OP-01065 Solution claimed

Is Thompson's group F amenable?

Thompson's group $F$ is the group of piecewise linear homeomorphisms of $[0, 1]$ whose breakpoints are dyadic rationals and whose slopes are powers of $2$.

Problem. Is $F$ amenable?

Source: Usually credited to R. Geoghegan (1979); see J. W. Cannon, W. J. Floyd and W. R. Parry (1996). Listed as G.2 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).

Imported from math.uwaterloo.ca/~j37peter/problems.html.

Classification

Primary: GR — Group Theory
Secondary: OA
Keywords: amenability, Peterson's operator algebra problems, Thompson's group F

References

@article{cannon1996,
  author  = {Cannon, J. W. and Floyd, W. J. and Parry, W. R.},
  title   = {Introductory notes on {R}ichard {T}hompson's groups},
  journal = {Enseign. Math. (2)},
  volume  = {42},
  number  = {3-4},
  pages   = {215--256},
  year    = {1996},
  doi     = {10.5169/seals-87877},
}
@misc{peterson-g2,
  author       = {Peterson, Jesse},
  title        = {Open problems in operator algebras},
  howpublished = {Web page},
  year         = {2026},
  url          = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
  note         = {Problem G.2; version of 15 May 2026},
}
@article{guba2023,
  author  = {Guba, Victor},
  title   = {Amenability problem for {T}hompson's group {F}: state of the art},
  journal = {Groups Complex. Cryptol.},
  volume  = {15},
  number  = {1},
  year    = {2023},
  doi     = {10.46298/jgcc.2023.15.1.11315},
}
@article{juschenko2018,
  author  = {Juschenko, Kate and Matte Bon, Nicol{\'a}s and Monod, Nicolas and de la Salle, Mikael},
  title   = {Extensive amenability and an application to interval exchanges},
  journal = {Ergodic Theory Dynam. Systems},
  volume  = {38},
  number  = {1},
  pages   = {195--219},
  year    = {2018},
  doi     = {10.1017/etds.2016.32},
}
@misc{moore2011,
  author        = {Moore, Justin Tatch},
  title         = {A note on {S}havgulidze's papers concerning the amenability problem for {T}hompson's group {F}},
  year          = {2011},
  eprint        = {1102.0747},
  archivePrefix = {arXiv},
}

Resolution claims

Thompson's group $F$ is nonamenable v1 GitHub ↗

OpenAI

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

Disproof · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: math @ 7ef5c0c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01065, "Is Thompson's group F amenable?", clopen.solutions.