{
  "id": "2207.13072",
  "version": "v1",
  "published": "2022-07-26T17:42:53.000Z",
  "updated": "2022-07-26T17:42:53.000Z",
  "title": "Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties",
  "authors": [
    "Vladimir L. Popov"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are nonlinear and contain the braid group $B_n$ on $n$ strands for $n\\geqslant 3$, and are nonamenable for $n\\geqslant 2$. As an application, it is proved that for $n\\geqslant 3$, every Cremona group of rank $\\geqslant 3n-1$ contains the groups ${\\rm Aut}(F_n)$ and $B_n$. This bound is 1 better than the one published earlier by the author; with respect to $B_n$ the order of its growth rate is one less than that of the bound following from the paper by D. Krammer. The basis of the construction are triplets $(G, R, n)$, where $G$ is a connected semisimple algebraic group and $R$ is a closed subgroup of its maximal torus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T17:42:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "14M99",
      "20F28",
      "20F29",
      "20F38"
    ],
    "keywords": [
      "free group",
      "embeddings",
      "rational affine algebraic varieties",
      "automorphism group contains",
      "connected semisimple algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}