EJMT Abstract

The special linear group SL(2,ℝ), the group of 2 × 2 real matrices with determinant one, is one of the most important and fundamental mathematical objects not only in mathematics but also in physics. In this paper, we propose a three-dimensional model of SL(2,ℝ) in ℝ³, which is realized by embedding SL(2,ℝ) into the unit 3-sphere. In this model, the set of symmetric matrices of SL(2,ℤ) forms a hyperbolic pattern on the unit disk, like the islands floating on the sea named SL(2,ℝ). The structure of this hyperbolic pattern is described in the upper half-plane H. The upper half-plane H also enables us to generate symmetric matrices of SL(2,ℝ) with three circles. Furthermore, the well-known fact H = SL(2,ℝ)/SO(2) is visualized as S¹ fibers of Hopf fibration in the unit 3-sphere. With this three-dimensional model in ℝ³, we can have a concrete image of SL(2,ℝ) and its noncommutative group structure. This kind of visualization might bring great benefits for the readers who have background not only in mathematics, but also in all areas of science.