{
  "id": "1609.08900",
  "version": "v1",
  "published": "2016-09-28T13:14:34.000Z",
  "updated": "2016-09-28T13:14:34.000Z",
  "title": "Rank gradient of sequences of subgroups in a direct product",
  "authors": [
    "Nikolay Nikolov",
    "Zvi Shemtov",
    "Mark Shusterman"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "For a sequence $\\{U_n\\}_{n = 1}^\\infty$ of finite index subgroups of a direct product $G = A \\times B$ of finitely generated groups, we show that $$\\lim_{n \\to \\infty} \\frac{\\min\\{|X| : \\langle X \\rangle = U_n\\}}{[G : U_n]} = 0$$ once $[A : A \\cap U_n], [B : B \\cap U_n] \\to \\infty$ as $n \\to \\infty$. Our proof relies on the classification of finite simple groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-28T13:14:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20F69",
      "20D05",
      "20F65",
      "37A20"
    ],
    "keywords": [
      "direct product",
      "rank gradient",
      "finite index subgroups",
      "finite simple groups",
      "proof relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}