Abstract

This paper is a survey on automorphism groups of compact unbordered non-orientable surfaces with dianalytic structures, called non-orientable Riemann surfaces. We deal with different aspects of groups acting on these surfaces, as the maximum order and minimum genus problems, the determination of the symmetric crosscap number of a finite group and the fixed point set of an automorphism. The guiding theme linking the sections of this survey comes from the Riemann uniformization theorem, which relates these surfaces to the combinatorial group theory of non-Euclidean crystallographic groups. Some other areas are also related to these surfaces, amongst which we mention real algebraic curves.

Introduction

It is easy to see that a classical analytic structure existing on orientable topological surfaces cannot exist on non-orientable ones. One can, however, relax a bit the notion of analyticity of a structure allowing the complex conjugation to be involved for transition functions between charts with overlapping domains. Such structures, called dianalytic, can exist on non-orientable closed surfaces. Sometimes they are also called conformal – motivated by the fact that the transition functions are angle-preserving but not necessarily orientation-preserving – while usually for classical structures of Riemann surfaces the term analytic is reserved. This way we obtain the notion of unbordered non-orientable Klein surface. These surfaces were called non-orientable Riemann surfaces by Singerman in [35], and we will keep here this convention. Similarly, a self-homeomorphism of such a surface is said to be an automorphism if its local forms are either analytic or the composite of complex conjugation with an analytic function.

The topic we present here is at the intersection of several important disciplines of mathematics: theory of functions of one complex variable (classical Riemann surfaces), algebraic geometry (algebraic curves), low-dimensional topology (mapping class groups, three-dimensional hyperbolic manifolds), Galois theory (automorphisms of algebraic function fields in one variable), differential geometry (moduli spaces, complex surfaces), combinatorics (regular maps on surfaces) and combinatorial group theory. This last is the main tool used to prove the results we present here, and that makes this survey suitable to this volume.

Abstract

Every finite group acts as a group of automorphisms of some compact bordered Klein surface of algebraic genus g ≥ 2. The same group G may act on different genera and so it is natural to look for the minimum genus on which G acts. This is the minimum genus problem for the group G. On the other hand, for a fixed integer g ≥ 2, there are finitely many abstract groups acting as a group of automorphisms of some compact bordered Klein surface of algebraic genus g. The condition g ≥ 2 assures that all such groups are finite. So it makes sense to look for the largest order of groups G acting on some surface of genus g when g is fixed and G runs over a prescribed family F of groups. This is the maximum order problem for the family F. There is a significant amount of research dealing with these two problems (or with some of their variations), and the corresponding results are scattered in the literature. The purpose of this survey is to gather some of these results, paying special attention to important families of finite groups.

Introduction

A natural extension of the definition of a compact Riemann surface, which is orientable and has no boundary, is to allow dianalytic transition functions, that is, functions which are either analytic or the composite of complex conjugation with an analytic function.

Abstract

In 1966 Harvey found necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a smooth epimorphism onto a cyclic group. As a consequence, he solved the minimum genus and the maximum order problems for the family of cyclic groups (the latter already solved by Wiman). This was a seminal work in the study of groups of automorphisms of compact Riemann surfaces. Since then, much research has been conducted to extend Harvey's theorem to other classes of finite groups and, in particular, to solve the minimum genus and maximum order problems for these classes. In this survey we present some results obtained so far.

Introduction

In 1966 Bill Harvey published Cyclic groups of automorphisms of a compact Riemann surface in the Quarterly Journal of Mathematics, [Har]. The main result in it gives necessary and suficient conditions on the signature of a Fuchsian group for it to admit a smooth epimorphism onto a cyclic group. This allowed him to solve the minimum genus and the maximum order problems for the family of cyclic groups, although the latter had already been solved by Wiman [Wim]. Some years earlier, at the beginning of the sixties, A. M. Macbeath [Mcb1] revisited the Poincaré uniformization theorem and settled the basis of the combinatorial study of compact Riemann surfaces and their groups of automorphisms. This combinatorial point of view opened the door to a fruitful research, subsequently developed by Macbeath's students Bill Harvey, Colin Maclachlan and David Singerman.

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.