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

Abstract

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.

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

Cited by in corpus (8)

• Dessins d'Enfants in $\mathcal{N}=2$ Generalised Quiver Theories
• On computing Belyi maps
• Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison
• Quantum Statistical Mechanics of the Absolute Galois Group
• Three arithmetic sites
• A new $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant of dessins d'enfants
• Shabat Polynomials and Monodromy Groups of Trees Uniquely Determined by their Passport
• On the rank of the Thurston pullback map