{
  "id": "1507.06908",
  "version": "v1",
  "published": "2015-07-24T16:25:33.000Z",
  "updated": "2015-07-24T16:25:33.000Z",
  "title": "Determining solubility for finitely generated groups of PL homeomorphisms",
  "authors": [
    "Collin Bleak",
    "Tara Brough",
    "Susan Hermiller"
  ],
  "comment": "24 pages, two figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson's group $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that the derived length of $G$ is bounded above by the number of $G$-orbit classes of the breakpoints of any finite set of generators. Finally, we specify a set of `computable' subgroups of $PL_+(I)$ (which includes R. Thompson's group $F$) and we give an algorithm which determines in finite time whether or not any given finite subset $X$ of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of $\\langle X\\rangle$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-24T16:25:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F10",
      "20E07",
      "20F16",
      "37E05"
    ],
    "keywords": [
      "finitely generated groups",
      "pl homeomorphisms",
      "determining solubility",
      "thompsons group",
      "finitely generated subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150706908B"
    }
  }
}