{
  "id": "convergence-groups",
  "title": "Convergence groups",
  "description": "Convergence groups on compact metric spaces, from the definition by sequences: every element is elliptic, parabolic or loxodromic, loxodromic elements have attracting and repelling fixed points and a compact fundamental domain, and the stabilizer of a loxodromic fixed point is virtually infinite cyclic; convergence actions and minimal actions pass to equivariant quotients, a convergence group on a continuum acts as a convergence group on its dendrite quotient, the points of the completion of the cut points have supports in the continuum that control which of them a group element fixes, under conditions on the group the dendrite quotient is not a point, and when the group fixes a point of the dendrite quotient the cut points form the vertices of a tree on which the group acts. A group with a finite generating set and a one-ended Cayley graph has the fixed-point property for trees that these results assume.",
  "units": [
    {
      "id": "convergence-actions-and-their-elements",
      "title": "Convergence actions and their elements",
      "reader": "convergence-actions-and-their-elements.html",
      "source": "src/convergence-actions-and-their-elements.md",
      "source_sha256": "5d69c6ff51a55316844e120178e557edc533dc7f9b8814c454ccff222d27f7d1",
      "status": "author_self_checked"
    },
    {
      "id": "quotients-and-minimal-actions",
      "title": "Quotients and minimal actions",
      "reader": "quotients-and-minimal-actions.html",
      "source": "src/quotients-and-minimal-actions.md",
      "source_sha256": "de639ec7ba65ec6b7f40e38d782c4cde329489f2e7770353f150f990af9b2bbd",
      "status": "author_self_checked"
    },
    {
      "id": "supports-and-fixed-points-in-the-completion",
      "title": "Supports and fixed points in the completion",
      "reader": "supports-and-fixed-points-in-the-completion.html",
      "source": "src/supports-and-fixed-points-in-the-completion.md",
      "source_sha256": "98e4a29223c5e412adad3991f8ffcb40670108e8ecdd1ec813766b55c2bb9075",
      "status": "author_self_checked"
    },
    {
      "id": "loxodromic-elements-and-the-completion",
      "title": "Loxodromic elements and the completion",
      "reader": "loxodromic-elements-and-the-completion.html",
      "source": "src/loxodromic-elements-and-the-completion.md",
      "source_sha256": "355c4c1912aafd199773c12bbc77e6c4d86bf75cf8da6da048e9628780aea00f",
      "status": "author_self_checked"
    },
    {
      "id": "trees-and-classes-of-finite-intervals",
      "title": "Trees and classes of finite intervals",
      "reader": "trees-and-classes-of-finite-intervals.html",
      "source": "src/trees-and-classes-of-finite-intervals.md",
      "source_sha256": "2e133a867a0770a01858ae63b472b014d05d4ae70d555853f1ee3bf74bd4edb5",
      "status": "author_self_checked"
    },
    {
      "id": "the-dendrite-quotient-is-not-a-point",
      "title": "The dendrite quotient is not a point",
      "reader": "the-dendrite-quotient-is-not-a-point.html",
      "source": "src/the-dendrite-quotient-is-not-a-point.md",
      "source_sha256": "29dfb0fad2d60d7cc995a5f945525605bdc5b2d9b07dde4c320b26763032c2aa",
      "status": "author_self_checked"
    },
    {
      "id": "the-tree-of-cut-points",
      "title": "The tree of cut points",
      "reader": "the-tree-of-cut-points.html",
      "source": "src/the-tree-of-cut-points.md",
      "source_sha256": "72a6f41b5f98593431666184340abb3d855981f91076bba20267a6ea6e50c143",
      "status": "author_self_checked"
    },
    {
      "id": "the-fixed-point-property-for-groups-with-one-end",
      "title": "The fixed-point property for groups with one end",
      "reader": "the-fixed-point-property-for-groups-with-one-end.html",
      "source": "src/the-fixed-point-property-for-groups-with-one-end.md",
      "source_sha256": "86b55ab413fe5ed60d34c1aef41d19104bd33ae74a6043495313769496a25d3c",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Point-Set Topology (C90): compact metric spaces",
    "groups, subgroups and index",
    "from the second lesson on: the course Pretrees (cut points of a connected space, the completion, the dendrite quotient of a continuum)",
    "for the sixth lesson: ordinal numbers and transfinite induction (core course Mathematical Logic, Set Theory, and Computability), through the lesson The dense quotient of a pretree",
    "the course proves every other fact it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}