  In this paper we define the notion of infinite or bounded fibre-like geodesic
cylinder in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, develop a
method to determine its volume and total surface area. We prove that the common
part of the above congruent fibre-like cylinders with the base plane are
Euclidean circles and determine their radii.
  Using the former classified infinite or bounded congruent regular prism
tilings with generating groups $\mathbf{pq2_1}$ we introduce the notions of
cylinder packings, coverings and their densities. Moreover, we determine the
densest packing, the thinnest covering cylinder arrangements in
$\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, their densities, their
connections with the extremal hyperbolic circle arrangements and with the
extremal fibre-like cylinder arrangements in $\mathbf{H}X\mathbf{R}$ space
  In our work we use the projective model of
$\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ introduced by E. Moln\'ar in
\cite{M97}.
