Symmetries of Riemann surfaces from a combinatorial point of view (Q2741166)

A symmetric Riemann surface \(X\) of genus \(g > 1\) corresponds to a complex curve \(C_{X}\) which can be defined over the reals and symmetries nonconjugated in the group \(\text{Aut}^{\pm}(X)\) of all automorphisms of \(X\) correspond to nonequivalent real forms of \(C_{X}.\) The aim of this lecture is a brief introduction to the combinatorial aspects of this theory together with samples of results and proofs.NEWLINENEWLINEThe natural questions that arise here are the following: 1) Does a Riemann surface \(X\) admit a symmetry? 2) How many nonconjugated symmetries may a given Riemann surface \(X\) admit? 3) What can one say about the topology o these symmetries, e. g. about the number of their ovals or about their separability?NEWLINENEWLINEContents of this paper: 1. On combinatorial approach in studies of automorphisms of Riemann surfaces. 2. Symmetries, symmetric Riemann surfaces and their groups of automorphisms. 3. Subgroups of NEC groups. 4. Is a Riemann surface symmetric ? 5. Quantitative studies of symmetries of Riemann surfaces. 6. Qualitative studies of symmetries of Riemann surfaces.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00037].