David Singerman

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In this work we explore the relationship between dessins and Riemann surfaces. From the work of Belyi and Grothendieck [1], [8], Jones/Singerman [10, 11], or Malgoire/Voisin [16], it follows that underlying a dessin H on a surface X, there is a canonical complex structure on X. This makes X into a Riemann surface R(H). In general, it is difficult to study(More)
Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of(More)
Macbeath gave a formula for the number of xed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the xed-point set of each non-identity element of a cyclic group of automorphisms(More)
This paper explains some facts probably known to experts and implicitely contained in the literature about dessins d’enfants but which seem to be nowhere explicitely stated. The 1skeleton of every regular Cori hypermap is the Cayley graph of its automorphism group, embedded in the underlying orientable surface. Conversely, every Cayley graph of a finite(More)