Belyi-extending maps and the Galois action on dessins d'enfants

Details Belyi-extending maps and the Galois action on dessins d'enfants Belyi-extending maps and the Galois action on dessins d'enfants

2003-04-30

arxiv.org/abs/math/0304489v2

Publications of the Research Institute for Mathematical Sciences, 42 (2006), no. 3, 721--738

Mathematics

Number Theory

13 pages, 7 figures; submitted for publication; revisions are that the paper now deals only with Galois invariants of dessins,

Scientific paper

10.2977/prims/1166642157

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 the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller 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.

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