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Taut branched surfaces from veering triangulations. (English) Zbl 1396.57033
Let \(M\) be a closed, irreducible and atoroidal \(3\)-manifold. In this paper, the author studies taut oriented branched surfaces in \(M\) spanning faces of the Thurston norm ball of \(M\). More precisely, he is interested in a question of U. Oertel [Lond. Math. Soc. Lect. Note Ser. 112, 253–272 (1986; Zbl 0628.57011)]: when is it possible to find a single taut oriented branched surface which carries representatives of all integral classes in the cone spanned by a face \(\sigma\)? The paper specialises in the case when \(M\) is hyperbolic and \(\sigma\) is fibered. In this context, L. Mosher [J. Differ. Geom. 34, No. 1, 1–36 (1991; Zbl 0754.58031)] proved that there exists such a branched surface if the vertices of \(\sigma\) have positive intersection with the singular orbits of the Fried suspension flow corresponding to \(\sigma\).
In this paper, the author extends Mosher’s result by developing a completely different approach, via Agol’s veering triangulations. Let \(c\) be the union of singular orbits of the Fried suspension flow, then the non-compact manifold \(M' := M \setminus c\) has a canonical veering triangulation \(\tau\). The \(2\)-skeleton of \(\tau\) is a taut oriented branched surface which spans a fibered face of the Thurston norm ball of \(M'\). The author carefully studies this branched surface in \(M'\) and then extends it to a face-spanning taut oriented branched surface in \(M\). To control such extension, a condition on the veering triangulation is sufficient. In particular, it is satisfied when the second Betti number of \(M\) is less than four.
MSC:
57M99 General low-dimensional topology
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References:
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