Hyperbolic generators for Fuchsian groups

Abstract

The standard system of generators for a Fuchsian group contains, in general, some elliptic and parabolic transformations. It is not, however, well known that one can always choose a system of generators consisting only of hyperbolic transformations.2 We will obtain this result by establishing a stronger statement: there exists a convex noneuclidean fundamental polygon, whose sides are identified by hyperbolic transformations. Clearly, fundamental other than the Fricke polygons are needed. Such are considered in [6] where most of the facts which we shall need are developed. We will often omit details and proofs when they would be only slight modifications of those in [6]. 1. Preliminaries. A finitely generated discontinuous group G of conformal self-mappings of the unit disc U is called a Fuchsian group. Let ir denote the canonical mapping of U onto U/G. Then U/G, with the induced conformal structure, is a Riemann surface. Given integers g, h, n, I and P1, * * *, v. satisfying