Fibre-Like Cylinders, Their Packings and Coverings in \(\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}\) Space

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in \(\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}\) $\widetilde{\mathbf{S}{\mathbf{L}}_2\mathbf{R}}$ 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 \(\mathbf {pq2_1}\) $\mathbf{pq}{\mathbf{2}}_{\mathbf{1}}$ we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in \(\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}\) $\widetilde{\mathbf{S}{\mathbf{L}}_2\mathbf{R}}$ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in \(\textbf{H}^3\) ${\mathbf{H}}^3$ and \(\textbf{H}^2\!\times \!\textbf{R}\) ${\mathbf{H}}^2\times \mathbf{R}$ spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of \(\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}\) $\widetilde{\mathbf{S}{\mathbf{L}}_2\mathbf{R}}$ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).