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

The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a $3$-dimensional space of constant curvature was settled by B"or"oczky and Florian for the hyperbolic space $\text{HYP}$ in \cite{BF64} and by proving the famous Kepler conjecture by Hales \cite{H} for the Euclidean space $\text{EUC}$. 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) $$\text{SXR},\text{HXR},\text{SLR},\text{NIL},\text{SOL},$$ where a transitive symmetry group of the ball packing is assumed, 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 packing with equal balls of the hyperbolic space $\text{HYP}$. However, it attains e.g. at horoball packing of ${\overline{\text{bH}}}^3$ where the ideal centres of horoballs lie on the absolute figure of ${\overline{\text{bH}}}^3$ inducing the regular ideal simplex tiling $(3,3,6)$ by its Coxeter-Schl"afli symbol. In this work we present a geodesic ball packing in the $\text{SXR}$ geometry whose density is $\approx 0.87499429$. The extremal configuration is described in Theorem 2.8, Our conjecture and further remarks are summarized in Section 3.