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

Published online by Cambridge University Press:  28 September 2006

BERNARD PERRON
Affiliation:
Laboratoire de Topologie, Université de Bourgogne, BP 47870 21078 – Dijon Cedex, France. e-mail: [email protected]
DALE ROLFSEN
Affiliation:
Pacific Institute for the Mathematical Sciences and Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2. e-mail: [email protected]

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 $\Sigma$.

Type
Research Article
Copyright
2006 Cambridge Philosophical Society

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