Algebraic & Geometric Topology Volume 24, issue 5 (2024)

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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

DOI: 10.2140/agt.2024.24.2721

Abstract

Let $B_{4}$ (resp. ${PB}_{4}$ ) be the braid group (resp. the pure braid group) on $4$ strands and ${N F I}_{{PB}_{4}} (B_{4})$ be the poset whose elements are finite-index normal subgroups $N$ of $B_{4}$ that are contained in ${PB}_{4}$ . We introduce $G T$ –shadows, which may be thought of as “approximations” to elements of the profinite version $\hat{G T}$ of the Grothendieck–Teichmüller group (Drinfeld 1991). We prove that $G T$ –shadows form a groupoid whose objects are elements of the underlying set ${N F I}_{{PB}_{4}} (B_{4})$ . $G T$ –shadows coming from elements of $\hat{G T}$ satisfy various additional properties and we investigate these properties. We establish an explicit link between $G T$ –shadows and the group $\hat{G T}$ . Selected results of computer experiments on $G T$ –shadows are presented. In the appendix we give a complete description of $G T$ –shadows in the abelian setting. We also prove that, in the abelian setting, every $G T$ –shadow comes from an element of $\hat{G T}$ . Objects very similar to $G T$ –shadows were introduced by D Harbater and L Schneps (1997). A variation of the concept of $G T$ –shadows for the gentle version of $\hat{G T}$ 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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