{
  "id": "a-hyperbolic-group-that-is-not-residually-finite",
  "title": "A hyperbolic group that is not residually finite",
  "description": "A torsion-free word-hyperbolic group that is not residually finite: the fundamental group of a finite complex of Euclidean triangles with three vertices, built from punctured lines over a finite field, in which one fixed nonidentity element dies in every finite quotient and in every finite-dimensional linear representation. The proof: a criterion for complexes whose vertex links have no short closed walks, through reduced planar fillings, angular Gauss-Bonnet, thin diagrams and a stability lemma, giving torsion-freeness, hyperbolicity and nontrivial closed paths; then a comparison of ranks over the reals and over a fixed prime field, through arrays of permutation matrices, local factorizations and a polynomial certificate over the lines. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "triangle-complexes-and-their-links",
      "title": "Triangle complexes and their links",
      "reader": "triangle-complexes-and-their-links.html",
      "source": "src/triangle-complexes-and-their-links.md",
      "source_sha256": "0993f90a40a5c17713f3285fa9d4886684f0db49fae48bf15f4791c4f15779d3",
      "status": "author_self_checked"
    },
    {
      "id": "polygons-and-plane-graphs",
      "title": "Polygons and plane graphs",
      "reader": "polygons-and-plane-graphs.html",
      "source": "src/polygons-and-plane-graphs.md",
      "source_sha256": "df0f98e6a4bb890538128f2920c644329f694e5e9b2406a940149bbcf2ba35bc",
      "status": "author_self_checked"
    },
    {
      "id": "reduced-planar-fillings",
      "title": "Reduced planar fillings",
      "reader": "reduced-planar-fillings.html",
      "source": "src/reduced-planar-fillings.md",
      "source_sha256": "6f6b3bf8fba0060817f38c0eda8ad76efabf2f27533078c99e936e63cc69ae3c",
      "status": "author_self_checked"
    },
    {
      "id": "angular-curvature-closed-paths-and-torsion",
      "title": "Angular curvature, closed paths and torsion",
      "reader": "angular-curvature-closed-paths-and-torsion.html",
      "source": "src/angular-curvature-closed-paths-and-torsion.md",
      "source_sha256": "9e28fa6552631af46a7d8a86ccfba0593f96c0c7cef44596127d9d224f359971",
      "status": "author_self_checked"
    },
    {
      "id": "thin-diagrams-and-hyperbolicity",
      "title": "Thin diagrams and hyperbolicity",
      "reader": "thin-diagrams-and-hyperbolicity.html",
      "source": "src/thin-diagrams-and-hyperbolicity.md",
      "source_sha256": "ea646bd7779bafc6e69921f2f5d5559a83d1212dc515bd19aea2cc6e8b82ad9a",
      "status": "author_self_checked"
    },
    {
      "id": "punctured-lines-and-the-complex",
      "title": "Punctured lines and the complex",
      "reader": "punctured-lines-and-the-complex.html",
      "source": "src/punctured-lines-and-the-complex.md",
      "source_sha256": "0c32c1b9f8d040cb997bd991331a9ef9356b984124186d1fc3360244ccd7792d",
      "status": "author_self_checked"
    },
    {
      "id": "the-rank-obstruction",
      "title": "The rank obstruction",
      "reader": "the-rank-obstruction.html",
      "source": "src/the-rank-obstruction.md",
      "source_sha256": "c4dead7a31512622fbb2c9a520b1bfcc02afbf327d4392faccc931e5aee94e1f",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Constant-coefficient equations and solvability, lesson Polygonal regions and the directions on a circle: the index of a closed polygonal path and its jump across an edge (Lemma 2.1) and the polygonal Jordan separation theorem (Theorem 3.1)",
    "course AG-CA, lesson The Nullstellensatz and Jacobson rings: the integers form a Jacobson ring (Proposition 3.2); finite-type algebras over Jacobson rings are Jacobson with finite residue field extensions (Theorem 4.2); used for Malcev's theorem",
    "core course Algebraic Topology: fundamental groups and homotopy of loops",
    "core course Real Analysis (Lebl, Basic Analysis I): the Lebesgue covering lemma (Lemma 7.4.10), uniform continuity on compact metric spaces (Theorem 7.5.11), compactness and connectedness of metric spaces",
    "linear algebra over fields (ranks, determinants, Gram-Schmidt, tensor products of matrices), finite groups and finite fields"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}