{
  "id": "1004.0222",
  "version": "v5",
  "published": "2010-04-01T19:57:27.000Z",
  "updated": "2011-04-17T03:00:58.000Z",
  "title": "Generalizing Magnus' characterization of free groups to some free products",
  "authors": [
    "Khalid Bou-Rabee",
    "Brandon Seward"
  ],
  "comment": "11 pages, complete rewrite",
  "categories": [
    "math.GR"
  ],
  "abstract": "A residually nilpotent group is \\emph{$k$-parafree} if all of its lower central series quotients match those of a free group of rank $k$. Magnus proved that $k$-parafree groups of rank $k$ are themselves free. In this note we mimic this theory with finite extensions of free groups, with an emphasis on free products of the cyclic group $C_p$, for $p$ an odd prime. We show that for $n \\leq p$ Magnus' characterization holds for the $n$-fold free product $C_p^{*n}$ within the class of finite-extensions of free groups. Specifically, if $n \\leq p$ and $G$ is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as $C_p^{*n}$, then $G \\cong C_p^{*n}$. We also show that such a characterization does not hold in the class of finitely generated groups. That is, we construct a rank 2 residually nilpotent group $G$ that shares all its lower central series quotients with $\\ffp$, but is not $\\ffp$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-04-17T03:00:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E26"
    ],
    "keywords": [
      "free group",
      "free product",
      "residually nilpotent group",
      "generalizing magnus",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.0222B"
    }
  }
}