2010/04/21 by Adam Clay, Clay, Adam, Dale Rolfsen +1
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #math.AT #math.GR
paper · pdf · doi:10.48550/arxiv.1004.3615
Minor changes from first version
arxiv created 2010/05/27 · arxiv updated 2010/05/28
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.