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

Abstract

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 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 $$\mathbf {pq2_1}$$ pq 2 1 we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in $$\textbf{H}^3$$ H 3 and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × 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 $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).