It is shown that, if Σ is a closed orientable surface and ϕ : Σ → Σ a homeomorphism, then one can find an ordering of π 1 (Σ) which is invariant under left-and rightmultiplication, as well as under ϕ * : π 1 (Σ) → π 1 (Σ), provided all the eigenvalues of the map induced by ϕ on the integral first homology groups of Σ 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

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... y 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 Σ.