Abstract
A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.
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Acknowledgements
The authors are very grateful to the referee for his (her) vigilant reading of the paper, very helpful comments, for pointing out an error in the first version of Theorem 6.1 and remarking some inaccuracies still in the revised version. Our gratitude is also directed to Professor Ernst-Ulrich Gekeler, the Managing Editor, for pointing out linguistic defects of the paper, for allowing its resubmission and finally to Aaron Wootton for the reading of the final version of the paper and linguistic improvements.
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C. Baginski was supported by the grant S/W I /3/2008 of Bialystok University of Technology, Bialystok, Poland.
G. Gromadzki was supported by the Research Grant N N201 366436 of the Polish Ministry of Sciences and Higher Education.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bagiński, C., Gromadzki, G. Minimal genus problem for pseudo-real Riemann surfaces. Arch. Math. 95, 481–492 (2010). https://doi.org/10.1007/s00013-010-0186-1
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DOI: https://doi.org/10.1007/s00013-010-0186-1