{
  "id": "2010.09673",
  "version": "v1",
  "published": "2020-10-19T17:08:24.000Z",
  "updated": "2020-10-19T17:08:24.000Z",
  "title": "Effective finite generation for [IA_n,IA_n] and the Johnson kernel",
  "authors": [
    "Mikhail Ershov",
    "Daniel Franz"
  ],
  "comment": "35 pages, 4 figures",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "Let $G_n$ denote either $Aut(F_n)$, the automorphism group of a free group of rank $n$, or $Mod(\\Sigma_n^1)$, the mapping class group of an orientable surface of genus $n$ with $1$ boundary component. In both cases $G_n$ admits a natural filtration $\\{G_n(k)\\}_{k=1}^{\\infty}$ called the Johnson filtration. The first terms of this filtration $G_n(1)$ are the subgroup of $IA$-automorphisms and the Torelli subgroup, respectively. It was recently proved for both families of groups that for each $k$, the $k^{\\rm th}$ term $G_n(k)$ is finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $[IA_n,IA_n]$, the second term of the Johnson filtration of $Aut(F_n)$, and an almost explicit finite generating set for the Johnson kernel, the second term of the Johnson filtration of $Mod(\\Sigma_n^1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-19T17:08:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F28",
      "20F65",
      "57M07"
    ],
    "keywords": [
      "effective finite generation",
      "johnson kernel",
      "explicit finite generating set",
      "johnson filtration",
      "second term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}