The layered triangulations I described in my last post reminded me of another construction that uses the same idea of layering tetrahedra onto a triangulated torus in order to change the boundary pattern. David Futer and Francois Gueritaud [1] use this idea to construct ideal triangulations of (most) punctured surface bundles.

Start with an ideal triangulation of a punctured torus (i.e. triangulate a torus with two triangles, then remove the vertex). Any two of the edges cut the torus onto a rectangle, and the third edge forms a diagonal of this rectangle. Take a tetrahedron, choose one of its edges and glue this edge and the two adjecent triangles to the diagonal edge and its two adjacent triangles. The result is a slightly bulging/thickened punctured torus. The triangulation on the top of the bulging torus comes from the bottom by removing the diagonal edge of the rectangle and replacing it with the other diagonal. (This is called a diagonal exchange move.) If we keep gluing in tetrahedra along different edges of the top, we eventually get a space homeomorphic to a punctured torus cross an interval. If we glue the top triangulation to the bottom, we get a punctured torus bundle. The order in which we glued in tetrahedra determines a sequence of diagonal exchange moves that defines the final gluing map and induces an ideal triangulation on the resulting manifold.

I think this construction is beautifully simple, but David and Francois had even more impressive things in mind. Because they understand the structure of the triangulation so well, they are able to apply angle-structure techniques to show that (for monodromies meeting appropriate combinatorial conditions) one can assign angles to the edges of the tetrahedra so that the angles around each edge add up to 360 and each tetrahedron looks like an ideal hyperbolic tetrahedron. With some more heavy machinery, this implies the existence of a hyperbolic structure on the torus bundle. As I understand it, they are currently working on extending the techniques (along with a third author who I can’t remember at the moment) to determine exactly which Dehn fillings on these punctured torus bundles produce hyperbolic manifolds. This doesn’t seem to be on the arXiv yet, but I’m looking forward to it.

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