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

 

Characteristic subsurfaces and Dehn filling


Authors: Steve Boyer, Marc Culler, Peter B. Shalen and Xingru Zhang
Journal: Trans. Amer. Math. Soc. 357 (2005), 2389-2444
MSC (2000): Primary 57M25, 57M50, 57M99
Published electronically: October 28, 2004
MathSciNet review: 2140444
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Abstract: Let $M$ be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in $M$ which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of $M(\alpha)$ has no non-abelian free subgroup, and if $M(\beta)$ is a reducible manifold which is not homeomorphic to $S^1 \times S^2$ or $P^3 \char93 P^3$, then $\Delta(\alpha, \beta)\le 5$. Under the same condition on $M(\beta)$, it is shown that if $M(\alpha)$ is Seifert fibered, then $\Delta(\alpha, \beta)\le 6$. Moreover, in the latter situation, character variety techniques are used to characterize the topological types of $M(\alpha)$ and $M(\beta)$ in case the bound of $6$ is attained.


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Additional Information

Steve Boyer
Affiliation: Département de Mathématiques, Université du Québec, Montréal, P. O. Box 8888, Postal Station Centre-ville Montréal, Québec, Canada H3C 3P8
Email: boyer@math.uqam.ca

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

Xingru Zhang
Affiliation: Department of Mathematics, SUNY at Buffalo, Buffalo, New York 14260-2900
Email: xinzhang@math.buffalo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03576-7
PII: S 0002-9947(04)03576-7
Received by editor(s): December 6, 2002
Received by editor(s) in revised form: December 2, 2003
Published electronically: October 28, 2004
Additional Notes: The first author was partially supported by NSERC grant OGP0009446 and FCAR grant ER-68657
The second and third authors were partially supported by NSF grant DMS 0204142
The fourth author was partially supported by NSF grant DMS 0204428.
Article copyright: © Copyright 2004 American Mathematical Society