An algorithm that decides, from a combinatorial box decomposition of a pseudo-Anosov flow on a closed 3-manifold, whether the flow has 'perfect fits' - stable and unstable leaves that become asymptotic in the universal cover, which is exactly the obstruction to a canonical veering triangulation. It combines a routine that certifies perfect fits (enumerating periodic orbits by symbolic dynamics and testing Fenley's freely-homotopic-orbit criterion via the conjugacy problem for 3-manifold groups) with one that certifies their absence by directly building the veering triangulation, yielding decidability of the orbit-equivalence problem for pseudo-Anosov flows without perfect fits.
A pseudo-Anosov flow on a closed 3-manifold has 'perfect fits' when stable and unstable leaves become asymptotic in the universal cover; perfect fits are exactly the obstruction to the flow correspon…