A group action on a bifoliated plane with no infinite product regions comes from a pseudo-Anosov or expansive flow on a 3-manifold exactly when a certain space of leaf pairs admits a properly discontinuous, cocompact group action.
Agol, Ideal triangulations of pseudo- A nosov mapping tori , Topology and geometry in dimension three, Contemp
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Reconstructing flows from the orbit space
A group action on a bifoliated plane with no infinite product regions comes from a pseudo-Anosov or expansive flow on a 3-manifold exactly when a certain space of leaf pairs admits a properly discontinuous, cocompact group action.