{
  "id": "0907.5010",
  "version": "v1",
  "published": "2009-07-28T22:05:45.000Z",
  "updated": "2009-07-28T22:05:45.000Z",
  "title": "The Schur multiplier, profinite completions and decidability",
  "authors": [
    "Martin R Bridson"
  ],
  "comment": "6 pages no figures. To appear in the Bulletin London Math Soc",
  "doi": "10.1112/blms/bdp133",
  "categories": [
    "math.GR"
  ],
  "abstract": "We fix a finitely presented group $Q$ and consider short exact sequences $1\\to N\\to G\\to Q\\to 1$ with $G$ finitely generated. The inclusion $N\\to G$ induces a morphism of profinite completions $\\hat N\\to \\hat G$. We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\\subset G$, can determine whether or not $\\hat P\\to\\hat G$ is an isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-28T22:05:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "20F10"
    ],
    "keywords": [
      "profinite completions",
      "schur multiplier",
      "decidability",
      "short exact sequences",
      "proper subgroups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.5010B"
    }
  }
}