Abstract
Let B3 be the Artin braid group on 3 strands and PB3 be the corresponding pure braid group. In this paper, we construct the groupoid GTSh of GT-shadows for a (possibly more tractable) version GTˆ0 of the Grothendieck-Teichmueller group GTˆ introduced in paper [12] by D. Harbater and L. Schneps. We call this group the gentle version of GTˆ and denote it by GTˆgen. The objects of GTSh are finite index normal subgroups N of B3 satisfying the condition N≤PB3. Morphisms of GTSh are called GT-shadows and they may be thought of as approximations to elements of GTˆgen. We show how GT-shadows can be obtained from elements of GTˆgen and prove that GTˆgen 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.
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