Skip to main content Accessibility help
×
Home

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)

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.

Copyright

References

Hide All
[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 http://arxiv.org/abs/1410.5774.
[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 http://arxiv.org/abs/1409.5730.
[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).

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed