Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular points of the two invariant foliations.

Ian Agol has introduced a canonical layered ideal triangulation of such manifolds through a periodic splitting sequence of train tracks. Such triangulations are characterized by a combinatorial property called veeringness.

Another layered structure can be obtained by considering the flow of incomplete euclidean metrics associated to the Teichmuller line determined by the monodromy. Given one of these metrics, you consider the Dirichlet decomposition with respect to the set of singular points . This changes finitely many times along the geodesic segment determined by the monodromy, and allows us to produce a layered cell decomposition (it is not clear from the definition that it should always be a triangulation), see http://ldtopology.wordpress.com/2009/08/20/canonical-triangulations-of-surface-bundles/.

If we restrict to the case where the fiber is a once punctured torus, the two constructions are the same, and produce the usual Floyd-Hatcher monodromy ideal triangulation.

I am looking for an example of a manifold such that Agol's construction and the Dirichlet domain construction produce different results.

Thank you very much!