A Three-Dimensional Model of $SL(2, \mathbb{R})$ and the Hyperbolic Pattern of $SL(2, \mathbb{Z})$ (Research Journal of Mathematics & Technology, 2022)
The special linear group $SL(2,\mathbb{R})$, the group of $2 × 2$ real matrices with determinant one,is one of the most important and fundamental mathematical objects not only in mathematics butalso in physics. In this paper, we propose a three-dimensional model of $SL(2, \mathbb{R})$ in $\mathbb{R}^3,$ whichis realized by embedding $SL(2,\mathbb{R})$ into the unit $3$-sphere. In this model, the set of symmetricmatrices of $SL(2,\mathbb{Z})$ forms a hyperbolic pattern on the unit disk, like the islands floating on thesea named $SL(2,\mathbb{R}).$ 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 $SL(2,\mathbb{R})$ withthree circles. Furthermore, the well-known fact $H = SL(2, \mathbb{R})/SO(2)$ is visualized as $S^1$ fibersof Hopf fibration in the unit $3$-sphere. With this three-dimensional model in $\mathbb{R}^3,$ we can have aconcrete image of $SL(2,\mathbb{R})$ and its noncommutative group structure. This kind of visualizationmight bring great benefits for the readers who have background not only in mathematics, but alsoin all areas of science.