GT-shadows for the gentle version of the Grothendieck-Teichmueller group

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.