-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

DOI:
https://doi.org/10.1090/S0002-9939-1993-1116260-4

MathSciNet review:
1116260

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Abstract: The Kleinian group is shown to have minimal covolume among all Kleinian groups containing torsion of order (the associated hyperbolic orbifold is also the minimal volume cusped orbifold). This follows from: Any cocompact Kleinian group with torsion of order has covolume at least . As a consequence, any compact hyperbolic manifold with a symmetry of order (with fixed points) has volume at least . 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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DOI:
https://doi.org/10.1090/S0002-9939-1993-1116260-4

Article copyright:
© Copyright 1993
American Mathematical Society