{
  "id": "1611.00432",
  "version": "v1",
  "published": "2016-11-02T01:03:25.000Z",
  "updated": "2016-11-02T01:03:25.000Z",
  "title": "Group actions, Teichmüller spaces and cobordisms",
  "authors": [
    "Boris N. Apanasov"
  ],
  "comment": "25 pages, 8 figures. arXiv admin note: text overlap with arXiv:1510.08951",
  "categories": [
    "math.GT"
  ],
  "abstract": "We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary $\\partial M$. In addition to the standard conformal ergodic action of a uniform hyperbolic lattice on the round sphere $S^{n-1}$ and its quasiconformal deformations in $S^n$, we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed $n$-ball into $S^n$), on non-trivial $(n-1)$-knots in $S^{n+1}$, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from \"non-standard\" components of its varieties of representations (faithful or with large kernels of defining homomorphisms).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-02T01:03:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "group actions",
      "teichmüller spaces",
      "complete hyperbolic structure",
      "fundamental group",
      "unusual actions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}