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arXiv · 2008.00066

What are GT-shadows?

Vasily A. Dolgushev·Khanh Q. Le·Aidan A. Lorenz
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Abstract

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 objects are finite index normal subgroups of $B_4$ that are contained in $PB_4$. In this paper, we introduce GT-shadows which may be thought of as "approximations" to elements of the profinite version $\widehat{GT}$ of the Grothendieck-Teichmueller group (see V. Drinfeld, Algebra i Analiz, 1990). We prove that GT-shadows form a groupoid whose objects are elements of $NFI_{PB_4}(B_4)$. We show that GT-shadows coming from elements of $\widehat{GT}$ satisfy various additional properties and we investigate these properties. We establish an explicit link between GT-shadows and the group $\widehat{GT}$ (see Theorem 3.8). We also present selected results of computer experiments on GT-shadows. In the appendix of this paper, 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 $\widehat{GT}$. Objects very similar to GT-shadows were introduced in a paper by D. Harbater and L. Schneps in 1997. A variation of the concept of $GT$-shadows for the coarse version of $\widehat{GT}$ was studied in papers by P. Guillot.

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Vasily A. Dolgushev, Khanh Q. Le, Aidan A. Lorenz. 2021-10-31. What are GT-shadows?. https://doi.org/10.2140/agt.2024.24.2721

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