Introduction

We begin by considering two familiar ways of constructing Riemann surfaces. First, we take a power series P converging on some disc D0 with centre Z0, and expand P about some point z1 in Do other than z0. In general, P will converge in a disc D1 extending beyond D0? and if we continue this process indefinitely we obtain a maximal Riemann surface on which the analytic continuation of P is defined. Of course, if we return to a region where P is already defined, but with different values, we create a new ‘sheet’ of the surface; thus we are led to the notion of a Riemann surface constructed from a given power series: this is the Weierstrass approach. A more modern approach is simply to define a Riemann surface as a complex analytic manifold but either way, there is the problem of showing that these two definitions are equivalent. It is easy enough to see that a Riemann surface obtained by analytic continuation is an analytic manifold, so we must focus our efforts on showing that every analytic manifold supports an analytic function. One solution to this problem lies in showing first that every such manifold arises as the quotient by a group action, and second, that we can construct functions invariant under this group action. As a by-product of a study of these groups we obtain important and very detailed quantitative information about the geometric nature of the general Riemann surface.