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: We show that if is a closed hyperbolic 3-manifold and if has a non-abelian free quotient, then the volume of is greater than . If, in addition, contains no genus- surface groups, then the volume of is greater than . Using these results we show that if there are infinitely many primitive homology classes in which are not represented by fibroids, then the volume of is greater than .

**[ACCS]**J. W. Anderson, R. Canary, M. Culler, P. B. Shalen,*Free kleinian groups and volumes of hyperbolic -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