The transverse finiteness conjecture

Given a fixed finite depth foliation F of an atoroidal manifold M, Gabai and Mosher described how to build a pseudo-Anosov flow on M that is almost transverse to F. The construction is somewhat flexible, and hence some foliations are transverse to multiple inequivalent pseudo-Anosov flows. However, Mosher’s Transverse Finiteness Conjecture (1996) asserts that a given finite depth F can be almost transverse to only finitely many pseudo-Anosov flows. In joint work with Michael Landry, we prove this conjecture. We also prove a related rigidity result: if F is almost transverse to some pseudo-Anosov flow without perfect fits, then it is in fact the unique pseudo-Anosov flow transverse to F. In our proofs, we organize possible flows by reconstructing their universal circles inductively through the depths of F, using Cannon—Thurston maps to control the possible choices that arise in the reconstruction.