Abstract
AbstractWe 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.
Publisher
Cambridge University Press (CUP)
Reference28 articles.
1. [24] Peters T. On L-spaces and non left-orderable 3-manifold groups. Preprint, available via http://arxiv.org/abs/0903.4495.
2. [26] Rolfsen D. Knots and links. Amer. Math. Soc. (2003).
3. Invariant ordering of surface groups and 3-manifolds which fibre over $S^1$
4. [18] Navas A. On the dynamics of (left) orderable groups. Preprint, available via http://arxiv.org/abs/0710.2466.
5. [17] Mura R. B. and Rhemtulla A. Orderable groups (Marcel Dekker Inc., 1977). Lecture Notes in Pure and Applied Mathematics, Vol. 27.
Cited by
19 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献