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

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, $\approx0.80529$, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Moln\'{a}r demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4, \boldsymbol{V}_4, \mathbf{R})$. We use this projective model of $\mathbf{H}^2\!\times\!\mathbf{R}$ to compute and visualize the locally optimal geodesic ball arrangements.