{
  "id": "2211.13458",
  "version": "v1",
  "published": "2022-11-24T07:52:59.000Z",
  "updated": "2022-11-24T07:52:59.000Z",
  "title": "On the stable cohomology of $\\mathrm{GL}(n, \\mathbb{Z})$, $\\mathrm{Aut}(F_n)$ and $\\mathrm{IA}_n$",
  "authors": [
    "Kazuo Habiro",
    "Mai Katada"
  ],
  "comment": "47 pages",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "abstract": "Borel's stability and vanishing theorem gives the stable cohomology of $\\mathrm{GL}(n,\\mathbb{Z})$ with coefficients in algebraic $\\mathrm{GL}(n,\\mathbb{Z})$-representations. We improve the stable range in two ways by using ideas of Borel and Kupers-Miller-Patzt. By combining the improved Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-24T07:52:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F28",
      "20J06"
    ],
    "keywords": [
      "stable cohomology",
      "stable rational cohomology",
      "conjectural algebraic structure",
      "hochschild-serre spectral sequence",
      "borels stability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}