Abstract

Ideal triangulations were introduced by Thurston as a tool for studying hyperbolic three-manifolds. Taut ideal triangulations were introduced by Lackenby as a tool for studying ``optimal’’ representatives of second homology classes.

After these applications in geometry and topology, it is time for dynamics. Veering triangulations (taut ideal triangulations with certain decorations) were introduced by Agol to study the mapping tori of pseudo-Anosov homeomorphisms. Gueritaud gave an alternative construction, and then Agol and Gueritaud generalised it to give veering triangulations for three-manifolds admitting pseudo-Anosov flows (without perfect fits).

We prove the converse of their result: that is, from any veering triangulation we produce a canonical dynamic pair of branched surfaces (in the sense of Mosher). These give flows on appropriate Dehn fillings of the original manifold. Furthermore, our construction and that of Agol—Gueritaud are inverses. This then gives a ``perfect’’ combinatorialisation of pseudo-Anosov flow (without perfect fits).

This is joint work with Henry Segerman.