New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in ${\mathbf{H}}^2\times \mathbf{R}$ space

Abstract

In this paper, we present a new record for the densest geodesic congruent ball packing configurations in ${\mathbf{H}}^2\times \mathbf{R}$ geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on ${\mathbf{H}}^2$ and some translation components on the real fibre direction $\mathbf{R}$ that can be determined by the corresponding Frobenius congruences. Moreover, we developed a procedure to determine the optimal radius for the densest geodesic ball packing configurations related to the considered screw motion groups. The highest packing density, $\approx 0.80529$, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Molnár demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere ${\mathcal{P}\mathcal{S}}^3({\mathbf{V}}^4,{\mathbf{V}}_4,\mathbf{R})$. We use this projective model of ${\mathbf{H}}^2\times \mathbf{R}$ to compute and visualize the locally optimal geodesic ball arrangements.

27 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2311.12260