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

Abstract

In \cite{Sz11} we have generalized the notion of the simplicial density function for horoballs in the extended hyperbolic space ${\bar{\mathbf{H}}}^n,(n\ge 2)$, where we have allowed {\it congruent horoballs in different types} centered at the 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 tetrahedra in hyperbolic $n$-space ${\bar{\mathbf{H}}}^n$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, {\it the well known Böröczky density upper bound for "congruent horoball" packings of ${\bar{\mathbf{H}}}^n$ does not remain valid for $n\ge 4$,} but these locally optimal ball arrangements do not 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 in different types.

arXiv comment14 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1105.4315