On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups

TL;DR

This paper introduces Kleinian Sphere Packings and Bugs, extending the concept of crystallographic packings to all dimensions, and proves that superintegral packings originate from Q-arithmetic lattices, addressing key questions from prior work.

Contribution

It generalizes Kleinian sphere packings to higher dimensions, extends the Arithmeticity Theorem, and clarifies the origins of integral packings and their relation to arithmetic lattices.

Findings

Superintegral Kleinian packings exist in all dimensions.

The Arithmeticity Theorem extends to Kleinian packings.

Integral packings originate from non-uniform lattices.

Abstract

We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral" such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions from [KN19]: (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.