{
  "id": "2401.06870",
  "version": "v1",
  "published": "2024-01-12T19:52:40.000Z",
  "updated": "2024-01-12T19:52:40.000Z",
  "title": "GT-shadows for the gentle version of the Grothendieck-Teichmueller group",
  "authors": [
    "Vasily A. Dolgushev",
    "Jacob J. Guynee"
  ],
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "Let $B_3$ be the Artin braid group on 3 strands and $PB_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $GTSh$ of GT-shadows for a (possibly more tractable) version $GT_0$ of the Grothendieck-Teichmueller group $GT$ introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of $GT$ and denote it by $GT_{gen}$. The objects of $GTSh$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N \\subset PB_3$. Morphisms of $GTSh$ are called GT-shadows and they may be thought of as approximations to elements of $GT_{gen}$. We show how GT-shadows can be obtained from elements of $GT_{gen}$ and prove that $GT_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GTSh$. Using this result, we get a criterion for identifying genuine GT-shadows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-12T19:52:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grothendieck-teichmueller group",
      "gentle version",
      "finite index normal subgroups",
      "artin braid group",
      "corresponding pure braid group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}