Fibre-Like Cylinders, Their Packings and Coverings in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} Space

Abstract

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} 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 pq21\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf {pq2_1}$$\\end{document} we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in H3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^3$$\\end{document} and H2×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^2\\!\ imes \\!\ extbf{R}$$\\end{document} 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 SL2R~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\ extbf{S}\ extbf{L}_2\ extbf{R}}$$\\end{document} introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).