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

TL;DR

This paper investigates the densest local horoball packings within totally asymptotic regular simplices in hyperbolic space, revealing that classical density bounds do not hold in higher dimensions and providing explicit formulas for these packings.

Contribution

It introduces a generalized density concept for horoballs of different types in hyperbolic space and identifies locally optimal packings that surpass classical bounds in dimensions four and higher.

Findings

B"or"oczky density bound does not hold for n≥4.

Explicit formulas for densities of locally optimal packings.

Locally optimal packings cannot be extended to entire hyperbolic 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 B\"or\"oczky density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^n$ does not remain valid for $n\ge4$,} 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.