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Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations. (English) Zbl 1013.57010

The author considers a compact 3-manifold \(M\) which fibres over the circle \(S^1\) with fibre \(S\) and pseudo-Anosov monodromy \(\Psi: S\to S\). This monodromy preserves a unique the Teichmüller geodesic in Teichmüller space for \(S\), and the points on this geodesics induce complex structures \(J_s\) on the fibres \(S_s\).
The tangent bundle of the fibres is annihilated by a closed nowhere vanishing 1-form \(\omega\) which defines an integer cohomology class \([\omega]\in H^1(M,\mathbb{Z})\) and determines a transverse invariant measure for the fibre foliation. More generally, any closed nowhere vanishing 1-form \(\omega\) on \(M\) determines a measured foliation \({\mathcal F}\) on \(M\). The author generalizes to this situation earlier results of Teichmüller, Bers and Thurston. He shows that there is a complex structure \(J\) on the leaves of \({\mathcal F}\), a unit speed “monodromy” flow \((M,{\mathcal F})\times \mathbb{R}\to (M,{\mathcal F})\) and an expansion constant \(k> 1\) such that \(f_t\) maps leaves by Teichmüller mappings with expansion factor \(k^{|k|}\). Moreover, all these data are uniquely determined by the cohomology class of the defining 1-form.
The main new tool for the proof is a polynomial \(\Theta_F\) of a so-called fibred face of the unit ball in \(H^1(M,\mathbb{R})\) equipped with the Thurston norm. This norm assigns to a cohomology class \(\phi\) the infimum of the absolute value of the Euler characteristic of a surface dual to \(\phi\). The unit ball for this norm is a polyhedron. A fibred face \(F\) of this ball (i.e. one which contains a fibre of a fibration) determines a 2-dimensional lamination \(L\subset M\) transverse to the fibre \(S\), with \(S\cap L\) equal to the expanding lamination for the monodromy \(\Psi: S\to S\). The polynomial \(\Theta_F\) is defined using the module of transversals for \(L\).
It is shown how to determine the expansion constant for the measured foliations defined by the classes in the cone \(\mathbb{R}_+ F\). The main theorem follows from a careful analysis of the polynomial and a result of Blank and Laudenbach who proved that any two measured foliations representing the same cohomology class are isotopic.

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F60 Teichmüller theory for Riemann surfaces
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
57N10 Topology of general \(3\)-manifolds (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57R30 Foliations in differential topology; geometric theory
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