Revista Matemática Iberoamericana


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Volume 24, Issue 3, 2008, pp. 921–939
DOI: 10.4171/RMI/560

Published online: 2008-12-31

Reflections of regular maps and Riemann surfaces

Adnan Melekoğlu[1] and David Singerman[2]

(1) Adnan Menderes University, Aydin, Turkey
(2) University of Southampton, UK

A compact Riemann surface of genus $g$ is called an M-surface if it admits an anti-conformal involution that fixes $g+1$ simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus $g ϯ 1$ there is a unique M-surface of genus $g$ that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix $g$ curves.

Keywords: Regular map, Riemann surface, Platonic surface, M-surface, (M−1)-surface

Melekoğlu Adnan, Singerman David: Reflections of regular maps and Riemann surfaces. Rev. Mat. Iberoamericana 24 (2008), 921-939. doi: 10.4171/RMI/560