Wada dessins associated with finite projective spaces and Frobenius compatibility

Cristina Sarti

Abstract


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 Pm(Fn). Usually, the automorphism group of these dessins is a cyclic Singer group Σl permuting transitively the vertices. However, in some cases, a second group of automorphisms Φf exists. It is a cyclic group generated by the Frobenius automorphism. We show under which conditions Φf is a group of automorphisms acting freely on the edges of the considered dessins.


Keywords


Dessins d'enfants, Wada dessins, bipartite graphs, graph embeddings, difference sets, finite geometries, Riemann surfaces, Frobenius automorphism, Singer groups

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DOI: https://doi.org/10.26493/1855-3974.122.404

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications