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Small optimal Margulis numbers force upper volume bounds


Author: Peter B. Shalen
Journal: Trans. Amer. Math. Soc. 365 (2013), 973-999
MSC (2010): Primary 57M50
DOI: https://doi.org/10.1090/S0002-9947-2012-05657-1
Published electronically: July 25, 2012
MathSciNet review: 2995380
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Abstract: If $ \lambda $ is a positive real number strictly less than $ \operatorname {log}3$, there is a positive number $ V_\lambda $ such that every orientable hyperbolic $ 3$-manifold of volume greater than $ V_\lambda $ admits $ \lambda $ as a Margulis number. If $ \lambda <(\operatorname {log}3)/2$, such a $ V_\lambda $ can be specified explicitly and is bounded above by

$\displaystyle \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ),$

where $ \operatorname {log}$ denotes the natural logarithm. These results imply that for $ \lambda <\operatorname {log}3$, an orientable hyperbolic $ 3$-manifold that does not have $ \lambda $ as a Margulis number has a rank-$ 2$ subgroup of bounded index in its fundamental group and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if $ \lambda <(\operatorname {log}3)/2$.

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Additional Information

Peter B. Shalen
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: shalen@math.uic.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05657-1
Received by editor(s): October 13, 2010
Received by editor(s) in revised form: June 16, 2011
Published electronically: July 25, 2012
Additional Notes: This work was partially supported by NSF grant DMS-0906155
Dedicated: Dedicated to José Montesinos on the occasion of his 65th birthday
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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