Kleinian sphere packings, reflection groups, and arithmeticity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929311429165056 |
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| author | Bogachev, Nikolay Kolpakov, Alexander Kontorovich, Alex |
| author_facet | Bogachev, Nikolay Kolpakov, Alexander Kontorovich, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01973 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Kleinian sphere packings, reflection groups, and arithmeticity Bogachev, Nikolay Kolpakov, Alexander Kontorovich, Alex Geometric Topology Combinatorics Group Theory 52C17, 20F55, 22E40 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. |
| title | Kleinian sphere packings, reflection groups, and arithmeticity |
| topic | Geometric Topology Combinatorics Group Theory 52C17, 20F55, 22E40 |
| url | https://arxiv.org/abs/2203.01973 |