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.