Geometric Galois Actions: The Grothendieck-Teichmüller group GT : a survey
Geometric Galois Actions: The Grothendieck-Teichmüller group GT : a survey
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几何伽罗瓦动作：Grothendieck-Teichmüller 群 GT：一项调查
DOI：
10.1017/cbo9780511758874.014
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Leila Schneps
中科院分区：
文献类型：
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作者：
Leila Schneps
在这个简短的调查中，我们回顾了 Grothendieck-Teichmüller 群 ĜT 的精确版本的定义。 Grothendieck-Teichmüller群最初是由Drinfel'd在[D]中定义的（碰巧不是在有限的情况下，而是在离散的、亲`和k-亲单能的情况下；这些定义都是类似的）。我们在迄今为止已知的精确版本上陈述了大部分结果，并给出了所有证明的参考。 （应该注意的是，有几个深刻的结果和猜想仅涉及 Grothendieck-Teichmüller 群的专业版本；我们在这里不讨论它们。）ĜT 最有趣的方面之一是它包含绝对伽罗瓦群 Gal(Q/Q) 作为子群，其方式与两个群在算术几何对象上的自然行为兼容，例如在 Q 上定义的某些覆盖和在 Q 上定义的簇的基本群；这两个群实际上是同构的可能性是一个令人兴奋的可能性，它提出了一种考虑 Gal(Q/Q) 的新方法。它反映了格洛腾迪克在 Esquisse d’un 计划中的建议，即通过研究绝对伽罗瓦群 Gal(Q/Q) 在他所谓的 Teichmüller 塔上的自然作用来研究绝对伽罗瓦群 Gal(Q/Q) 的组合性质，描述如下：
In this short survey, we recall the definition of the profinite version of the Grothendieck-Teichmüller group ĜT . The Grothendieck-Teichmüller group was originally defined by Drinfel’d in [D] (as it happens, not in the profinite case, but in the discrete, pro-` and k-pro-unipotent cases; these definition are all analogous). We state most of the results on the profinite version ĜT known to date, and give references to all the proofs. (It should be noted that there are several deep results and conjectures concerning only the pro` version of the Grothendieck-Teichmüller group; we do not discuss them here.) One of the most interesting aspects of ĜT is the fact that it contains the absolute Galois group Gal(Q/Q) as a subgroup in a way compatible with natural actions of both groups on arithmetico-geometric objects such as certain covers defined over Q and fundamental groups of varieties defined over Q; the possibility that these two groups are actually isomorphic is an exciting one, suggesting a new way of considering Gal(Q/Q). It reflects Grothendieck’s suggestion in the Esquisse d’un Programme of studying the combinatorial properties of the absolute Galois group Gal(Q/Q) by studying its natural action on what he calls the Teichmüller tower, described as follows: