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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singular surfaces, mod 2 homology, and hyperbolic volume, I


Authors: Ian Agol, Marc Culler and Peter B. Shalen
Journal: Trans. Amer. Math. Soc. 362 (2010), 3463-3498
MSC (2000): Primary 57M50
Published electronically: February 2, 2010
MathSciNet review: 2601597
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Abstract: If $ M$ is a simple, closed, orientable $ 3$-manifold such that $ \pi_1(M)$ contains a genus-$ g$ surface group, and if $ H_1(M;\mathbb{Z}_2)$ has rank at least $ 4g-1$, we show that $ M$ contains an embedded closed incompressible surface of genus at most $ g$. As an application we show that if $ M$ is a closed orientable hyperbolic $ 3$-manifold of volume at most $ 3.08$, then the rank of $ H_1(M;\mathbb{Z}_2)$ is at most $ 6$.


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Additional Information

Ian Agol
Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, California 94720-3840
Email: ianagol@math.berkeley.edu

Marc Culler
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607-7045
Email: culler@math.uic.edu

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

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04362-X
PII: S 0002-9947(10)04362-X
Received by editor(s): July 3, 2005
Received by editor(s) in revised form: February 2, 2008
Published electronically: February 2, 2010
Additional Notes: This work was partially supported by NSF grants DMS-0204142 and DMS-0504975
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.