{
  "id": "2106.11552",
  "version": "v1",
  "published": "2021-06-22T05:39:00.000Z",
  "updated": "2021-06-22T05:39:00.000Z",
  "title": "Finitely generated subgroups of free groups as formal languages and their cogrowth",
  "authors": [
    "Arman Darbinyan",
    "Rostislav Grigorchuk",
    "Asif Shaikh"
  ],
  "comment": "35 pages, 8 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-22T05:39:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E07",
      "68Q45",
      "68Q70"
    ],
    "keywords": [
      "finitely generated subgroups",
      "free group",
      "formal languages",
      "groups explicitly construct ergodic automaton",
      "minimal deterministic finite automaton"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}