{
  "id": "2207.04759",
  "version": "v1",
  "published": "2022-07-11T10:41:31.000Z",
  "updated": "2022-07-11T10:41:31.000Z",
  "title": "Ideals of equations for elements in a free group and Stallings folding",
  "authors": [
    "Dario Ascari"
  ],
  "comment": "26 pages, 14 figures",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $F$ be a finitely generated free group and let $H\\le F$ be a finitely generated subgroup. Given an element $g\\in F$, we study the ideal $\\mathfrak{I}_g$ of equations for $g$ with coefficients in $H$, i.e. the elements $w(x)\\in H*\\langle x\\rangle$ such that $w(g)=1$ in $F$. The ideal $\\mathfrak{I}_g$ is a normal subgroup of $H*\\langle x\\rangle$, and we provide an algorithm, based on Stallings folding operations, to compute a finite set of generators for $\\mathfrak{I}_g$ as a normal subgroup. We provide an algorithm to find an equation in $\\mathfrak{I}_g$ with minimum degree, i.e. an equation $w(x)$ such that its cyclic reduction contains the minimum possible number of occurrences of $x$ and $x^{-1}$; this answers a question of A. Rosenmann and E. Ventura. More generally, we provide an algorithm that, given $d\\in\\mathbb{N}$, determines whether $\\mathfrak{I}_g$ contains equations of degree $d$ or not, and we give a characterization of the set of all the equations of that specific degree. We define the set $D_g$ of all integers $d$ such that $\\mathfrak{I}_g$ contains equations of degree $d$; we show that $D_g$ coincides, up to a finite set, either with the set of non-negative even numbers or with the set of natural numbers. Finally, we provide examples to illustrate the techniques introduces in this paper. We discuss the case where $\\text{rank}(H)=1$. We prove that both kinds of sets $D_g$ can actually occur. The examples also show that the equations of minimum possible degree aren't in general enough to generate the whole ideal $\\mathfrak{I}_g$ as a normal subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-11T10:41:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F70",
      "20E05",
      "20F65"
    ],
    "keywords": [
      "normal subgroup",
      "finite set",
      "contains equations",
      "cyclic reduction contains",
      "generated free group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}