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.

This book contains extended notes of most of the lectures given at the (instructional) Conference on Topics on Riemann Surfaces and Fuchsian Groups held in Madrid in 1998 to mark the 25th anniversary of the Universidad Nacional de Educación a Distancia. We wish to thank all contributors for their talks and texts, and in a special way to Alan F. Beardon for the excellent introduction. All papers have been refereed and we also thank to the referees for their work.

Introduction

Let X be a compact Riemann surface of genus g. A symmetry of X is an anticonformal involution T : X → X. The topological nature of a symmetry T is determined by properties of its fixed point set F(T). F(T) consists of k disjoint Jordan curves, where 0 ≤ k ≤ g + 1 (Harnack's theorem). X – F(T) has either one or two components. It consists of one component if X/〈T〉 is non-orientable and two components if X/〈T〉 is orientable. Let T be a symmetry of X and suppose that in F(T) there are k disjoint Jordan curves, then we shall say (see [4]) that the species of T is +k if X – F(T) has two components and –k if X – F(T) has one component.

We shall say that a Riemann surface X is an M (respectively M – 1) Riemann surface if it admits a symmetry with g + 1 fixed curves (respectively g fixed curves). S. M. Natanzon in [9] and [10] announced some properties about the topological nature of the symmetries of M and M – 1 Riemann surfaces and, in the hyperelliptic case, the classification of such symmetries up to conjugation in the automorphism group. Later, in [11] and [12] the above results were proved by topological techniques.