Kleinian sphere packings, reflection groups, and arithmeticity

Nikolay Bogachev; Alexander Kolpakov; Alex Kontorovich

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arXiv · 2203.01973

Kleinian sphere packings, reflection groups, and arithmeticity

Nikolay Bogachev·Alexander Kolpakov·Alex Kontorovich

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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.

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Nikolay Bogachev, Alexander Kolpakov, Alex Kontorovich. 2022-03-03. Kleinian sphere packings, reflection groups, and arithmeticity. https://arxiv.org/abs/2203.01973

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