On combinatorial approach in studies of automorphisms of Riemann surfaces

Throughout the lecture a Riemann surface is meant to be a compact Riemann surface of genus g ≥ 2 and a symmetry of such surface, an antiholomorphic involution. The reader who has attended the lecture must be convinced of the importance of studies of symmetries of Riemann surfaces and the role that the groups of their automorphisms play there. Up to certain extent that lecture illustrates how to use results concerning this subject and in this one we shall mainly show how to get them. A symmetric Riemann surface X corresponds to a complex curve CX which can be defined over the reals and symmetries nonconjugated in the group Aut±(X) of all automorphisms of X correspond to nonequivalent real forms of CX. The aim of this lecture is a brief introduction to the combinatorial aspects of this theory together with samples of results and proofs. The most natural questions that arise here are the following:

• does a Riemann surface X admit a symmetry?

• how many nonconjugated symmetries may a given Riemann surface X admit?

• what can one say about the topology of these symmetries, e.g. about the number of their ovals or about their separability?

Riemann surfaces form a category with the holomorphic maps as morphisms and this category is closed under the quotients with respect to the action of groups of holomorphic automorphisms.