# 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.

Contents Full-Text PDF