Wilson's graph operations on Wada dessins

Cristina Sarti (University of Frankfurt, Germany)

Dessins d'enfants may be defined as bipartite graphs embedded in Riemann's surfaces. They determine in a unique way the conformal and algebraic structure of the surface of the embedding. Knowing the combinatorial properties of the embedded dessin may help describing the surface e.g. in term of defining equations. This task is easier if the embedded dessin has a 'large' automorphism group. In this talk, we present new results concerning the construction of dessins for which the underlying graph illustrates the incidence structure of points and hyperplanes of projective spaces over finite fields. In particular, we concentrate on a special type of dessins so-called Wada dessins which under some conditions have a 'large' orientation-preserving automorphism group. We show that applying to them algebraic operations we call 'mock' Wilson operations we may obtain new dessins. We consider the automorphism group of the new dessins and we explain how they are topologically related to the dessins we started with.