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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
Bulletin of the London Mathematical Society – Wiley
Published: Sep 1, 1989
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