Revista Matemática Iberoamericana

Reflections of regular maps and Riemann surfaces

Adnan Melekoğlu and David Singerman

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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.

Article information

Rev. Mat. Iberoamericana, Volume 24, Number 3 (2008), 921-939.

First available in Project Euclid: 9 December 2008

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Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]

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), no. 3, 921--939.

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