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

Starting with a pseudo-Anosov flow $\phi$ on a closed hyperbolic $3$-manifold $M$ and an embedded surface $S\subset M$ that is (almost) transverse to $\phi$, we relate the hyperbolic geometry of $M$ (e.g. volume, circumference, short geodesics) to dynamical invariants of $\phi$ encoded by the curve graph of $S$.