论文标题
什么是GT-Shadows？
What are GT-shadows?
论文作者
论文摘要
令$ b_4 $（分别为$ pb_4 $）为4个链的辫子组（纯辫子组），$ nfi_ {pb_4}（b_4）$是poset，其对象是$ pb_4 $中包含的$ b_4 $的对象的$ b_4 $的poset。在本文中，我们介绍了GT-Shadows，可以将其视为“近似值”，以介绍Grothendieck-Teichmueller Group的Profinite版本$ \ widehat {gt} $（参见V. Drinfeld，Elgebra I Analiz，1990年）。我们证明gt-shadows形成了一个groupoid，其对象是$ nfi_ {pb_4}（b_4）$的元素。我们表明，来自$ \ widehat {gt} $元素的gt-shadows满足了各种其他属性，我们研究了这些属性。我们在GT-Shadows和Group $ \ wideHat {gt} $之间建立了一个明确的链接（请参阅Theorem 3.8）。我们还介绍了GT-Shadows上计算机实验的选定结果。在本文的附录中，我们对Abelian环境中的GT-Shadows进行了完整的描述。我们还证明，在Abelian设置中，每个GT阴影都来自$ \ wideHat {gt} $的元素。 D. Harbater和L. Schneps在1997年的一篇论文中引入了与GT-Shadows非常相似的对象。
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.
