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

MathSciNet review:
1422858

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Abstract | References | Similar Articles | Additional Information

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]**Marc Culler and Peter B. Shalen,*Paradoxical decompositions, 2-generator Kleinian groups, and volumes of hyperbolic 3-manifolds*, J. Amer. Math. Soc.**5**(1992), no. 2, 231–288. MR**1135928**, 10.1090/S0894-0347-1992-1135928-4**[CS2]**Marc Culler and Peter B. Shalen,*The volume of a hyperbolic 3-manifold with Betti number 2*, Proc. Amer. Math. Soc.**120**(1994), no. 4, 1281–1288. MR**1205485**, 10.1090/S0002-9939-1994-1205485-6**[CS3]**Marc Culler and Peter B. Shalen,*Volumes of hyperbolic Haken manifolds. I*, Invent. Math.**118**(1994), no. 2, 285–329. MR**1292114**, 10.1007/BF01231535**[H]**Marshall Hall Jr.,*Coset representations in free groups*, Trans. Amer. Math. Soc.**67**(1949), 421–432. MR**0032642**, 10.1090/S0002-9947-1949-0032642-4**[KS]**Abraham Karrass and Donald Solitar,*On finitely generated subgroups of a free group*, Proc. Amer. Math. Soc.**22**(1969), 209–213. MR**0245655**, 10.1090/S0002-9939-1969-0245655-2**[T]**William P. Thurston,*A norm for the homology of 3-manifolds*, Mem. Amer. Math. Soc.**59**(1986), no. 339, i–vi and 99–130. MR**823443****[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:
http://dx.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