Aaron Wootton, University of Portland

S. Allen Broughton, Rose-Hulman Institute of
Technology

May 24, 2010

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* -> P^{1}(**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.

- Slides of the talk (PDF file 599 K)