{
  "id": "1609.06253",
  "version": "v1",
  "published": "2016-09-20T16:59:02.000Z",
  "updated": "2016-09-20T16:59:02.000Z",
  "title": "Geometry of the word problem for 3-manifold groups",
  "authors": [
    "Mark Brittenham",
    "Susan Hermiller",
    "Tim Susse"
  ],
  "comment": "30 pages",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-20T16:59:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F10",
      "57M05",
      "68Q42"
    ],
    "keywords": [
      "word problem",
      "fundamental group",
      "strongly coset automatic",
      "autostackable respecting",
      "finite state automata"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}