{
  "id": "2405.11725",
  "version": "v1",
  "published": "2024-05-20T02:04:35.000Z",
  "updated": "2024-05-20T02:04:35.000Z",
  "title": "First examples of non-abelian quotients of the Grothendieck-Teichmueller group that receive surjective homomorphisms from the absolute Galois group of rational numbers",
  "authors": [
    "Ivan Bortnovskyi",
    "Vasily A. Dolgushev",
    "Borys Holikov",
    "Vadym Pashkovskyi"
  ],
  "comment": "comments are welcome",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element $K$ of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to $K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$ in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-20T02:04:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "absolute galois group",
      "receive surjective homomorphisms",
      "grothendieck-teichmueller group",
      "rational numbers",
      "non-abelian quotients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}