Thompson's group F and amenability
OpenAI's proof (2026) that Thompson's group F is not amenable, answering Geoghegan's 1979 question. The course develops F through dyadic partitions of the unit interval: normalized restrictions, images of fine uniform partitions, elements moving pairs of separated dyadic intervals, and covariance. It constructs an explicit Lipschitz map of the closed unit ball of a Hilbert space into its unit sphere that moves every point by at least 1/2, through a curve with a bounded, uniformly separated tail. It then colors dyadic partitions recursively with this map and proves that one finite set of elements of F has a uniformly large relative boundary on every finite subset, so that F has no Følner sets.
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Thompson's group F and dyadic partitions
- A Lipschitz map of the Hilbert ball that moves every point
- Thompson's group F is not amenable
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Means, Følner sets, and regular representations (Measured relations and operator algebras)
- Unitarizability implies amenability (Amenability and unitarizability)
- The mean value theorem and the Bolzano–Weierstrass theorem of real analysis; Brouwer's fixed-point theorem, used only in remarks (core course Algebraic Topology, D60, Lecture 30).