Result 248, Group theory

Thompson's group F is nonamenable

Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.

Lean formalization Proof

The bigger picture

Why it matters

Can an infinite group admit a uniform average that respects its symmetries? The manuscript claims the answer is no for Thompson's group F, settling a longstanding question about this particular group of interval transformations.

What changes?

Thompson's group F consists of increasing, continuous rearrangements of the unit interval that are linear on finitely many pieces. Their breakpoints are fractions with power-of-two denominators, and their slopes are powers of two; the group operation is composition. Amenability means that all subsets can receive probabilities in a way that respects finite addition and is unchanged by multiplying by a group element. The manuscript reports that F is nonamenable, claiming to confirm Geoghegan's conjecture.

What does that help mathematicians do?

A concrete consequence of the claimed result is that no probability assignment to all subsets of F can simultaneously give F probability one, add probabilities of two disjoint sets, and remain unchanged under translation. Researchers therefore could rule out arguments that require such an invariant probability assignment on F, rather than continuing to search for one.

Are there practical applications?

The immediate value is foundational: the claim would determine whether this explicitly defined group supports symmetry-preserving averaging, a basic structural question in group theory. The supplied abstract identifies resolution of the amenability problem as its consequence. It does not describe an algorithm, quantitative bounds, or a practical application beyond mathematics.

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

Thompson's group F is nonamenable

September 23, 2026 13 pages Main result formalized in Lean

We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Thompsons-group-F-is-nonamenable-September-23-2026,
  author = {{OpenAI}},
  title = {{Thompson's group $F$ is nonamenable}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf}{OAI:Thompsons-group-F-is-nonamenable-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/248.md.

Thompson's group F is nonamenable

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

Scope

The amenability problem for Thompson's group FF asks whether it admits a positive normalized left-invariant mean on bounded real functions. The formalized result rules out such a mean for the standard group of dyadic piecewise-linear homeomorphisms of the interval, proving nonamenability. No explicit boundary constant or prescribed generating set is given.

Comparator links

Result Comparator statement
Nonamenability of Thompson's group FF ThompsonNonamenability.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.