Galois actions on regular dessins and Fuchsian group covers

Aaron Wootton, University of Portland
S. Allen Broughton, Rose-Hulman Institute of Technology

May 24, 2010

Conference on Riemann Surfaces and Dessins d’Enfants
On the Occasion of Jürgen Wolfart’s 65th Birthday

Title: Galois actions on regular dessins and Fuchsian group covers

Abstract: It is well known that a regular Belyi function B : S -> P1(C) on a surface S determines a regular dessin D on S and a realization of the surface as quotient S = H/U of the hyperbolic plane. The group U is a normal, torsion free subgroup of a triangle group T. The group G = T/U is a group of automorphisms of S, specifically the group of covering transformation of B. Also, the surface S has defining equations with coefficients in a number field K. An element of the absolute Galois group determines a Galois conjugate surface S' by acting on the coefficients of a defining equation of S. There is an associated dessin D', and subgroup U' contained in T such that S' = H/U'. The goal of this talk is to establish as much as possible about D' and U' from the knowledge of the pair T,U. In certain well-known examples the dessin D' is obtained by Wilson operations. Here we also consider construction of the dessin when Wilson operations are not valid. We illustrate the methods with low genus surfaces and cyclic n-gonal surfaces.

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