Wada dessins associated with finite projective spaces and Frobenius compatibility - Sarti, Cristina (Ars mathematica contemporanea, 2011)
Dessins d'enfants (hypermaps) are useful to describe algebraic properties of the Riemann surfaces they are embedded in. In general, it is not easy to describe algebraic properties of the surface of the embedding starting from the combinatorial properties of an embedded dessin. However, this task becomes easier if the dessin has a large automorphism group. In this paper we consider a special type of dessins, so-called Wada dessins. Their underlying graph illustrates the incidence structure of finite projective spaces ?$\Bbb P^m (\Bbb F_n)$?. Usually, the automorphism group of these dessins is a cyclic Singer group ?$Σ _\ell$? permuting transitively the vertices. However, in some cases, a second group of automorphisms ?$Φ_f$? exists. It is a cyclic group generated by the Frobenius automorphism. We show under what conditions ?$Φ_f$? is a group of automorphisms acting freely on the edges of the considered dessins.
Keywords: bipartite graphs; dessins d'enfants; difference sets; dvodelni grafi; finite geometries; Frobenius automorphism; Frobeniusov automorfizem; graph embeddings; končne geometrije; Riemann surfaces; Riemannove ploskve; Singer groups; Singerjeve grupe; vložitve grafov; Wada dessins