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Residual nilpotence and ordering in one-relator groups and knot groups

Published online by Cambridge University Press:  14 January 2015

I. M. CHISWELL ,
A. M. W. GLASS  and
JOHN S. WILSON
I. M. CHISWELL
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS. e-mail: [email protected]
A. M. W. GLASS
Affiliation:
Queens' College, Cambridge CB3 9ET. e-mail: [email protected]
JOHN S. WILSON
Affiliation:
Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG. e-mail: [email protected]
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 275 - 288
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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