  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 Ozsv\'ath and Szab\'o.
