On Schottky Groups with Applications to Kleinian Groups

Since every compact Riemann surface can be uniformized by a Schottky group, it is natural to look at the space of Schottky groups, or Schottky space, as a space of moduli for Riemann surfaces. The main purpose of this paper is to investigate Schottky groups and the boundary of Schottky space. In the course of this investigation, we prove that a group of Mobius transformations, which is a limit of a sequence of Schottky groups of genus g, is a free group on g generators without elliptic transformations. We generalize this result to groups which are limits of general kleinian groups. The author would like to thank Professor L.Bers for his patient guidance and encouragement in the preparation of this manuscript. These results have been announced in the Bulletin of the American Mathematical Society.