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|Title:||Perfect and acyclic subgroups of finitely presentable groups|
|Authors:||Berrick, A.J. |
|Source:||Berrick, A.J., Hillman, J.A. (2003-12). Perfect and acyclic subgroups of finitely presentable groups. Journal of the London Mathematical Society 68 (3) : 683-698. ScholarBank@NUS Repository. https://doi.org/10.1112/S0024610703004587|
|Abstract:||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 Z 2. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in S 2 × S 1. 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.|
|Source Title:||Journal of the London Mathematical Society|
|Appears in Collections:||Staff Publications|
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