What are GT–shadows ?

Dolgushev, Vasily A¹ ; Le, Khanh Q² ; Lorenz, Aidan A³

¹ Department of Mathematics, Temple University, Philadelphia, PA, United States
² Department of Mathematics, Rice University, Houston, TX, United States
³ Department of Mathematics, Vanderbilt University, Nashville, TN, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 5, pp. 2721-2777

Résumé

Let B ⁡ 4 (resp.  PB ⁡ 4) be the braid group (resp. the pure braid group) on 4 strands and NFIPB ⁡ 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 GT^ of the Grothendieck–Teichmüller group (Drinfeld 1991). We prove that GT–shadows form a groupoid whose objects are elements of the underlying set NFIPB ⁡ 4(B ⁡ 4). GT–shadows coming from elements of GT^ satisfy various additional properties and we investigate these properties. We establish an explicit link between GT–shadows and the group 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 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 GT^ was studied by P Guillot (2016 and 2018).

Détail

DOI : 10.2140/agt.2024.24.2721

Keywords: Grothendieck–Teichmueller group, operads, braid groups, the absolute Galois group of rational numbers