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PreprintCC BY 4.0arXiv

Every countable group embeds in a group of type FPn

  • Laurent Bartholdi
  • Roman Mikhailov
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Abstract

For every integer n≥2, we prove that every countable group embeds in a group of type FPn. We also construct a group of type Fn containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type Fn if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov. An appendix combines the construction with the controlled-filling argument in a subsequently released OpenAI preprint to prove that every countable group embeds in a two-generator group of type FP∞.

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Cite

BibTeXRIS
BibTeX
@misc{bartholdi2026every,
  title = {{Every countable group embeds in a group of type $\mathrmFP_n$}},
  author = {Laurent Bartholdi and Roman Mikhailov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.37562},
  eprint = {2609.37562},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37562},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Every countable group embeds in a group of type FP_n
AU  - Laurent Bartholdi
AU  - Roman Mikhailov
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.37562
UR  - https://arxiv.org/abs/2609.37562
AB  - For every integer n≥2, we prove that every countable group embeds in a group of type FP_n. We also construct a group of type F_n containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type F_n if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov. An appendix combines the construction with the controlled-filling argument in a subsequently released OpenAI preprint to prove that every countable group embeds in a two-generator group of type FP_∞.
ER  -