We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck’s dessins d’enfants. We deﬁne a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of “new-type” relations on the injection of the absolute Galois group into the Grothendieck–Teichmüller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We give an example of a new invariant differing on two dessins which have the same values for the other readily computable invariants.
Cite this article
Melanie Matchett Wood, Belyi-Extending Maps and the Galois Action on Dessins d’Enfants. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, pp. 721–737