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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Line arrangements in $\mathbb{H}^3$


Author: Peter Milley
Journal: Proc. Amer. Math. Soc. 133 (2005), 3115-3120
MSC (2000): Primary 57M60, 51M09; Secondary 57M50
Published electronically: April 20, 2005
MathSciNet review: 2159793
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Abstract: If $M=\mathbb{H}^3/G$ is a hyperbolic manifold and $\gamma\subset M$ is a simple closed geodesic, then $\gamma$ lifts to a collection of lines in $\mathbb{H}^3$ acted upon by $G$. In this paper we show that such a collection of lines cannot contain a particular type of subset (called a bad triple) unless $G$ has orientation-reversing elements. This fact allows us to extend certain lower bounds on hyperbolic volume to the non-orientable case.


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Additional Information

Peter Milley
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
Address at time of publication: Department of Mathematics, University of California–Riverside, Riverside, California 92521-0135
Email: milley@math.princeton.edu, milley@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07875-5
PII: S 0002-9939(05)07875-5
Keywords: Hyperbolic geometry, non-orientable manifolds
Received by editor(s): April 15, 2004
Received by editor(s) in revised form: June 3, 2004
Published electronically: April 20, 2005
Additional Notes: The author was supported in part by NSF Grants DMS-9505253 and DMS-0071852.
The author would like to thank David Gabai for his comments and support, and the reviewer for his comments and corrections.
Dedicated: Dedicated to my wife, Cheryl
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.