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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!

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2 Answers 2

up vote 5 down vote accepted

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).

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Nice example, François! –  Dave Futer Jun 28 '13 at 18:13
Nice example indeed. At this point, it might be reasonable to ask if all or some of these different layered structures which Francois has built admit a positive angle structure as in the case of Agol's triangulations. It seems reasonable, as all the trivial obstructions to this do not occur. Still, I don't see directly how to generalize the proof in the "Agol/square" case, which heavily relies on the veeringness property, to the general setting. –  leone slavich Aug 15 '13 at 0:01

I believe the two constructions should coincide. Here's why.

There is an interesting and very close variant of the Dirichlet domain construction you're describing. Take a singular flat metric on the fiber surface $F$, in which the stable foliation is vertical, and the unstable foliation horizontal. Now, consider a maximal rectangle $R \subset F$, with sides along the leaves of the foliations, and whose interior is disjoint from the singularities. By maximality, this rectangle must have a singular point along each of its 4 sides, which defines a quadrilateral in $F$.

Thickening the rectangle gives a tetrahedron. Let's say, by convention, that the diagonal connecting horizontal sides is above the diagonal connecting vertical sides (this matches the verticality of the stable foliation). It's not hard to check that these tetrahedra glue up to give a natural, combinatorially canonical triangulation of $F \times [0,1]$, which is invariant under the monodromy.

Guéritaud has proved that the above construction produces the same canonical triangulation as Agol's train track construction. I don't believe his proof is written up, but an announcement (including a summary of the construction) appears in this Oberwolfach report:


Now, it remains to ask: does Guéritaud's construction produce the the same result as your Dirichlet domain construction? It seems to me that they should coincide. For Guéritaud, triangles in $F$ come from rectangles in the singular flat metric with two singular points on adjacent sides and one singular point in the opposite corner. It seems likely that after rescaling in the horizontal or vertical direction, those 3 points would lie on a circle whose interior is free of other singular points, i.e. those 3 points define a cell of the Dirichlet domain. Then, one would need to check that as one moves along the Teichmuller line, i.e. rescales in the horizontal and vertical direction, the inscribed circles with empty interior transform in the right way.

The above paragraph is not a proof, of course. But the common setting of singular flat metrics should enable one to find either a proof or a counterexample. Update: I see that François has constructed a counterexample.

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Dave, thank you for your reply.I'm still not totally convinced that the two constructions are the same.The veeringness condition can be rephrased in terms of slopes of the ideal edges: an edge is left veering if its slope is positive, and right veering if it is negative.At a certain point along the Teichmuller line, we might have a maximal circle, with no singular points in the interior, which circumscribes an ideal triangle whose edges all have positive slopes. Such a triangle cannot appear in the Agol-Gueritaud construction. There might be an obstruction to this happening which I don't see. –  leone slavich Jun 25 '13 at 11:53

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