Symmetries of quasiplatonic Riemann surfaces

Abstract

We state and prove a corrected version of a theorem of Singerman, which relates the existence of symmetries (anticonformal involutions) of a quasiplatonic Riemann surface $\mathcal{S}$ (one uniformised by a normal subgroup $N$ of finite index in a cocompact triangle group $\Delta$) to the properties of the group $G=\Delta /N$. We give examples to illustrate the revised necessary and sufficient conditions for the existence of symmetries, and we relate them to properties of the associated dessins d'enfants, or hypermaps.