Hyperbolic Riemann Surfaces with the Trivial Group of Automorphisms

Abstract

In [2] for the first time there have been given a complete proof of the following statement of A. Hurwitz: For any integer g > 2 there exists a compact Riemann surface of genus g, whose group of all conformai automorphisms is trivial. Then Greenberg [3] has shown that for g > 2 almost all points in the Teichmuller space Tg, except perhaps for an analytic subset, correspond to Riemann surfaces with the trivial group of conformal automorphisms. Nevertheless, only a few constructive examples of such Riemann surfaces are known. One of them is given by Accola [1]. However, the method of Accola does not let us describe analytically the fundamental set of a Fuchsian group which uniformizes that surface. In the present paper, announced in [5], we construct in an explicit way the fundamental set of a Fuchsian group which uniformizes a compact Riemann surface with the trivial group of conformai automorphisms.