{
  "id": "the-finite-energy-boundary-of-a-group",
  "title": "The finite-energy boundary of a group",
  "description": "Finite-energy functions on the Cayley complex of a finitely presented group, and the boundary they give a one-ended group: a function with connected level sides, a compact metric space containing the group with square summable edge lengths, a boundary that is a continuum, paths near boundary points, the action of the group on the boundary as a convergence group with dense orbits, the dimension and first Čech cohomology of the boundary, a continuum without cut points inside the boundary, and the proof that this continuum is locally connected. The result: under named hypotheses on the group, every finite-energy gradient is a limit of gradients of finitely supported functions. This is the part of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree that takes place on the group.",
  "units": [
    {
      "id": "finite-energy-functions-on-a-cayley-complex",
      "title": "Finite-energy functions on a Cayley complex",
      "reader": "finite-energy-functions-on-a-cayley-complex.html",
      "source": "src/finite-energy-functions-on-a-cayley-complex.md",
      "source_sha256": "8d74423e01057dbe9ed469611b09fc713c655112c6d53efd61ff6a2515880da4",
      "status": "author_self_checked"
    },
    {
      "id": "the-compact-space-and-its-boundary",
      "title": "The compact space and its boundary",
      "reader": "the-compact-space-and-its-boundary.html",
      "source": "src/the-compact-space-and-its-boundary.md",
      "source_sha256": "c0b62213209ed69f6dd89b12792b73d038b1a0d84ef01a38a29fccc623c3c403",
      "status": "author_self_checked"
    },
    {
      "id": "the-convergence-action",
      "title": "The convergence action",
      "reader": "the-convergence-action.html",
      "source": "src/the-convergence-action.md",
      "source_sha256": "349b7b92759d2ddaa538571da555a3bad7512026451a7beac4ee82ab36b0f5df",
      "status": "author_self_checked"
    },
    {
      "id": "dimension-and-cohomology-of-the-boundary",
      "title": "Dimension and cohomology of the boundary",
      "reader": "dimension-and-cohomology-of-the-boundary.html",
      "source": "src/dimension-and-cohomology-of-the-boundary.md",
      "source_sha256": "995843ed8dfbb315a20abb42985d6a4d35c46bb0edb22a9c8e24843f713af3c5",
      "status": "author_self_checked"
    },
    {
      "id": "a-block-of-the-boundary",
      "title": "A block of the boundary",
      "reader": "a-block-of-the-boundary.html",
      "source": "src/a-block-of-the-boundary.md",
      "source_sha256": "dd7e28fb28a551e336b040dc7e49129cb0c43cb336fa7498743421f151a04b24",
      "status": "author_self_checked"
    },
    {
      "id": "a-continuum-without-cut-points-and-its-graph",
      "title": "A continuum without cut points and its graph",
      "reader": "a-continuum-without-cut-points-and-its-graph.html",
      "source": "src/a-continuum-without-cut-points-and-its-graph.md",
      "source_sha256": "729af4e14fb4ff8ccc7f18fb59976f1987a3a741835a67c5a14bbd233bac8c17",
      "status": "author_self_checked"
    },
    {
      "id": "component-labels-and-annular-depth",
      "title": "Component labels and annular depth",
      "reader": "component-labels-and-annular-depth.html",
      "source": "src/component-labels-and-annular-depth.md",
      "source_sha256": "4ce10e6810daa189cd6065c67c7e58bcd31c3d90153b8e2385ef1d68b51973a1",
      "status": "author_self_checked"
    },
    {
      "id": "vertices-of-large-depth",
      "title": "Vertices of large depth",
      "reader": "vertices-of-large-depth.html",
      "source": "src/vertices-of-large-depth.md",
      "source_sha256": "0addf9fe4ef9341adfe8392d48d4514bb29758de403bfef9f51e37b7bac2ed62",
      "status": "author_self_checked"
    },
    {
      "id": "the-continuum-is-locally-connected",
      "title": "The continuum is locally connected",
      "reader": "the-continuum-is-locally-connected.html",
      "source": "src/the-continuum-is-locally-connected.md",
      "source_sha256": "fd991e55d2671fd516443f86eb679e81b940dfcfe3d0159c3399f333dcecebe1",
      "status": "author_self_checked"
    },
    {
      "id": "the-theorem-and-its-hypotheses",
      "title": "The theorem and its hypotheses",
      "reader": "the-theorem-and-its-hypotheses.html",
      "source": "src/the-theorem-and-its-hypotheses.md",
      "source_sha256": "3cfbd6fa994493d546acc90b9c3e4e31d7d915ac5a183fb4de9097e82d9399a0",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Functional Analysis (D20): closed subspaces of a Hilbert space, the Riesz representation theorem",
    "core course Measure and Integration (D10): Tonelli's theorem, dominated convergence",
    "free groups and presentations of groups",
    "for the third lesson: the course Convergence groups (lesson Convergence actions and their elements)",
    "for the fourth lesson: the course Continua, arcs and cut points, and covering dimension (Definition 2.1 of Covering dimension and finite trivializing covers)",
    "for the fifth lesson: the course Convergence groups (lesson The tree of cut points) and, through it, the course Pretrees",
    "for the tenth lesson: the course Convergence groups (lesson The fixed-point property for groups with one end)",
    "the course proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}