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

TL;DR

This paper explores the relationship between the hyperbolic geometry of 3-manifolds and the dynamical properties of pseudo-Anosov flows via the curve graph of embedded surfaces.

Contribution

It establishes a connection between hyperbolic geometric invariants and dynamical invariants derived from the curve graph in the context of pseudo-Anosov flows.

Findings

Hyperbolic volume relates to dynamical invariants of the flow.

Short geodesics are connected to the curve graph properties.

Circumference measurements reflect flow dynamics.

Abstract

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