Result 253, Group theory

An infinite finitely presented simple amenable group

Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.

Lean formalization Proof

The bigger picture

Why it matters

Can an infinite symmetry system have a finite rulebook while satisfying both simplicity and amenability? The manuscript reports that it can, settling an existence question about which structural constraints on groups can coexist.

What changes?

The manuscript claims to construct one infinite, finitely presented, simple, amenable group. A group is a system of reversible operations with a consistent composition rule. Finitely presented means finitely many generators and defining relations describe it completely. Simple means it has no normal subgroups other than itself and the identity subgroup, so it has no smaller nontrivial quotient group. Amenable means it admits a translation-invariant, finitely additive probability measure on all its subsets. The claim is existence, not classification.

What does that help mathematicians do?

In the reported example, every structure-preserving map to a finite group must be trivial. Simplicity forces such a map either to collapse everything or to be injective, and an infinite group cannot inject into a finite one. Thus finite-group images cannot distinguish its elements, despite its finite presentation. The claimed construction shows that amenability does not prevent this combination.

Are there practical applications?

Its immediate value is foundational: it would establish that a finite rulebook, simplicity, amenability and infinitude are compatible. Researchers could therefore rule out any proposed general obstruction based solely on those four properties. The supplied abstract describes an existence result, not a practical algorithm or deployment; broader uses would require further information about the construction.

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.

Manuscript

An infinite finitely presented simple amenable group

September 23, 2026 58 pages Main result formalized in Lean

We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026,
  author = {{OpenAI}},
  title = {{An infinite finitely presented simple amenable group}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf}{OAI:An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/253.md.

An infinite finitely presented simple amenable group

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

Scope

The formalized result answers the existence question affirmatively: there is one group that is infinite, finitely presented, simple, and amenable. Amenability is expressed by the Følner condition.

The later general claims about central kernels and enlargements across families are not included.

Comparator links

Result Comparator statement
Infinite finitely presented simple amenable group SimpleAmenable.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.