{
  "id": "math/0703703",
  "version": "v1",
  "published": "2007-03-23T15:46:59.000Z",
  "updated": "2007-03-23T15:46:59.000Z",
  "title": "Residual $p$ properties of mapping class groups and surface groups",
  "authors": [
    "Luis Paris"
  ],
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "Let $\\mathcal M (\\Sigma, \\mathcal P)$ be the mapping class group of a punctured oriented surface $(\\Sigma, \\mathcal P)$ (where $\\mathcal P$ may be empty), and let $\\mathcal T_p(\\Sigma,\\mathcal P)$ be the kernel of the action of $\\mathcal M (\\Sigma, \\mathcal P)$ on $H_1 (\\Sigma \\setminus \\mathcal P, \\mathbb F_p)$. We prove that $\\mathcal T_p(\\Sigma, \\mathcal P)$ is residually $p$. In particular, this shows that $\\mathcal M (\\Sigma, \\mathcal P)$ is virtually residually $p$. For a group $G$ we denote by $\\mathcal I_p(G)$ the kernel of the natural action of ${\\rm Out} (G)$ on $H_1(G, \\mathbb F_p)$. In order to achieve our theorem, we prove that, under certain conditions ($G$ is conjugacy $p$-separable and has Property A), the group $\\mathcal I_p(G)$ is residually $p$. The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy $p$-separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy $p$-separable is, from a technical point of view, the main result of the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-23T15:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F38"
    ],
    "keywords": [
      "mapping class group",
      "surface groups",
      "free groups",
      "natural action",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3703P"
    }
  }
}