Math-2429 · mathematics · Group theory

Amenability of Thompson Group and Interval Homeomorphism Groups

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.

$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.

Statement recorded in the supplied source.

Is Thompson's group $F$ amenable? Is the group $\mathrm{PL}_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the interval amenable?

Formulation references: [S1], [S2].

Short English statement

Is Thompson's group $F$ amenable? Is the group $\mathrm{PL}_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the interval amenable?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003415 (OWR-15216-021). Catalogue attribution under CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
S2 Original problem formulation and mathematical context. https://doi.org/10.4171/owr/2017/11.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2429-A1 · Literature report · LLM-generated · unknown · 2026-10-06

OpenAI claim summary for Math-2429

Theorem 1.1 proves F nonamenable. Since F is a subgroup of PL_+(I), amenability passing to subgroups proves PL_+(I) nonamenable as well; both questions receive negative answers. Within these assumptions, this is a claimed negative answer to the stated question.

Amenability of Thompson Group and Interval Homeomorphism Groups

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. Thompson group $F$ is claimed to be nonamenable.

Mathematical statement of the claimed result

Result 1.

For Thompson's group $F$, consisting of increasing piecewise-linear homeomorphisms of $[0,1]$ with dyadic breakpoints and slopes powers of two, the source claims \[ F\text{ is nonamenable}. \] This is a statement about the standard discrete group $F$.

Outline of the argument

Recursive colors on dyadic partitions are designed to behave under affine transport. Finite averages of these colors give a uniform boundary discrepancy under a fixed set of group moves, contradicting the existence of Følner sets.

Formal verification

For family 248, the documented statement rules out a positive normalized invariant mean for standard dyadic Thompson $F$; it does not specify an explicit expansion constant. The listed Comparator files are ThompsonNonamenability.lean. Named declarations include OAI.ThompsonNonamenability.thompson_F_nonamenable_composition. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

OpenAI, Thompson's group F is nonamenable, 2026-09-23; citation key OAI:Thompsons-group-F-is-nonamenable-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
Family 248: Lean scope.
Comparator ThompsonNonamenability.json.
Comparator ThompsonNonamenability.lean.
OpenAI catalogue entry.

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Record details

Subjects: Group theory
Collection: Top Open Problems
Mathematics rank: 2429
UnsolvedMath: Catalogue entry