Abstract
A spherical point of a Kleinian group Γ is a point of ℍ3 that is stabilized by a spherical triangle subgroup of Γ. 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 H3. These bound from below the edge lengths of the singular graph. An elliptic element of a Kleinian group is simple if the translates of its axis under the group Γ form a disjoint collection of hyperbolic lines. Here we announce that the minimal covolume Kleinian group contains no simple elliptics of order p ≥ 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.
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Research supported in part by grants from the N.Z. Marsden Fund, the U.S. National Science Foundation and the Volkswagen-Stiftung (RiP-Program at the Mathematisches Forschungsin- stitut in Oberwolfach for FWG and GJM).
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Gehring, F.W., Marshall, T.H. & Martin, G.J. Recent Results in the Geometry of Kleinian Groups. Comput. Methods Funct. Theory 2, 249–256 (2003). https://doi.org/10.1007/BF03321019
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DOI: https://doi.org/10.1007/BF03321019