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PreprintCC BY 4.0arXiv

Arithmetic Polyhedra

  • Daniel Allcock
  • Pat Devlin
  • Anna Felikson
  • Alex Kontorovich
  • Ian Whitehead
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Abstract

The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three ``seed'' polyhedra.

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Cite

BibTeXRIS
BibTeX
@misc{allcock2026arithmetic,
  title = {{Arithmetic Polyhedra}},
  author = {Daniel Allcock and Pat Devlin and Anna Felikson and Alex Kontorovich and Ian Whitehead},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05349},
  eprint = {2609.05349},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05349},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Arithmetic Polyhedra
AU  - Daniel Allcock
AU  - Pat Devlin
AU  - Anna Felikson
AU  - Alex Kontorovich
AU  - Ian Whitehead
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05349
UR  - https://arxiv.org/abs/2609.05349
AB  - The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three ``seed'' polyhedra.
ER  -