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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: I.M.Chiswell@qmul.ac.uk
A. M. W. GLASS
Affiliation:
Queens' College, Cambridge CB3 9ET. e-mail: amwg@dpmms.cam.ac.uk
JOHN S. WILSON
Affiliation:
Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG. e-mail: John.Wilson@maths.ox.ac.uk
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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

REFERENCES

[1]Baumslag, G.Some aspects of groups with unique roots. Acta Math. 104 (1960), 217–303.Google Scholar
[2]Baumslag, G.Groups with the same lower central sequence as a relatively free group. I. The groups. Trans. Amer. Math. Soc. 129 (1967), 308–321.Google Scholar
[3]Clay, A., Desmarais, C. and Naylor, P. Testing bi-orderability of knot groups. Preprint, available at http://arxiv.org/abs/1410.5774.Google Scholar
[4]Clay, A. and Rolfsen, D.Ordered groups, eigenvalues, knots, surgery and L-spaces. Math. Proc. Camb. Phil. Soc. 152 (2012), 115–129.Google Scholar
[5]Cohen, D. E.Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14 (Cambridge University Press, Cambridge, 1989).Google Scholar
[6]Cohn, P. M.Algebra. Volume 1 (John Wiley & Sons, London–New York–Sydney, 1974).Google Scholar
[7]Crowell, R. H. and Fox, R. H.Introduction to knot theory. Graduate Texts in Mathematics 57 (Springer-Verlag, New York-Heidelberg, 1977).Google Scholar
[8]Fuchs, L.Partially Ordered Algebraic Systems (Pergamon Press, New York, 1963).Google Scholar
[9]Glass, A. M. W.Partially ordered groups. Series in Algebra 7 (World Scientific Pub. Co., Singapore, 1999).Google Scholar
[10]Linnell, P. A., Rhemtulla, A. H. and Rolfsen, D. P. O.Invariant group orderings and Galois conjugates. J. Algebra 319 (2008), 4891–4898.Google Scholar
[11]Lyndon, R. C. and Schupp, P. E.Combinatorial group theory. Classics in Mathematics (Springer-Verlag, Heidelberg, 2001).Google Scholar
[12]Magnus, W., Karrass, A. and Solitar, D.Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, 2nd revised edition (Dover Publications, New York, 1976).Google Scholar
[13]Botto Mura, R. and Rhemtulla, A. H.Orderable groups. Lecture Notes in Pure and Appl. Alg. 27 (Marcel Dekker, New York, 1977).Google Scholar
[14]Naylor, G. and Rolfsen, D. Generalised torsion in knot groups. Preprint, available at http://arxiv.org/abs/1409.5730.Google Scholar
[15]Perron, B. and Rolfsen, D.On orderability of fibred knot groups. Math. Proc. Camb. Phil. Soc. 135 (2003), 147–153.CrossRefGoogle Scholar
[16]Robinson, D. J. S.A course in the theory of groups. Graduate Texts in Math. 80 (Springer–Verlag, Heidelberg, 1982).CrossRefGoogle Scholar
[17]Rolfsen, D.Knots and links. Mathematics Lecture Series 7 (Publish or Perish, Berkeley 1976).Google Scholar