Residual nilpotence and ordering in one-relator groups and knot groups
Residual nilpotence and ordering in one-relator groups and knot groups
复制标题
单关系群和结群中的剩余幂零性和排序
DOI：
10.1017/s0305004114000644
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发表时间：
2014
影响因子：
0.8
通讯作者：
John S. Wilson
中科院分区：
文献类型：
--
作者：
I. Chiswell;A. Glass;John S. Wilson
设G = λ x，t| w是一个单关系子群，其中w是x，t中的一个词。如果w是x的共轭的乘积，则与w相关联，存在整数上的多项式Aw（X），在G是纽结群的情况下，它是纽结的亚历山大多项式。我们证明了，如果Aw（X）的所有根都是真实的且正的，则G是双序的;如果G是双序的，则至少有一个根是真实的且正的.这揭示了双有序性的某些结群和一个问题的粘土和罗尔夫森。其中一个结果依赖于G. Baumslag关于根到群的附加，这可能有独立的兴趣。
Abstract Let G = 〈x, t | w〉 be a one-relator group, where w is a word in x, t. If w is a product of conjugates of x then, associated with w, there is a polynomial Aw(X) over the integers, which in the case when G is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on w, that if all roots of Aw(X) are real and positive then G is bi-orderable, and that if G is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.