2003/05/06 by Alice Chang, Chang, Alice, Jie Qing +3 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AP #math.DG #msc:30F40 #msc:53A30 #msc:58J60
paper · pdf · doi:10.48550/arxiv.math/0305083
18 pages, 1 figure
arxiv created 2003/05/06 · arxiv updated 2009/11/30
In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper of Bishop and Jones to show that conformal finiteness implies geometric finiteness unless 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 a metric of finite geometry on the Kleinian manifold Ω(Γ)/Γ.