Moscow Mathematical Journal

Volume 18, Issue 3, July–September 2018 pp. 421–436.

The Groups Generated by Maximal Sets of Symmetries of Riemann Surfaces and Extremal Quantities of their Ovals

Authors: Grzegorz Gromadzki (1) and Ewa Kozłowska-Walania (1)
Author institution:(1) Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Summary:

Given g ≥ 2, there are formulas for the maximal number of non-conjugate symmetries of a Riemann surface of genus g and the maximal number of ovals for a given number of symmetries. Here we describe the algebraic structure of the automorphism groups of Riemann surfaces, supporting such extremal configurations of symmetries, showing that they are direct products of a dihedral group and some number of cyclic groups of order 2. This allows us to establish a deeper relation between the mentioned above quantitative (the number of symmetries) and qualitative (configurations of ovals) cases.

2010 Math. Subj. Class. Primary: 30F99; Secondary: 14H37, 20F.

Keywords: Automorphisms of Riemann surfaces, symmetric Riemann surfaces, real forms of complex algebraic curves, Fuchsian and NEC groups, ovals of symmetries of Riemann surfaces, separability of symmetries, Harnack-Weichold conditions.

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