On finiteness of Kleinian groups in general dimension

Abstract

Abstract. In this paper we provide a criteria for geometric finiteness of Kleiniangroups in general dimension. We formulate the concept of conformal finiteness forKleinian groups in space of dimension higher than two, which generalizes the notionof analytic finiteness in dimension two. Then we extend the argument in the paperof Bishop and Jones to show that conformal finiteness implies geometric finitenessunless the set of limit points is of Hausdorff dimension n. Furthermore we show that,for a given Kleinian group Γ, conformal finiteness is equivalent to the existence of ametric of finite geometry on the Kleinian manifold Ω(Γ)/Γ. 1991 Mathematics Subject Classification: Primary 53A30; Secondary 30F40,58J60, 53C21.§0. IntroductionA discrete subgroup Γ of the group of conformal transformations of the unitsphere S n is called a Kleinian group if there is a non-empty domain Ω(Γ) of dis-continuity in S n . The group Γ also acts as a subgroup of the group of hyperbolicisometries of the unit ball B n+1