Volumes and geodesic ball packings to the regular prism tilings in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space

After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups $\mathbf{pq2_1}$. $\SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. In this paper we consider geodesic spheres and balls in $\SLR$ (even in $\mathbf{SL_{\mathrm{2}}R})$, if their radii $\rho\in [0, \frac{\pi}{2})$, and determine their volumes. Moreover, we consider the prisms of the above space and compute their volumes, define the notion of the geodesic ball packing and its density. We develop a procedure to determine the densities of the densest geodesic ball packings for the tilings, or in this paper more precisely, for their generating groups $\mathbf{pq2_1}$ (for integer rotational parameters $p,q$; $3\le p, \frac{2p}{p-2} <q$). We look for those parameters $p$ and $q$ above, where the packing density large enough as possible. Now our record is $0.567362$ for $(p, q) = (8, 10)$. These computations seem to be important, since we do not know optimal ball packing, namely in the hyperbolic space $\HYP$. We know only the density upper bound 0.85326, realized by horoball packing of $\HYP$ to its ideal regular simplex tiling. Surprisingly, for the so-called translation ball packings under the same groups $\mathbf{pq2_1}$ in \cite{MSzV13} we have got larger density $0.841700$ for $(p, q) = (5, 10000 \rightarrow \infty)$ close to the above upper bound. We use for the computation and visualization of the $\SLR$ space its projective model introduced by the first author in \cite{M97}.