{
  "id": "2009.13489",
  "version": "v1",
  "published": "2020-09-28T17:27:49.000Z",
  "updated": "2020-09-28T17:27:49.000Z",
  "title": "Homological Filling Functions with Coefficients",
  "authors": [
    "Xingzhe Li",
    "Fedor Manin"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in \"Asymptotic invariants of infinite groups\", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n \\geq 1$ and every pair of coefficient groups $A, B \\in \\{\\mathbb{Z},\\mathbb{Q}\\} \\cup \\{\\mathbb{Z}/p\\mathbb{Z} : p\\text{ prime}\\}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T17:27:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "57M07"
    ],
    "keywords": [
      "asymptotic invariants",
      "cayley graph",
      "main theorem",
      "asymptotic behavior",
      "define homological filling functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}