Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space

TL;DR

This paper finds the optimal horoball packing densities for specific Coxeter tilings in hyperbolic 3-space, revealing multiple configurations that achieve the universal density bound and linking these to special mathematical functions and hyperbolic volumes.

Contribution

It explicitly determines the optimal horoball packings for Koszul-type Coxeter tilings in hyperbolic 3-space and connects these densities to special functions and geometric invariants.

Findings

Optimal packings attain the universal density bound of approximately 0.853276.

Multiple Coxeter tilings realize the extremal packing densities.

Results relate packing densities to Lobachevsky function and hyperbolic manifold volumes.

Abstract

We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,\Lambda\!\left(\tfrac{\pi}{3}\right) \right)^{-1} \;\approx\; 0.853276, \] where $\Lambda$ denotes the Lobachevsky function. These results show that extremal packing densities in $\mathbb{H}^3$ are realized by multiple explicit Coxeter tilings and are closely tied to special values of $L$-functions and hyperbolic manifold volumes.