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Volumes of hyperbolic Haken manifolds, II


Authors: Marc Culler and Peter B. Shalen
Journal: Proc. Amer. Math. Soc. 125 (1997), 3059-3067
MSC (1991): Primary 57M50; Secondary 57N10
DOI: https://doi.org/10.1090/S0002-9939-97-04101-4
MathSciNet review: 1422858
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if $M$ is a closed hyperbolic 3-manifold and if $\pi _{1}(M)$ has a non-abelian free quotient, then the volume of $M$ is greater than $0.92$. If, in addition, $\pi _{1}(M)$ contains no genus-$2$ surface groups, then the volume of $M$ is greater than $1.02$. Using these results we show that if there are infinitely many primitive homology classes in $H_{2}(M;\mathbb {Z})$ which are not represented by fibroids, then the volume of $M$ is greater than $0.83$.


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

  • [ACCS] J. W. Anderson, R. Canary, M. Culler, P. B. Shalen, Free kleinian groups and volumes of hyperbolic $3$-manifolds, J. Diff. Geom. 43 (1996), 738-782. CMP 97:02
  • [CHS] M. Culler, S. Hersonsky and P. B. Shalen, The first Betti number of the smallest closed hyperbolic 3-manifold, preprint.
  • [CS1] M. Culler and P. B. Shalen, Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds, J. Amer. Math. Soc. 5 (1992), 231-288. MR 93a:57017
  • [CS2] M. Culler and P. B. Shalen, The volume of a hyperbolic 3-manifold with Betti number 2, Proc. Amer. Math. Soc. 120 (1994), 1281-1288. MR 94f:57012
  • [CS3] M. Culler and P. B. Shalen, Volumes of hyperbolic Haken manifolds, I, Invent. math 118 (1994), 285-329. MR 95g:57023
  • [H] M. Hall, Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421-432. MR 11:322e
  • [KS] A. Karass and D. Solitar, On finitely generated subgroups of a free group, Proc. Amer. Math. Soc. 22 (1969), 209-213. MR 39:6961
  • [T] W. P. Thurston, A norm for the homology of 3-manifolds, Memoirs of the Amer. Math. Soc. No. 339, Volume 59, 1986, pp. 99-130. MR 88h:57014
  • [W] J. Weeks, Hyperbolic structures on three-manifolds, Thesis, Princeton University, 1985.

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Additional Information

Marc Culler
Affiliation: Department of Mathematics (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607-7045

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

DOI: https://doi.org/10.1090/S0002-9939-97-04101-4
Received by editor(s): November 6, 1995
Additional Notes: This research was partially supported by National Science Foundation grant DMS9302520.
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society

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