Cyclic projective planes and Wada dessins (Q5931106)

A dessin on an orientable compact surface \(X\) is a bipartite graph \(\mathcal D\) embedded into \(X\) such that the complement of \(\mathcal D\) is the disjoint union of simply connected domains. If every vertex of \(\mathcal D\) lies on the border of every cell of \(\mathcal D\) then \(\mathcal D\) is called a Wada dessin. Given a dessin \(\mathcal D\) on \(X\), one naturally can provide \(X\) with the structure of an algebraic curve defined over some number field. There exists a meromorphic Belyi function \(\beta\) on \(X\), ramified at most above \(0,1, \infty\), and the open vertices of \(\mathcal D\) are the connected components of \(\beta^{-1}(0,1)\). NEWLINENEWLINENEWLINEThe authors investigate the case that \(\mathcal D\) arises from the incidence graph of a cyclic projective plane \(\mathbb P\) and that the Singer group \(Z_l\) induces automorphisms of \(\mathcal D\). It turns out that all important properties of \(\mathcal D\) depend on cyclic orderings of the difference set for \(\mathbb P\). In particular, if the order of the Singer group \(l\) is a prime, then the incidence graph of \(\mathbb P\) admits an embedding as a Wada dessin, and there is some evidence that the same holds true for general \(l \not= 21\).