[GRAPHICS]and B-alpha =[GRAPHICS].The subgroup G(alpha), of SL2(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. (C) 2021 Elsevier B.V. All rights reserved.