{
  "id": "thompsons-group-f-and-amenability",
  "title": "Thompson's group F and amenability",
  "description": "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.",
  "units": [
    {
      "id": "thompsons-group-f-and-dyadic-partitions",
      "title": "Thompson's group F and dyadic partitions",
      "reader": "thompsons-group-f-and-dyadic-partitions.html",
      "source": "src/thompsons-group-f-and-dyadic-partitions.md",
      "source_sha256": "4e777dbd60b4257c1509b71105076dc202ead72926da2244ec41291d7c126d46",
      "status": "author_self_checked"
    },
    {
      "id": "a-lipschitz-map-of-the-hilbert-ball-that-moves-every-point",
      "title": "A Lipschitz map of the Hilbert ball that moves every point",
      "reader": "a-lipschitz-map-of-the-hilbert-ball-that-moves-every-point.html",
      "source": "src/a-lipschitz-map-of-the-hilbert-ball-that-moves-every-point.md",
      "source_sha256": "2d7acb6f27ace580ee0e8905779396b5c43da73260ead5744706fb191b0f0970",
      "status": "author_self_checked"
    },
    {
      "id": "thompsons-group-f-is-not-amenable",
      "title": "Thompson's group F is not amenable",
      "reader": "thompsons-group-f-is-not-amenable.html",
      "source": "src/thompsons-group-f-is-not-amenable.md",
      "source_sha256": "fc516b7eab474bf50241c23f903aada6effca9d90cf98f0b1e87f51b9a59a466",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "OA-ERGODIC lesson means-folner-sets-and-regular-representations (Section 1, amenability by invariant means; Theorem 3.1, a countable discrete group is amenable exactly when it has a Folner sequence; Theorem 2.1 in a remark)",
    "core course Algebraic Topology (D60), Lecture 30, Theorem 30.1 (Brouwer's fixed-point theorem; used only in remarks of the second lesson)",
    "real analysis: the mean value theorem and the Bolzano-Weierstrass theorem"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}