On the classification of Kleinian groups: I—Koebe groups

In this paper we exhibit a class of Klein]an groups called Koebe groups, and prove that every un]formization of a closed Riemann surface can be realized by a unique Koebe group.The first examples of these groups were due to Klein [6] who constructed them using his combination theorem.Koebe [7, 8] proved a general un]formization theorem for Klein]an groups constructed in this fashion.Our existence theorem is based on the generalized combination theorems [11,12], and uses Bers' technique of variation of parameters using quasiconformal mappings [3].The uniqueness theorem is a generalization of Koebe's original proof [7], but with weaker hypotheses.Detailed statements of theorems, and outlines of proofs appear in section 2. The theorems are all formulated in terms of Klein]an groups; equivalent formulations in terms of un]formizations of Riemann surfaces appear in [13], where our main result was first announced.