JournalsrmiVol. 24, No. 3pp. 921–939

Reflections of regular maps and Riemann surfaces

  • Adnan Melekoğlu

    Adnan Menderes University, Aydin, Turkey
  • David Singerman

    University of Southampton, UK
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Abstract

A compact Riemann surface of genus gg is called an M-surface if it admits an anti-conformal involution that fixes g+1g+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 gϯ1g ϯ 1 there is a unique M-surface of genus gg that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix gg curves.

Cite this article

Adnan Melekoğlu, David Singerman, Reflections of regular maps and Riemann surfaces. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 921–939

DOI 10.4171/RMI/560