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Ordered groups, eigenvalues, knots, surgery and L-spaces

Published online by Cambridge University Press:  22 September 2011

ADAM CLAY  and
DALE ROLFSEN
ADAM CLAY
Affiliation:
Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, CanadaH3C 3P8. e-mail: [email protected]
DALE ROLFSEN
Affiliation:
Pacific Institute for the Mathematical Sciences and Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z2. e-mail: [email protected]
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Abstract

We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard–Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 115 - 129
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Boyd, D. W.Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24 (4) (1981), 453–469.CrossRefGoogle Scholar
[2]Boyer, S., Rolfsen, D. and Wiest, B.Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble) 55 (1) (2005), 243–288.Google Scholar
[3]Calegari, D.Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. (Oxford University Press, 2007).CrossRefGoogle Scholar
[4]Calegari, D. and Dunfield, N. M.Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (1) (2003), 149–204.Google Scholar
[5]Dehornoy, P.Braid groups and left distributive operations. Trans. Amer. Math. Soc. 345 (1) (1994), 115–150.Google Scholar
[6]Dehornoy, P., Dynnikov, I., Rolfsen, D. and Wiest, B.Ordering Braids, volume 148 of Surveys and Monographs. (American Mathematical Society, 2008).CrossRefGoogle Scholar
[7]Fintushel, R. and Stern, R. J.Constructing lens spaces by surgery on knots. Math. Z. 175 (1) (1980), 33–51.Google Scholar
[8]Ghys, É.Groups acting on the circle. Enseign. Math. (2) 47 (3-4) (2001), 329–407.Google Scholar
[9]Gruenberg, K. W.Relation modules of finite groups. American Mathematical Society, 1976. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 25.CrossRefGoogle Scholar
[10]Hironaka, E.The Lehmer polynomial and pretzel links. Canad. Math. Bull. 44 (4) (2001), 440–451.Google Scholar
[11]Hölder, O.Die Axiome Quantität und die Lehre vom Mass. Ber. Verh. Sächs. Wiss. Leipzig, Math.-Phis. 53 (1901), 1–64.Google Scholar
[12]Howie, J. and Short, H.The band-sum problem. J. London Math. Soc. (2) 31 (3) (1985), 571–576.CrossRefGoogle Scholar
[13]Ito, T. Braid ordering and knot genus. Preprint, available via http://arxiv.org/abs/0805.2042v3.Google Scholar
[14]Kirby, R. Problems in low-dimensional topology. In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 35–473 (Amer. Math. Soc., 1997). Updated version available from the author's website.Google Scholar
[15]Linnell, P. A., Rhemtulla, A. H. and Rolfsen, D. P. O.Invariant group orderings and Galois conjugates. J. Algebra 319 (12) (2008), 4891–4898.CrossRefGoogle Scholar
[16]Malyutin, A. V. and Netsvetaev, N. Yu.Dehornoy order in the braid group and transformations of closed braids. Algebra i Analiz 15 (3) (2003), 170–187.Google Scholar
[17]Mura, R. B. and Rhemtulla, A.Orderable groups (Marcel Dekker Inc., 1977). Lecture Notes in Pure and Applied Mathematics, Vol. 27.Google Scholar
[18]Navas, A. On the dynamics of (left) orderable groups. Preprint, available via http://arxiv.org/abs/0710.2466.Google Scholar
[19]Ni, Y.Knot Floer homology detects fibred knots. Invent. Math. 170 (3) (2007), 577–608.CrossRefGoogle Scholar
[20]Ozsváth, P. and Szabó, Z.Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2) 159 (3) (2004), 1159–1245.CrossRefGoogle Scholar
[21]Ozsváth, P. and Szabó, Z.On knot Floer homology and lens space surgeries. Topology 44 (6) (2005), 1281–1300.Google Scholar
[22]Perron, B. and Rolfsen, D.On orderability of fibred knot groups. Math. Proc. Camb. Phil. Soc. 135 (1) (2003), 147–153.CrossRefGoogle Scholar
[23]Perron, B. and Rolfsen, D.Invariant ordering of surface groups and 3-manifolds which fibre over S 1. Math. Proc. Camb. Phil. Soc. 141 (2) (2006), 273–280.Google Scholar
[24]Peters, T. On L-spaces and non left-orderable 3-manifold groups. Preprint, available via http://arxiv.org/abs/0903.4495.Google Scholar
[25]Roberts, R., Shareshian, J. and Stein, M.Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation. J. Amer. Math. Soc. 16 (3) (electronic) (2003), 639–679.CrossRefGoogle Scholar
[26]Rolfsen, D.Knots and links. Amer. Math. Soc. (2003).CrossRefGoogle Scholar
[27]Rolfsen, D. Mappings of nonzero degree between 3-manifolds: a new obstruction. In Advances in topological quantum field theory, volume 179 of NATO Sci. Ser. II Math. Phys. Chem. pages 267–273 (Kluwer Academic Publisher, 2004).Google Scholar
[28]Watson, L. Involutions of 3-manifolds and Khovanov homology. PhD dissertation, available via http://www.math.ucla.edu/lwatson/pages/pdf/thesisPhD-final.pdf.Google Scholar