|
Singular surfaces, mod 2 homology, and hyperbolic volume, I
Author(s):
Ian
Agol;
Marc
Culler;
Peter
B.
Shalen
Journal:
Trans. Amer. Math. Soc.
362
(2010),
3463-3498.
MSC (2000):
Primary 57M50
Posted:
February 2, 2010
MathSciNet review:
2601597
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
If  is a simple, closed, orientable  -manifold such that  contains a genus- surface group, and if  has rank at least  , we show that  contains an embedded closed incompressible surface of genus at most  . As an application we show that if  is a closed orientable hyperbolic  -manifold of volume at most  , then the rank of  is at most  .
References:
-
- 1.
- Ian Agol, Tameness of hyperbolic
 -manifolds, arXiv:math/0405568v1[math.GT]. - 2.
- Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken
 -manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053-1077 (electronic). With an appendix by Nathan Dunfield. MR 2328715 - 3.
- James W. Anderson, Richard D. Canary, Marc Culler, and Peter B. Shalen, Free Kleinian groups and volumes of hyperbolic
 -manifolds, J. Differential Geom. 43 (1996), no. 4, 738-782. MR 1412683 (98c:57012) - 4.
- Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic
 -manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385-446 (electronic). MR 2188131 (2006g:57030) - 5.
- Marc Culler and Peter B. Shalen, Singular surfaces, mod
 homology, and hyperbolic volume, II, arXiv:math/070166v5[math.GT]. - 6.
- Marc Culler and Peter B. Shalen, Paradoxical decompositions,
 -generator Kleinian groups, and volumes of hyperbolic  -manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 231-288. MR 1135928 (93a:57017) - 7.
- David Gabai, Foliations and the topology of
 -manifolds, J. Differential Geom. 18 (1983), no. 3, 445-503. MR 723813 (86a:57009) - 8.
- John Hempel,
 -manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385 (2005e:57053) - 9.
- William Jaco and Peter B. Shalen, Peripheral structure of
 -manifolds, Invent. Math. 38 (1976), no. 1, 55-87. MR 0428332 (55:1357) - 10.
- William Meeks, III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621-659. MR 678484 (84f:53053)
- 11.
- C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1-26. MR 0090053 (19:761a)
- 12.
- Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/ 0303109v1[math.DG].
- 13.
- Peter B. Shalen and Philip Wagreich, Growth rates,
 -homology, and volumes of hyperbolic  -manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 2, 895-917. MR 1156298 (93d:57002) - 14.
- Arnold Shapiro and J. H. C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174-178. MR 0103474 (21:2242)
- 15.
- Jonathan Simon, Compactification of covering spaces of compact
 -manifolds, Michigan Math. J. 23 (1976), no. 3, 245-256 (1977). MR 0431176 (55:4178) - 16.
- John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12-19. MR 0121796 (22:12526)
- 17.
- -, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 0175956 (31:232)
- 18.
- Friedhelm Waldhausen, On irreducible
 -manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 0224099 (36:7146) - 19.
- Jeff Weeks and Craig Hodgson, ClosedCensusInvariants.txt, downloadable text file.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
57M50
Retrieve articles in all Journals with
MSC (2000):
57M50
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:
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
Posted:
February 2, 2010
Additional Notes:
This work was partially supported by NSF {grants DMS-0204142 and DMS-0504975}
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
