Aaron Wootton, University of Portland
S. Allen Broughton, Rose-Hulman Institute of
Technology
May 24, 2010
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.