Kleinian sphere packings, reflection groups, and arithmeticity

TL;DR

This paper advances the understanding of sphere packings by establishing the existence of crystallographic packings in higher dimensions, introducing a geometric doubling method, and analyzing integral properties using Lorentzian forms.

Contribution

It solves the existence problem for crystallographic sphere packings in certain higher dimensions and introduces a new geometric doubling technique for constructing packings.

Findings

Existence of crystallographic sphere packings in certain higher dimensions.

A geometric doubling procedure for generating sphere packings.

Analysis of properly integral packings using Lorentzian quadratic forms.

Abstract

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study "properly integral" packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher-dimensional hyperbolic polyhedra.