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

 

Canonical splittings of groups and 3-manifolds


Authors: Peter Scott and Gadde A. Swarup
Journal: Trans. Amer. Math. Soc. 353 (2001), 4973-5001
MSC (2000): Primary 57M07, 57N10, 20E06
Published electronically: July 25, 2001
MathSciNet review: 1852090
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Abstract:

We introduce the notion of a `canonical' splitting over $\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$ for a finitely generated group $G$. We show that when $G$ happens to be the fundamental group of an orientable Haken manifold $M$ with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related to the one given by the standard JSJ-decomposition of $M$. This leads to a new proof of Johannson's Deformation Theorem.


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Additional Information

Peter Scott
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: pscott@math.lsa.umich.edu

Gadde A. Swarup
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
Email: gadde@ms.unimelb.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02871-9
PII: S 0002-9947(01)02871-9
Keywords: 3-manifold, characteristic submanifold, Deformation Theorem, ends of groups, intersection number, JSJ decomposition, splittings of groups
Received by editor(s): August 12, 2000
Received by editor(s) in revised form: April 9, 2001
Published electronically: July 25, 2001
Additional Notes: The first author was partially supported by NSF grant DMS 034681.
Article copyright: © Copyright 2001 American Mathematical Society