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Journal of the London Mathematical Society 2003 68(3):683-698; doi:10.1112/S0024610703004587
© 2003 by London Mathematical Society
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© The London Mathematical Society

Perfect and Acyclic Subgroups of Finitely Presentable Groups

A. J. Berrick and J. A. Hillman

Department of Mathematics, National University of Singapore Singapore berrick{at}math.nus.edu.sg
School of Mathematics, University of Sydney NSW 2006, Australia jonh{at}maths.usyd.edu.au

Received 26 March 2002. Revision received 21 March 2003.

Acyclic groups of low dimension are considered. To indicate the results simply, let G' be the nontrivial perfect commutator subgroup of a finitely presentable group G. Then def(G)≤1. When def(G)=1, G' is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that G' be finitely generated); moreover, G/G' is then Z or Z2. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in S2x S1. In these geometric examples, G' cannot be finitely generated; in general, it cannot be finitely presentable. When G is a 3-manifold group it fails to be acyclic; on the other hand, if G' is finitely generated it has finite index in the group of a Q-homology 3-sphere.


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