The aim of this paper is to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space H 3 extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space H n , (n ≥ 2), and prove that, in this sense, the well known Böröczky-Florian density upper bound for "congruent horoball" packings of H 3 does not remain valid to the fully asymptotic tetrahedra. The density of this locally densest packing is ≈ 0.874994, may be surprisingly larger than the Böröczky-Florian density upper bound ≈ 0.853276 but our local ball arrangement seems not to have extension to the whole hyperbolic space. * Mathematics Subject Classification 2010: 52C17, 52C22, 52B15.

We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in [12] and [13]. The optimal hyperball packings of these tilings are generated by the "inscribed hyperspheres" whose metric data can be calculated by our method -based on the projective interpretation of the hyperbolic geometry -by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H 3 .

W. Thurston classified the eight simply connected three-dimensional maximal homogeneous Riemannian geometries (see Thurston and Levy 1997, Scott 1983). One of these is the S 2 × R geometry which is the direct product of the spherical plane S 2 and the real line R. The complete list of the space groups of S 2 × R is given by Farkas (Beitr Algebra Geom 42: [235][236][237][238][239][240][241][242][243][244][245][246][247][248][249][250] 2001). Farkas and Molnár (Proceedings of the Colloquium on Differential Geometry, Debrecen, Hungary, pp [105][106][107][108][109][110][111][112][113][114][115][116][117][118] 2001) have classified the S 2 × R manifolds by similarity and diffeomorphism. In this paper we investigate the geodesic balls of S 2 × R and compute their volume, define in this space the notion of geodesic ball packing and its density. Moreover, we determine the densest geodesic ball packing for generalized Coxeter space groups of S 2 × R. The density of the densest packing for these space groups is ≈ 0.82445423. Surprisingly, the kissing number of the balls in this packing is only 2 (!!). Molnár (Beitr Algebra Geom 38(2):261-288, 1997) has shown that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere PS 3 (V 4 , V 4 , R). In our work we shall use this projective model of S 2 × R geometry and in this manner the geodesic lines, geodesic spheres can be visualized on the Euclidean screen of the computer. This visualization will show also the arrangement of some geodesic ball packings for the above space groups.

The goal of this paper is to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space H 3 . Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known Böröczky-Florian density upper bound for "congruent horoball" packings of H 3 remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.