Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$ 6$-torsion and hyperbolic volume

Authors: F. W. Gehring and G. J. Martin
Journal: Proc. Amer. Math. Soc. 117 (1993), 727-735
MSC: Primary 30F40; Secondary 20H10, 57S30
MathSciNet review: 1116260
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Abstract: The Kleinian group $ \operatorname{PGL} (2,\,{O_3})$ is shown to have minimal covolume $ ( \approx 0.0846 \ldots )$ among all Kleinian groups containing torsion of order $ 6$ (the associated hyperbolic orbifold is also the minimal volume cusped orbifold). This follows from: Any cocompact Kleinian group with torsion of order $ 6$ has covolume at least $ \tfrac{1} {9}$. As a consequence, any compact hyperbolic manifold with a symmetry of order $ 6$ (with fixed points) has volume at least $ \tfrac{4} {3}$. These results follow from new collaring theorems for torsion in a Kleinian group arising from our generalizations of the Shimizu-Leutbecher inequality.

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Article copyright: © Copyright 1993 American Mathematical Society