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

  • I. M. CHISWELL (a1), A. M. W. GLASS (a2) and JOHN S. WILSON (a3)


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.



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[1]Baumslag, G.Some aspects of groups with unique roots. Acta Math. 104 (1960), 217303.
[2]Baumslag, G.Groups with the same lower central sequence as a relatively free group. I. The groups. Trans. Amer. Math. Soc. 129 (1967), 308321.
[3]Clay, A., Desmarais, C. and Naylor, P. Testing bi-orderability of knot groups. Preprint, available at
[4]Clay, A. and Rolfsen, D.Ordered groups, eigenvalues, knots, surgery and L-spaces. Math. Proc. Camb. Phil. Soc. 152 (2012), 115129.
[5]Cohen, D. E.Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14 (Cambridge University Press, Cambridge, 1989).
[6]Cohn, P. M.Algebra. Volume 1 (John Wiley & Sons, London–New York–Sydney, 1974).
[7]Crowell, R. H. and Fox, R. H.Introduction to knot theory. Graduate Texts in Mathematics 57 (Springer-Verlag, New York-Heidelberg, 1977).
[8]Fuchs, L.Partially Ordered Algebraic Systems (Pergamon Press, New York, 1963).
[9]Glass, A. M. W.Partially ordered groups. Series in Algebra 7 (World Scientific Pub. Co., Singapore, 1999).
[10]Linnell, P. A., Rhemtulla, A. H. and Rolfsen, D. P. O.Invariant group orderings and Galois conjugates. J. Algebra 319 (2008), 48914898.
[11]Lyndon, R. C. and Schupp, P. E.Combinatorial group theory. Classics in Mathematics (Springer-Verlag, Heidelberg, 2001).
[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).
[13]Botto Mura, R. and Rhemtulla, A. H.Orderable groups. Lecture Notes in Pure and Appl. Alg. 27 (Marcel Dekker, New York, 1977).
[14]Naylor, G. and Rolfsen, D. Generalised torsion in knot groups. Preprint, available at
[15]Perron, B. and Rolfsen, D.On orderability of fibred knot groups. Math. Proc. Camb. Phil. Soc. 135 (2003), 147153.
[16]Robinson, D. J. S.A course in the theory of groups. Graduate Texts in Math. 80 (Springer–Verlag, Heidelberg, 1982).
[17]Rolfsen, D.Knots and links. Mathematics Lecture Series 7 (Publish or Perish, Berkeley 1976).


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