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

Vasily A. Dolgushev; Jacob J. Guynee

paper

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

arXiv:2401.06870

Abstract

Let $B_{3}$ be the Artin braid group on 3 strands and $P B_{3}$ be the corresponding pure braid group. In this paper, we construct the groupoid $GT S h$ of GT-shadows for a (possibly more tractable) version $G T_{0}$ 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 $G T_{g e n}$ . The objects of $GT S h$ are finite index normal subgroups $N$ of $B_{3}$ satisfying the condition $N \subset P B_{3}$ . Morphisms of $GT S h$ are called GT-shadows and they may be thought of as approximations to elements of $G T_{g e n}$ . We show how GT-shadows can be obtained from elements of $G T_{g e n}$ and prove that $G T_{g e n}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GT S h$ . Using this result, we get a criterion for identifying genuine GT-shadows.