Abstract

A finite group G is said to be an M*-group if it is the group of automorphisms of a bordered compact Klein surface with maximal symmetry. M*-groups play an analogous role for Klein surfaces as Hurwitz groups do for Riemann surfaces. In this survey we present a summary of results on M*-groups. We first examine their properties and the known families of M*-groups. Then we study their structure to obtain new methods for constructing additional families. Finally we examine the relationship between Hurwitz groups, H*-groups and M*-groups.

Introduction

The study of Riemann and Klein surfaces with maximal automorphism groups has a long history. It is well known that a compact Riemann surface of genus g ≥ 2 admits at most 84(g – 1) automorphisms. Automorphism groups of Riemann surfaces with this maximal number of automorphisms are called Hurwitz groups. It is known that Hurwitz groups exist for infinitely many values of g and also do not exist for infinitely many g. The article by Conder [9] contains a nice survey of known results about Hurwitz groups. Corresponding problems concerning Klein surfaces have also received a good deal of attention and we present a summary of known results here.

A Klein surface is the orbit space of a Riemann surface under the action of a symmetry, that is, an anticonformal automorphism of order two. The algebraic genus of the Klein surface is defined to be the genus of its Riemann double cover.