Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: July 25, 2012
MathSciNet review: 2995380
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M50

Retrieve articles in all journals with MSC (2010): 57M50

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

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.

American Mathematical Society