{
  "id": "1612.01242",
  "version": "v1",
  "published": "2016-12-05T04:10:22.000Z",
  "updated": "2016-12-05T04:10:22.000Z",
  "title": "Properties of random nilpotent groups",
  "authors": [
    "Albert Garreta",
    "Alexei Miasnikov",
    "Denis Ovchinnikov"
  ],
  "comment": "25 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study random nilpotent groups of the form $G=N/\\langle\\langle R \\rangle \\rangle$, where $N$ is a non-abelian free nilpotent group with $m$ generators, and $R$ is a set of $r$ random relators of length $\\ell$. We prove that the following holds asymptotically almost surely as $\\ell\\to \\infty$: 1) If $r\\leq m-2$, then the ring of integers $\\mathbb{Z}$ is e-definable in $G/{it Is}(G_3)$, and systems of equations over $\\mathbb{Z}$ are reducible to systems of equations over $G$ (hence, they are undecidable). Moreover, $Z(G)\\leq {\\it Is}(G')$, $G/G_3$ is virtually free nilpotent of rank $m-r$, and $G/G_3$ cannot be decomposed as the direct product of two non-virtually abelian groups. 2) If $r=m-1$, then $G$ is virtually abelian. 3) If $r= m$, then $G$ is finite. 4) If $r\\geq m+1$, then $G$ is finite and abelian. In the last three cases, systems of equations are decidable in $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-05T04:10:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "properties",
      "study random nilpotent groups",
      "non-abelian free nilpotent group",
      "random relators",
      "virtually free nilpotent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}