From an Anosov flow on a 3-manifold, one can extract an action of the fundamental group of the manifold on a plane preserving a pair of transverse foliations, and on a compactification of the plane by an ideal circle. My talks will give an introduction to this picture and show a recent application, joint with Thomas Barthelmé and Steven Frankel, to the classification problem for Anosov flows. By proving rigidity results for certain discrete group actions on planes and their boundaries, we show that transitive (pseudo-)Anosov flows are determined up to orbit equivalence by the algebraic data of the set of free homotopy classes of periodic orbits. 
https://indico.math.cnrs.fr/event/7554/