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The Action of GT-Shadows on Child's Drawings

Vasily A. Dolgushev

Abstract

GT-shadows are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{GT}$ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB_4, that are normal in B_4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{GT}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, $\widehat{GT}$, and the absolute Galois group G_Q of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over rationals. Finally, we describe selected examples of non-Abelian child's drawings.

The Action of GT-Shadows on Child's Drawings

Abstract

GT-shadows are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB_4, that are normal in B_4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of . We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, , and the absolute Galois group G_Q of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over rationals. Finally, we describe selected examples of non-Abelian child's drawings.

Paper Structure

This paper contains 19 sections, 23 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Background material
  4. Reminder of $\mathsf{GT}$-shadows
  5. $\widehat{\mathsf{GT}}$ versus the groupoid $\mathsf{GTSh}$
  6. Permutation pairs, permutation triples and child's drawings
  7. Representing child's drawings by group homomorphisms
  8. The action of $\widehat{\mathsf{GT}}$ on child's drawings and the functor $\mathcal{A}: \widehat{\mathsf{GT}}_{\mathsf{NFI}} \to \mathsf{Dessin}$
  9. The action of $\mathsf{GT}$-shadows on child's drawings
  10. The hierarchy of orbits
  11. The monodromy group and the passport are invariant with respect to the action of $\mathsf{GT}$-shadows
  12. Abelian child's drawings
  13. $\mathsf{GTSh}$-orbit of an Abelian child's drawing is a singleton
  14. Examples of $\mathsf{GTSh}^{\heartsuit}$-orbits for (non-Abelian) child's drawings
  15. An example of degree $6$, genus $0$ and orbit size $2$
  16. ...and 4 more sections

Key Result

Proposition 2.1

For every $\mathsf{N} \in \mathsf{NFI}_{\mathrm{PB}_4}(\mathrm{B}_4)$ and every $[m,f] \in \mathsf{GT}(\mathsf{N})$, the group homomorphism T-F-2 is onto.

Figures (1)

  • Figure 1.1: The isomorphisms ${\alpha}$ and $\beta$

Theorems & Definitions (42)

  • Remark 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 32 more