Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space

Abstract

In Szirmai (Acta Mathematica Hungarica 136/1-2:39–55, 2012) we generalized the notion of simplicial density function for horoballs in the extended hyperbolic space ${\overline{\mathbf{H}}^3}$, where we allowed horoballs in different types centered at various vertices of a totally asymptotic tetrahedron. By this new aspect, in this paper we study the locally densest horoball packing arrangements and their densities with respect to totally asymptotic regular simplices in the hyperbolic n-space ${\overline{\mathbf{H}}^n}$ extended with its absolute figure, where the ideal centers of horoballs lie in the vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, the well known Böröczky–Florian density upper bound for “congruent ball and horoball” packings of ${\overline{\mathbf{H}}^3}$ does not remain valid for the analogous packing of ${\overline{\mathbf{H}}^n}$, for n ≥ 4. Although locally optimal ball arrangements do not seem to have extensions to the whole n-dimensional hyperbolic space. Moreover, we determine an explicit formula for the density of the above locally optimal horoball packings, allowing horoballs of different types (i.e. they are differently packed in their ideal simplex).