Perfect and acyclic subgroups of finitely presentable groups

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.