{
  "id": "2006.06371",
  "version": "v1",
  "published": "2020-06-11T12:41:55.000Z",
  "updated": "2020-06-11T12:41:55.000Z",
  "title": "Metabelian groups: full-rank presentations, randomness and Diophantine problems",
  "authors": [
    "Albert Garreta",
    "Leire Legarreta",
    "Alexei Miasnikov",
    "Denis Ovchinnikov"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study metabelian groups $G$ given by full rank finite presentations $\\langle A \\mid R \\rangle_{\\mathcal{M}}$ in the variety $\\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\\max\\{0, |A|-|R|\\}$ and a virtually abelian normal subgroup, and that if $|R| \\leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\\geq |A|$. We further prove that if $|R| \\leq |A|-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T12:41:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20F70",
      "20F10",
      "20F16",
      "20F69",
      "03B25",
      "03D35",
      "60G99"
    ],
    "keywords": [
      "diophantine problem",
      "full-rank presentations",
      "randomness",
      "full rank finite presentations",
      "metabelian groups satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}