Result 250, Group theory

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.

Lean formalization Proof

The bigger picture

Why it matters

Embedding a group inside another preserves its algebra while placing it in a larger setting. These manuscripts link the possibility of embedding in a finitely described simple group to whether equality of words can be decided algorithmically.

What changes?

The manuscripts report that a finitely generated group has decidable word problem, meaning an algorithm can test whether any word in its generators represents the identity, exactly when it embeds in a finitely presented simple group. Finite presentation means finitely many generators and defining relations; simplicity excludes nontrivial proper normal subgroups. For every such decidable group, the target can have type F-infinity: a classifying space with finitely many cells in each dimension.

What does that help mathematicians do?

The equivalence would rule out embedding any finitely generated group with undecidable word problem in a finitely presented simple group. Separately, another manuscript reports a single group of type F-infinity containing every finitely presented group. Its finitely generated subgroups are exactly, up to isomorphism, the recursively presented groups, whose defining relations can be enumerated algorithmically. This identifies precisely which finitely generated groups can occur inside that universal group; simplicity is not claimed for it.

Are there practical applications?

The immediate value is foundational: the claims connect algorithmic questions about group elements with algebraic embeddings and topological finiteness. Type F-infinity supplies a finite number of cells in each dimension, not necessarily finitely many overall. This strengthens the available setting for studying embedded groups, without asserting an efficient word-problem algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Finite algebraic envelopes and the Boone–Higman conjecture

September 23, 2026 35 pages

We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{Finite algebraic envelopes and the Boone--Higman conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf}{OAI:Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026}},
  year = {2026}
}

Simple F∞ overgroups of groups with decidable word problem

September 23, 2026 30 pages

Every finitely generated group with decidable word problem embeds in a simple group of type F∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026,
  author = {{OpenAI}},
  title = {{Simple $F_\infty$ overgroups of groups with decidable word problem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf}{OAI:Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026}},
  year = {2026}
}

A universal group of type F∞

September 23, 2026 18 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/250.md.

Boone–Higman embeddings with higher finiteness

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Boone–Higman conjecture characterizes finitely generated groups with decidable word problem by embeddings into finitely presented simple groups. The formalization proves the equivalence in full: a finitely generated group has a computable word problem for a finite generating set exactly when it embeds by an injective homomorphism into a finitely presented simple group.

The formalization proves the higher-finiteness strengthening of the Boone–Higman conjecture: every finitely generated group with decidable word problem embeds by an injective homomorphism into a nontrivial simple group of type F∞F_\infty. Here type F∞F_\infty means that the group has a classifying CW complex with finitely many cells in each dimension. No additional finiteness property of the original group is assumed.

The formalized result constructs one group HH of type F∞F_\infty containing every finitely presented group. Here type F∞F_\infty means that HH has a classifying space with finitely many cells in each dimension.

The group HH is fixed before the groups embedded into it are chosen. No word-problem assumption is imposed, and the classifying space need not be finite-dimensional or have finitely many cells in total. The paper's converse about recursively presented subgroups is not included.

Comparator links

Result Comparator statement
Boone–Higman equivalence BooneHigman.lean
Simple type F∞F_\infty overgroups of groups with decidable word problem SimpleOvergroups.lean
Universal group of type F∞F_\infty UniversalFInfinity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.