New Lower Bounds for Optimal Horoball Packing Density in Hyperbolic $n$-space for $6 \leq n \leq 9$

Abstract

Koszul type Coxeter simplex tilings exist in hyperbolic $n$-space $\mathbb{H}^n$ up to $ n = 9$, and their horoball packings achieve the highest known regular ball packing densities for $n = 3, 4, 5$. In this paper we determine the optimal horoball packing densities of Koszul simplex tilings in dimensions $6 \leq n \leq 9$, which give new lower bounds for optimal packing density in each dimension. The symmetries of the packings are given by Coxeter simplex groups, and a parameter related to the Busemann function gives an isometry invariant description of different optimal horoball packing configurations.