Uniqueness of gluings and virtual finiteness of pseudo-Anosov flows on graph manifolds

Abstract: In this article, we give a characterization of when two pseudo-Anosov flows obtained via gluings of pieces of pseudo-Anosov flows are orbit equivalent. As an application of this work, and the description of pseudo-Anosov flows in Seifert pieces due to Barbot and Fenley, we prove a ``virtual'' version of the Finiteness Conjecture for transitive pseudo-Anosov flows on graph-manifolds.