{
  "id": "2208.07145",
  "version": "v1",
  "published": "2022-08-15T12:42:43.000Z",
  "updated": "2022-08-15T12:42:43.000Z",
  "title": "On the Diophantine problem in some one-relator groups",
  "authors": [
    "Carl-Fredrik Nyberg-Brodda"
  ],
  "comment": "Preliminary version. Comments more than welcome. 27 pages, 87 references",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "abstract": "We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister--Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation $a^mb^n = 1$ is virtually a direct product of hyperbolic groups for all $m, n \\geq 0$, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the two-generator, one-relator groups $G_{m,n}$ with defining relation a commutator $[a^m, b^n] = 1$, where $m, n \\geq 1$. In doing so, we define and study a natural class of groups (RABSAGs), related to right-angled Artin groups (RAAGs). We reduce the Diophantine problem in the groups $G_{m,n}$ to the Diophantine problem in groups which are virtually certain RABSAGs. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group $G_{2,2}$ with the single defining relation $[a^2, b^2] = 1$. We use the recent classification by Gray & Howie of RAAG subgroups of one-relator groups to classify the RAAG subgroups of some RABSAGs, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups $\\operatorname{NG}(p,q)$, which are $(p+1)$-generated one-relator groups generalising the solvable Baumslag--Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-15T12:42:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F05",
      "20F10",
      "20F32",
      "20M05"
    ],
    "keywords": [
      "diophantine problem",
      "raag subgroups",
      "conclude decidability",
      "study newman groups",
      "submonoid membership problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}