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A candidate for the densest packing with equal balls in Thurston geometries. (English) Zbl 1301.52035

Summary: The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a \(3\)-dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space \(\mathbf H ^3\), and, with the proof of the famous Kepler conjecture, by Hales for the Euclidean space \(\mathbf E ^3\). The goal of this paper is to extend the problem of finding the densest geodesic ball (or sphere) packing for the other \(3\)-dimensional homogeneous geometries (Thurston geometries) \(\mathbf S ^2\times\mathbf R\), \(\mathbf H ^2\times\mathbf R\), \(\widetilde{\mathbf{SL}_2\mathbf R}\), \(\mathbf{Nil}\), \(\mathbf{Sol}\). In the following a transitive symmetry group of the ball packing is assumed, which is one of the discrete isometry groups of the considered space. Moreover, we describe a candidate of the densest geodesic ball packing. The greatest density until now is \(\approx 0.85327613\) that is not realized by a packing with equal balls of the hyperbolic space \(\mathbf H^3\). However, that is attained, e.g., by a horoball packing of \(\overline{\mathbf H}^3\) where the ideal centres of horoballs lie on the absolute figure of \(\overline{\mathbf H}^3\) inducing the regular ideal simplex tiling \((3,3,6)\) by its Coxeter-Schläfli symbol. In this work we present a geodesic ball packing in the \(\mathbf S^2\times\mathbf R\) geometry whose density is \(\approx 0.87757183\). The extremal configuration is described in Theorem 2.6. A conjecture for the densest ball packing in Thurston geometries and further remarks are summarized in Sect. 1.1, 1.2 and 2.3.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
53A35 Non-Euclidean differential geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces

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