{
  "id": "2210.16123",
  "version": "v1",
  "published": "2022-10-28T13:38:12.000Z",
  "updated": "2022-10-28T13:38:12.000Z",
  "title": "On the Dehn functions of a class of monadic one-relation monoids",
  "authors": [
    "Carl-Fredrik Nyberg-Brodda"
  ],
  "comment": "19 pages. Comments welcome!",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "We give an infinite family of monoids $\\Pi_N$ (for $N=2, 3, \\dots$), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $\\Pi_N$ is at least exponential. More precisely, we prove that the Dehn function $\\partial_N(n)$ of $\\Pi_N$ satisfies $\\partial_N(n) \\succeq N^{n/4}$. This answers negatively a question posed by Cain & Maltcev in 2013 on whether every monoid defined by a single relation of the form $bUa=a$ has quadratic Dehn function. Finally, by using the decidability of the rational subset membership problem in the metabelian Baumslag--Solitar groups $\\operatorname{BS}(1,n)$ for all $n \\geq 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each $\\Pi_N$ has decidable word problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-28T13:38:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M05",
      "20F10",
      "20F65"
    ],
    "keywords": [
      "monadic one-relation monoids",
      "rational subset membership problem",
      "metabelian baumslag-solitar groups",
      "quadratic dehn function",
      "decidable word problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}