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What are GT–shadows?

Vasily A Dolgushev, Khanh Q Le and Aidan A Lorenz

Algebraic & Geometric Topology 24 (2024) 2721–2777
Abstract

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).

Keywords
Grothendieck–Teichmueller group, operads, braid groups, the absolute Galois group of rational numbers
Mathematical Subject Classification
Primary: 14F35, 14H30, 18M60
Secondary: 14H57
References
Publication
Received: 25 July 2022
Revised: 8 April 2023
Accepted: 13 June 2023
Published: 19 August 2024
Authors
Vasily A Dolgushev
Department of Mathematics
Temple University
Philadelphia, PA
United States
Khanh Q Le
Department of Mathematics
Rice University
Houston, TX
United States
Aidan A Lorenz
Department of Mathematics
Vanderbilt University
Nashville, TN
United States

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