A candidate for the densest packing with equal balls in Thurston geometries

Abstract

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) \(\mathbf S ^2\!\times \!\mathbf R , \mathbf H ^2\!\times \!\mathbf R , \) \(\widetilde{\mathbf{S \mathbf L _2\mathbf R }}, \mathbf {Nil} , \mathbf {Sol} . \) 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.