Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 2 (2002), No. 1, 249--256 Copyright Heldermann Verlag 2002 Recent Results in the Geometry of Kleinian Groups Frederick W. Gehring fgehring@umich.edu, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A. Timothy H. Marshall tmarshall@wesleyan.edu, Department of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A. Gaven J. Martin martin@math.auckland.ac.nz, Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand. [Abstract-pdf] [Abstract-ps] A \emph{spherical point} of a Kleinian group $\Gamma$ is a point of $\mathbb{H}^3$ that is stabilized by a spherical triangle subgroup of $\Gamma$. Such points appear as vertices in the singular graph of the quotient hyperbolic $3$-orbifold. We announce here sharp lower bounds for the hyperbolic distances between such points in~$\mathbb{H}^3$. These bound from below the edge lengths of the singular graph. An elliptic element of a Kleinian group is \emph{simple} if the translates of its axis under the group~$\Gamma$ form a disjoint collection of hyperbolic lines. Here we announce that the minimal covolume Kleinian group contains no simple elliptics of order~\hbox{$p\geq 3$}. Applications of these estimates leads to sharp volume bounds for hyperbolic $3$-orbifolds whose singular set contains a spherical point. We are also able to present substantial progress to the problem of identifying the minimal covolume Kleinian group. Keywords: Kleinian group, discrete group, hyperbolic geometry, hyperbolic volume. MSC 2000: Primary 30F40, 30D50, 51M20, 57M50. [FullText-pdf (184 K)] [FullText-ps (240 K)]