{
  "id": "2307.16284",
  "version": "v1",
  "published": "2023-07-30T17:41:10.000Z",
  "updated": "2023-07-30T17:41:10.000Z",
  "title": "Arboreal Galois groups for quadratic rational functions with colliding critical points",
  "authors": [
    "Robert L. Benedetto",
    "Anna Dietrich"
  ],
  "comment": "28 pages, 4 figures",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be a field, and let $f\\in K(z)$ be rational function. The preimages of a point $x_0\\in P^1(K)$ under iterates of $f$ have a natural tree structure. As a result, the Galois group of the resulting field extensions of $K$ naturally embed into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup $M_{\\ell}$ that this so-called arboreal Galois group $G_{\\infty}$ must lie in if $f$ is quadratic and its two critical points collide at the $\\ell$-th iteration. After presenting a new description of $M_{\\ell}$ and a new proof of Pink's theorem, we state and prove necessary and sufficient conditions for $G_{\\infty}$ to be the full group $M_{\\ell}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-30T17:41:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "11R32",
      "14G25"
    ],
    "keywords": [
      "arboreal galois group",
      "quadratic rational functions",
      "colliding critical points",
      "natural tree structure",
      "th iteration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}