{
  "id": "2110.02188",
  "version": "v2",
  "published": "2021-10-05T17:26:58.000Z",
  "updated": "2023-06-12T16:54:10.000Z",
  "title": "Solving the membership problem for certain subgroups of $SL_2(\\mathbb{Z})$",
  "authors": [
    "Sandie Han",
    "Ariane M. Masuda",
    "Satyanand Singh",
    "Johann Thiel"
  ],
  "comment": "New version extends results to cases when $u=1$ and index computations/proofs are included",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "For positive integers $u$ and $v$, let $L_u=\\begin{bmatrix}1 & 0 \\\\u&1\\end{bmatrix}$ and $R_v=\\begin{bmatrix}1 & v \\\\ 0 & 1\\end{bmatrix}$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\\begin{bmatrix}a & c \\\\b&d\\end{bmatrix}$ in $G_{u,v}$ when $u,v\\geq 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v> 4$. Additionally, we compute $[\\mathscr{G}_{u,v}\\colon G_{u,v}]$ for $u,v\\geq 1$, extending a result of Chorna, Geller, and Shpilrain.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-12T16:54:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "membership problem",
      "short continued fraction representation",
      "positive integers",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}