Descent of orbit-equivalence finiteness from a finite cover

Determine whether, for a closed 3-manifold M and a family of transitive pseudo-Anosov flows on M whose lifts to a common finite cover are all orbit equivalent, the family on M belongs to at most finitely many orbit-equivalence classes.

Background

Question 3 asks whether orbit-equivalence finiteness can be descended from a finite cover to the original closed 3-manifold. A positive answer would convert the paper’s virtual finiteness theorem into an actual finiteness theorem for the relevant families of pseudo-Anosov flows.

The paper notes that the question has a positive answer on hyperbolic manifolds because such manifolds have finite mapping class group, but states that the toroidal case is wide open. The authors also report that they do not know any example in which the number of orbit-equivalence classes downstairs exceeds one.

References

Note that to deduce the Finiteness Conjecture from the virtual version we obtained, one could hope to answer positively the following question, which is interesting in and of itself: Question 3. Let M be a closed 3-manifold and {ϕi} a family of (transitive) pseudo-Anosov flows on M . Suppose that there exists a finite cover ˆM such that all the lifts ˆϕi are orbit equivalent in ˆM . Does there exist n such that {ϕi} belongs to at most n distinct orbit equivalent classes? Note that if the flows ˆϕi are orbit equivalent via a homeomorphism that is homotopic to the identity, then the ϕi themselves are orbit equivalent (see [BTZ26, Proposition 2.5]). It implies that the answer to Question 3 is positive on hyperbolic manifolds as they have finite mapping class group, but the toroidal case seems to be wide open. We do not even know of an example where n̸ = 1.

— Uniqueness of gluings and virtual finiteness of pseudo-Anosov flows on graph manifolds  (2608.13526 - Barthelmé et al., 13 Aug 2026) in Question 3 and the discussion immediately following it, Section 1.1, page 4

Prove the Finiteness Conjecture for pseudo-Anosov flows using contact geometry for closed hyperbolic manifolds. Extend to the toroidal case.

— On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry  (2609.03700 - Bowden et al., 3 Sep 2026) in Section 5, Outlook: Further Questions, Problem “General pseudo-Anosov Flows”