Computational Methods and Function Theory 2 (2002), No. 1, 249--256
Copyright Heldermann Verlag 2002
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.
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.
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