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Erschienen in:
2016 | OriginalPaper | Buchkapitel
8. Wilson Operations
verfasst von : Gareth A. Jones, Jürgen Wolfart
Erschienen in: Dessins d'Enfants on Riemann Surfaces
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Abstract
As shown in Chap. 3, a dessin uniquely determines all the relevant properties of its underlying Belyĭ surface. A key invariant is the moduli field of the corresponding algebraic curve, and a major step in determining this is to understand the action of the absolute Galois group \(\mathbb{G}\) on regular dessins. Under relatively mild conditions this action can be described combinatorially with some map and hypermap operations, the so-called Wilson (hole) operations, introduced around the same time as Belyĭ functions. However, their role in the understanding of Galois actions on dessins has only recently been discovered. We give several examples, based on the regular embeddings of complete graphs classified in Chap. 7 In the final section we consider the group of all operations on dessins, introduced by James, showing that it is isomorphic to the outer automorphism group of the free group of rank 2, and hence to \(\mathop{\mathrm{GL}}\nolimits _{2}(\mathbb{Z})\). As an example we consider the action of this group on the 19 regular dessins with automorphism group A
5.