## $6$-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.## References

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## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**117**(1993), 727-735 - MSC: Primary 30F40; Secondary 20H10, 57S30
- DOI: https://doi.org/10.1090/S0002-9939-1993-1116260-4
- MathSciNet review: 1116260