Perfect and Acyclic Subgroups of Finitely Presentable Groups |
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| Authors: | Berrick, A. J. Hillman, J. A. |
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| Affiliation: | 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 |
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| Abstract: | Acyclic groups of low dimension are considered. To indicatethe results simply, let G' be the nontrivial perfect commutatorsubgroup of a finitely presentable group G. Then def(G) 1. Whendef(G)=1, G' is acyclic provided that it has no integral homologyin dimensions above 2 (a sufficient condition for this is thatG' be finitely generated); moreover, G/G' is then Z or Z2. Naturalexamples are the groups of knots and links with Alexander polynomial1. 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-manifoldgroup it fails to be acyclic; on the other hand, if G' is finitelygenerated it has finite index in the group of a Q-homology 3-sphere. |
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