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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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