Coarse geometry and closing lemmas

Steven Frankel (Yale)

Abstract: A flow on a manifold is called quasigeodesic if each orbit is coarsely comparable to a geodesic. A flow is called pseudo-Anosov if it has a transverse attracting-repelling structure. If the ambient manifold is hyperbolic, these two conditions are mysteriously related. In fact, Calegari conjectures that every quasigeodesic flow on a closed hyperbolic manifold can be deformed into a pseudo-Anosov flow. The transverse attracting-repelling structure of a pseudo-Anosov flow lends its orbits a form of rigidity. For example, the Anosov closing lemma says that if a point returns close to itself after a long time, then there's a nearby point that returns exactly to itself. The goal of this talk is clarify the relationship between quasigeodesic and pseudo-Anosov flows. We will show that a quasigeodesic flow has a *coarse* transverse attracting-repelling structure, and use this to prove a closing lemma. In particular, we will show that every quasigeodesic flow on a closed hyperbolic manifold has periodic orbits, answering a question of Calegari's.