  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}.
