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 $\widehat{GT}$ 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.