{
  "id": "2207.00920",
  "version": "v1",
  "published": "2022-07-02T23:45:49.000Z",
  "updated": "2022-07-02T23:45:49.000Z",
  "title": "Stable rational homology of the IA-automorphism groups of free groups",
  "authors": [
    "Mai Katada"
  ],
  "comment": "55 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "The rational homology of the IA-automorphism group $\\operatorname{IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\\operatorname{IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\\operatorname{IA}_n$. We obtain a representation-stable $\\operatorname{GL}(n,\\mathbb{Q})$-subquotient of the Albanese homology of $\\operatorname{IA}_n$, which conjecturally coincides with the entire Albanese homology of $\\operatorname{IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\\operatorname{IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\\operatorname{IA}_n$ for $n\\ge 9$. We also study the Albanese homology of an analogue of $\\operatorname{IA}_n$ to the outer automorphism group of $F_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-02T23:45:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F28",
      "20J06"
    ],
    "keywords": [
      "stable rational homology",
      "ia-automorphism group",
      "free group",
      "entire third albanese homology",
      "outer automorphism group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}