Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility

\textit{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 \textit{Wada dessins}. Their underlying graph illustrates the incidence structure of finite projective spaces $\PR{m}{n}$. Usually, the automorphism group of these dessins is a cyclic \textit{Singer group} $\Sigma_\ell$ permuting transitively the vertices. However, in some cases, a second group of automorphisms $\Phi_f$ exists. It is a cyclic group generated by the \textit{Frobenius automorphism}. We show under what conditions $\Phi_f$ is a group of automorphisms acting freely on the edges of the considered dessins.