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

Abstract

Let $B_3$ be the Artin braid group on 3 strands and ${\mathrm{PB}}_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $\mathrm{GTSh}$ of $\mathrm{GT}$-shadows for a (possibly more tractable) version ${\widehat{\mathrm{GT}}}_0$ of the Grothendieck-Teichmueller group $\widehat{\mathrm{GT}}$ introduced in paper [12] by D. Harbater and L. Schneps. We call this group the gentle version of $\widehat{\mathrm{GT}}$ and denote it by ${\widehat{\mathrm{GT}}}_{gen}$. The objects of $\mathrm{GTSh}$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N\le {\mathrm{PB}}_3$. Morphisms of $\mathrm{GTSh}$ are called $\mathrm{GT}$-shadows and they may be thought of as approximations to elements of ${\widehat{\mathrm{GT}}}_{gen}$. We show how $\mathrm{GT}$-shadows can be obtained from elements of ${\widehat{\mathrm{GT}}}_{gen}$ and prove that ${\widehat{\mathrm{GT}}}_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $\mathrm{GTSh}$. Using this result, we get a criterion for identifying genuine $\mathrm{GT}$-shadows.