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A universal group of type F∞
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Abstract
We construct a single group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F∞ form of the higher-dimensional Higman embedding question.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Laurent Bartholdi and Roman Mikhailov, Every countable group embeds in a group of type $\mathrm{FP}_n$ (arXiv:2609.37562v2, 8 October 2026; first version 29 September 2026). Appendix A adapts this paper's controlled additivity, uniform fillings and product-coefficient arguments to obtain two-generator $FP_\infty$ overgroups of arbitrary countable groups, with ChatGPT Astra assistance disclosed in the arXiv record.
Formalization
- OpenAI math release, Lean scope notes for family 250 (lean/docs/250.md) (OpenAI math release, 2026-10-06). The release's scope notes state that the Lean development proves Theorem 1.1 (one group of type F∞ containing every finitely presented group) but not the converse in Corollary 1.2 about recursively presented subgroups; the comparator statement is
lean/ComparatorChallenges/UniversalFInfinity.lean. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Fan Wu, Xiaolei Wu, Mengfei Zhao, Zixiang Zhou, Embedding groups into boundedly acyclic groups (arXiv:2407.07703, to appear in J. London Math. Soc.). Proves that every group of type F_n embeds in a boundedly acyclic group of type F_n, improving a theorem of Fournier-Facio–Löh–Moraschini; the introduction mentions the Fournier-Facio–Löh–Moraschini strategy only through [FFZ2026].
Digests and expositions
- AI Proved What?, Decidable means simple inside? Three AI preprints claim the Boone–Higman conjecture (episode 250, October 2026). This AI-assisted video and transcript distinguish the three claimed results and outline the affine-action and higher-finiteness constructions. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 250
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-universal-group-of-type-F-infinity-September-23-2026,
author = {{OpenAI}},
title = {{A universal group of type $F_\infty$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf}{OAI:A-universal-group-of-type-F-infinity-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/250.md. The Lean statements have not been checked by people against the paper's claims.
Family 250: Boone–Higman embeddings with higher finiteness
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group.
Other papers in this family:
- Finite algebraic envelopes and the Boone–Higman conjecture · September 23, 2026
- Simple F∞ overgroups of groups with decidable word problem · September 23, 2026