What are GT–shadows ?

Let B 4 (resp.PB 4 ) be the braid group (resp.the pure braid group) on 4 strands and NFI PB 4 .B 4 / be the poset whose elements are finite-index normal subgroups N of B 4 that are contained in PB 4 .We introduce GT-shadows, which may be thought of as "approximations" to elements of the profinite version c GT of the Grothendieck-Teichmüller group (Drinfeld 1991).We prove that GT-shadows form a groupoid whose objects are elements of the underlying set NFI PB 4 .B 4 /.GT-shadows coming from elements of c GT satisfy various additional properties and we investigate these properties.We establish an explicit link between GT-shadows and the group c GT. Selected results of computer experiments on GT-shadows are presented.In the appendix we give a complete description of GT-shadows in the abelian setting.We also prove that, in the abelian setting, every GT-shadow comes from an element of c GT. Objects very similar to GT-shadows were introduced by D Harbater and L Schneps (1997).A variation of the concept of GT-shadows for the gentle version of c GT was studied by P Guillot (2016 and 2018).14F35, 14H30, 18M60; 14H57 1. Introduction 2721 2. GT-pairs and GT-shadows 2728 3. The Main Line functor ᏹᏸ and c GT 2750 4. Selected results of computer experiments 2760 Appendix A. The operad PaB and its profinite completion 2766 Appendix B. Charming GT-shadows in the abelian setting 2772 References 27751 We tacitly assume that our automorphisms act as identity on objects.