Pseudo-Anosov flows, hyperbolic geometry, and the curve graph

In this talk, I will discuss a joint work with Samuel J. Taylor, where we consider embedded non-fiber surfaces in a closed hyperbolic 3-manifold transverse to a pseudo-Anosov flow. Utilizing foundational work on Kleinian surface groups, we connect the hyperbolic geometry of the surface to the dynamics of the flow via the curve graph. More specifically, the flow gives a pair of special multicurves on the surface, and we show that they are also geometrically distinguished, enabling us to extract geometric information, such as the position of short curves and the volume, from the relation of the pair of multicurves in the curve graph. Our result also connects to recent study of the geometry of endperiodic mapping tori.