Revista Matemática Iberoamericana

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Volume 31, Issue 4, 2015, pp. 1403–1414
DOI: 10.4171/RMI/873

Published online: 2015-12-23

Symmetries of quasiplatonic Riemann surfaces

Gareth A. Jones[1], David Singerman[2] and Paul D. Watson[3]

(1) University of Southampton, UK
(2) University of Southampton, UK
(3) Peter Symonds College, Winchester, UK

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.

Keywords: Riemann surface, symmetry, triangle group, hypermap

Jones Gareth, Singerman David, Watson Paul: Symmetries of quasiplatonic Riemann surfaces. Rev. Mat. Iberoamericana 31 (2015), 1403-1414. doi: 10.4171/RMI/873