Invariant ordering of surface groups and 3-manifolds which fibre over $S^1$

Abstract

It is shown that, if $\Sigma$ is a closed orientable surface and $\varphi\!: \Sigma \to \Sigma$ a homeomorphism, then one can find an ordering of $\pi_1(\Sigma)$ which is invariant under left- and right-multiplication, as well as under $\varphi_* \colon \pi_1(\Sigma) \to \pi_1(\Sigma)$, provided all the eigenvalues of the map induced by $\varphi$ on the integral first homology groups of $\Sigma$ are real and positive. As an application, if $M^3$ is a closed orientable 3-manifold which fibres over the circle, then its fundamental group is bi-orderable if the associated homology monodromy has all eigenvalues real and positive. This holds, in particular, if the monodromy is in the Torelli subgroup of the mapping class group of $\Sigma$.