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Cofinite integral braces and crystallographic groups

  • Rafał Lutowski
  • Andrzej Szczepański
  • Anna Zamojska-Dzienio
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Abstract

A left brace is cofinite integral (CIB) if its additive group is free abelian of finite rank and its quotient by the socle is finite. Rump introduced such braces and showed that the multiplicative group of every CIB is a crystallographic group. We study CIBs from the point of view of crystallographic groups. We prove an analogue of the Bieberbach theorems: in each dimension there are only finitely many isomorphism classes of CIBs. For a fixed crystallographic group, we show that its CIB structures correspond to the orbits of a finite group action on a set of bijective 1-cocycles. We use this description to develop algorithms that enumerate them. Finally, we give a complete census of CIBs on crystallographic and Bieberbach groups up to dimension 6.

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Cite

BibTeXRIS
BibTeX
@misc{lutowski2026cofinite,
  title = {{Cofinite integral braces and crystallographic groups}},
  author = {Rafał Lutowski and Andrzej Szczepański and Anna Zamojska-Dzienio},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32352},
  eprint = {2610.32352},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32352},
}
RIS
TY  - UNPB
TI  - Cofinite integral braces and crystallographic groups
AU  - Rafał Lutowski
AU  - Andrzej Szczepański
AU  - Anna Zamojska-Dzienio
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32352
UR  - https://arxiv.org/abs/2610.32352
AB  - A left brace is cofinite integral (CIB) if its additive group is free abelian of finite rank and its quotient by the socle is finite. Rump introduced such braces and showed that the multiplicative group of every CIB is a crystallographic group. We study CIBs from the point of view of crystallographic groups. We prove an analogue of the Bieberbach theorems: in each dimension there are only finitely many isomorphism classes of CIBs. For a fixed crystallographic group, we show that its CIB structures correspond to the orbits of a finite group action on a set of bijective 1-cocycles. We use this description to develop algorithms that enumerate them. Finally, we give a complete census of CIBs on crystallographic and Bieberbach groups up to dimension 6.
ER  -