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!



Comparing different layered structures for fibered 3-manifolds: example request.
The two triangulations (or cellulations) do not coincide in general.


Context:
In fact, given a centrally symmetric convex shape $S$ in the plane, one could at any time $t$ on the Teichmüller line look for maximal copies of $S$ (up to scaling and translation, but not rotation) disjoint from the singularities. Such maximal copies will have some number of singularities in their boundary. The convex hull of those singularities gives a polygon (let us say a triangle since this is the generic case), and the union of these triangles (over all maximal copies of $S$ at time $t$) is a triangulation of the surface at time $t$. Letting $t$ vary, we will see (generically) a sequence of diagonal exchanges, forming a layered triangulation. The two examples considered so far are for $S$ a square (Agol) and a circle (Dirichlet). For other shapes $S$ we would get even more different triangulations. Also, if all singularities have even number of prongs we can drop the condition that $S$ be centrally symmetric.


Now the counter-example. Consider a flat surface $F$ obtained by gluing together opposite sides of a regular Euclidean $8$-gon. Call the vertices $N, NE, E, SE, S, SW, W, NW$ (like the cardinal directions); they are the singularities in the quotient. Stretch slightly in the $NE, SW$ direction, so that the disk circumscribed to $N, NE, E$ does not meet any other points: the triangle $N, NE, E$ will therefore appear in the Dirichlet decomposition. However, it is not in the Agol decomposition (in fact not even the edge from $N$ to $E$ is there) since any rectangle containing $N$ and $E$ contains $NE$.


It only remains to make sure we can find a pseudo-Anosov whose Teichmüller line passes through (or near) this metric. Since $F$ is a Veech surface, this is not difficult: there is a whole (non-cocompact) lattice of $SL(2, R)$ inducing mapping classes of $F$, which can therefore be chosen pseudo-Anosov with eigendirections close to vertical and horizontal (or to any other given pair of directions). 


