Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Bounding the Decomposition of a Poincaré Duality Group

Bounding the Decomposition of a Poincaré Duality Group BOUNDING THE DECOMPOSITION OF A POINCARE DUALITY GROUP M. J. DUNWOODY In this paper the following theorem is proved. THEOREM. Let G be a PD -group (n ^ 1) (with coefficient field either Z or Z) n 1 and let T be a G-tree such that for every eeET, G is a PD ~ -group. Suppose T has no compressible edges. There is an integer n(G) such that \G\VT\ ^ n(G) and \G\ET\ ^ n(G). This answers a question put to me by Peter Kropholler. Kropholler [8] has since used this result in his proof of an analogue of the torus decomposition theorem for certain PZ)"-groups. The theorem can be proved fairly easily by slightly modifying the proof of the Theorem in Section 4 of [1]. Alternatively a proof can be given using an argument similar to that of [4, Lemma III, 7.5]. The proof given here uses an idea that comes from [1]. The theorem is the algebraic analogue of the well-known theorem (Hempel [6, Lemma 13.2]) that there is a bound on the number of incompressible non-parallel disjoint surfaces embedded in a compact three-manifold. First we recall some of the definitions and results from the theory of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Bounding the Decomposition of a Poincaré Duality Group

Loading next page...
 
/lp/wiley/bounding-the-decomposition-of-a-poincar-duality-group-bYVj3vr041

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/21.5.466
Publisher site
See Article on Publisher Site

Abstract

BOUNDING THE DECOMPOSITION OF A POINCARE DUALITY GROUP M. J. DUNWOODY In this paper the following theorem is proved. THEOREM. Let G be a PD -group (n ^ 1) (with coefficient field either Z or Z) n 1 and let T be a G-tree such that for every eeET, G is a PD ~ -group. Suppose T has no compressible edges. There is an integer n(G) such that \G\VT\ ^ n(G) and \G\ET\ ^ n(G). This answers a question put to me by Peter Kropholler. Kropholler [8] has since used this result in his proof of an analogue of the torus decomposition theorem for certain PZ)"-groups. The theorem can be proved fairly easily by slightly modifying the proof of the Theorem in Section 4 of [1]. Alternatively a proof can be given using an argument similar to that of [4, Lemma III, 7.5]. The proof given here uses an idea that comes from [1]. The theorem is the algebraic analogue of the well-known theorem (Hempel [6, Lemma 13.2]) that there is a bound on the number of incompressible non-parallel disjoint surfaces embedded in a compact three-manifold. First we recall some of the definitions and results from the theory of

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Sep 1, 1989

There are no references for this article.