6-torsion and hyperbolic volume

Gehring, F. W.; Martin, G. J.

doi:10.1090/S0002-9939-1993-1116260-4

$6$-torsion and hyperbolic volume
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by F. W. Gehring and G. J. Martin PDF

Proc. Amer. Math. Soc. 117 (1993), 727-735 Request permission

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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