Fibre-like cylinders, their packings and coverings in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space

Abstract

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}.