Perfect and Acyclic Subgroups of Finitely Presentable Groups
Perfect and Acyclic Subgroups of Finitely Presentable Groups
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DOI：
10.1112/s0024610703004587
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发表时间：
2003-12
期刊：
影响因子：
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通讯作者：
A. Berrick;J. Hillman
中科院分区：
文献类型：
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作者：
A. Berrick;J. Hillman
考虑低维无环群。为了简单地表明结果，令 G′ 为有限可表示群 G 的非平凡完美交换子群。则 def(G)⩽1。当def(G)=1时，G'是非循环的，前提是它在2以上的维度上没有积分同源性（充分条件是G'是有限生成的）；此外，G/G'则为Z或Z2。自然的例子是具有亚历山大多项式 1 的结和链环组。基于 S2× S1 中的结，给出了进一步的构造。在这些几何例子中，G'不能是有限生成的；一般来说，它不可能是有限的可呈现的。当 G 是一个 3 流形群时，它不是无环群；另一方面，如果 G′ 是有限生成的，那么它在 Q 同调 3 球面群中的索引是有限的。
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 S2× 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.