{
  "id": "1903.03864",
  "version": "v1",
  "published": "2019-03-09T20:16:22.000Z",
  "updated": "2019-03-09T20:16:22.000Z",
  "title": "On the top homology group of Johnson kernel",
  "authors": [
    "Alexander A. Gaifullin"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "The action of the mapping class group $\\mathrm{Mod}_g$ of an oriented surface $\\Sigma_g$ on the lower central series of $\\pi_1(\\Sigma_g)$ defines the descending filtration in $\\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\\mathcal{I}_g$ and the Johnson kernel $\\mathcal{K}_g$. By a fundamental result of Johnson (1985), $\\mathcal{K}_g$ is the subgroup of $\\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $\\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\\mathcal{K}_g,\\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\\mathcal{K}_g,\\mathbb{Q})$ is not finitely generated as a module over the group ring $\\mathbb{Q}[\\mathcal{I}_g]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-09T20:16:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F34",
      "57M07",
      "20J05"
    ],
    "keywords": [
      "johnson kernel",
      "homology group",
      "free abelian subgroup",
      "lower central series",
      "mapping class group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}