Finiteness of the gluing procedure of pseudo-Anosov flows in dimension 3

One approach to study and classify pseudo-Anosov flows on toroidal 3-manifolds that was successfully developed over many years by Barbot and Fenley is to use the JSJ decomposition of the manifold that is adapted to the flow, and then describe the restriction to the pieces. 

What had been missing so far from this strategy was a condition on how one can uniquely recover the original flow on the given manifold from gluing the model obtained in each piece.

I will present a criterion to control the reconstruction, ensuring that only finitely many pseudo-Anosov flows can arise from the gluing procedure. This provides a key step toward the broader finiteness problem for pseudo-Anosov flows on a fixed 3-manifold. This is a joint work with Thomas Barthelmé.