The action of GT-shadows on child's drawings

Abstract

$\mathrm{GT}$-shadows [8] are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{\mathrm{GT}}$ introduced by V. Drinfeld in 1990. $\mathrm{GT}$-shadows form a groupoid $\mathrm{GTSh}$ whose objects are finite index subgroups of the pure braid group ${\mathrm{PB}}_4$, that are normal in $B_4$. The goal of this paper is to describe the action of $\mathrm{GT}$-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{\mathrm{GT}}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of $\mathrm{GTSh}$, $\widehat{\mathrm{GT}}$, 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 ${\mathrm{GTSh}}^{♡}$ of charming $\mathrm{GT}$-shadows. We use the action of $\mathrm{GT}$-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over $Q$. Finally, we describe selected examples of non-Abelian child's drawings.