Kleinian sphere packings, reflection groups, and arithmeticity

Bogachev, Nikolay; Kolpakov, Alexander; Kontorovich, Alex

doi:10.1090/mcom/3858

Kleinian sphere packings, reflection groups, and arithmeticity
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by Nikolay Bogachev, Alexander Kolpakov and Alex Kontorovich

Math. Comp. 93 (2024), 505-521

DOI: https://doi.org/10.1090/mcom/3858

Published electronically: July 26, 2023

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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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Bibliographic Information

Nikolay Bogachev
Affiliation: The Kharkevich Institute for Information Transmission Problems, Moscow, Russia; and Moscow Institute of Physics and Technology, Dolgoprudny, Russia
MR Author ID: 1210571
ORCID: 0000-0002-9289-5192
Email: nvbogach@mail.ru
Alexander Kolpakov
Affiliation: Institut de Mathématiques, Université de Neuchâtel, CH–2000 Neuchâtel, Switzerland
MR Author ID: 774696
ORCID: 0000-0002-6764-8894
Email: kolpakov.alexander@gmail.com
Alex Kontorovich
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854
MR Author ID: 704943
ORCID: 0000-0001-7626-8319
Email: alex.kontorovich@rutgers.edu
Received by editor(s): April 9, 2022
Received by editor(s) in revised form: August 8, 2022, and February 2, 2023
Published electronically: July 26, 2023
Additional Notes: The first author was partially supported by the Russian Science Foundation, grant no. 22-41-02028. The second author was partially supported by the Swiss National Science Foundation, project no. PP00P2-202667. The third author was partially supported by NSF grant DMS-1802119, BSF grant 2020119, and the 2020-2021 Distinguished Visiting Professorship at the National Museum of Mathematics.
Journal: Math. Comp. 93 (2024), 505-521
DOI: https://doi.org/10.1090/mcom/3858
MathSciNet review: 4654631