Glasnik Matematicki, Vol. 60, No. 2 (2025), 267-290. \( \)

REAL EQUATIONS FOR \(o\)–EXTREMAL RIEMANN SURFACES WITH ABELIAN AUTOMORPHISM GROUPS

Ewa Kozłowska-Walania and Peter Turbek

Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
e-mail:retrakt@mat.ug.edu.pl

Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street, Hammond, Indiana, 46323
e-mail:psturbek@pnw.edu

Abstract.   It is well known that the fixed point set of a Riemann surface of genus \(g\) under the action of a symmetry is either empty or consists of a disjoint set of at most \(g+1\) ovals. Bounds on the total number of fixed ovals given by a set of \(k\) non-conjugate symmetries are known. In this paper, for \(k \ge 4\), we calculate all the possible topological types of symmetries in such a maximal configuration, provided that the symmetries commute. We also find real equations for the Riemann surfaces that achieve these bounds where the symmetries are expressed as complex conjugation.

2020 Mathematics Subject Classification.   30F99, 14H37, 20F

Key words and phrases.   Riemann surface, symmetry of a Riemann surface, real form, automorphisms of Riemann surface, equations for Riemann surfaces, Fuchsian groups, Riemann uniformization theorem

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https://doi.org/10.3336/gm.60.2.06

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