On boundaries of teichmüller spaces and on kleinian groups, III

The Teichmfiller space of a marked Riemann surface S, of finite conformal type (g, n) with signature or of a finitely generated Fuchsian group of the first kind representing S, was shown to have a complex structure by Ahlfors [3] (see also Rauch [23]).Bers [7] later showed that it could be embedded as a bounded domain in the space of bounded quadratic differentials on S; this space has dimension 3g-3 §The Bers embedding of the Teichmfiller space lends itself naturally to questions about the boundary.In a two part, almostjoint paper, Bers [8] and Maskit [20] systematically examined that boundary.In particular, Bers showed that totally degenerate Kleinian groups are represented on the boundary.A totally degenerate Kleinian group has a region of discontinuity which is connected and simply connected.Using his deep construction techniques, Maskit gave an exhaustive list of those pathologies which might be expected to occur on the boundary and showed that they do indeed occur.We shall study those groups which are not pathological, in a sense to be defined in w 1, and are called regular.This class coincides with Bers' non-degenerate non-quasi-Fuchsian groups and Maskit's groups giving complete factorizations of S. Regularity is a planar concept although we will show in w 7 that regularity is equivalent to geometric finiteness.Since we will need to refer on many occasions to the earlier papers in this series, Bers [8] will be denoted B-I and Maskit [20] will be denoted B-II.The class of non-pathological boundary groups has been studied constructively by Maskit [20] and using the structure of their associated 3-manifolds, in the torsionfree case, by Marden [19] and Earle and Marden [12].The techniques used in the present work are twodimensional and non-constructive.They involve conformal and quasiconformal mappings and plane topology, although much of the work draws deeply, if not directly then in spirit, on the work of Maskit.