Quasigeodesic pseudo-Anosov flows in hyperbolic 3-manifolds and connections with large scale geometry

TL;DR

This paper establishes a topological and dynamical condition that characterizes when pseudo-Anosov flows in hyperbolic 3-manifolds are quasigeodesic, linking flow properties with large-scale geometric behavior.

Contribution

It provides a necessary and sufficient condition for quasigeodesic pseudo-Anosov flows based on orbit homotopy bounds and constructs a flow ideal boundary related to Gromov compactification.

Findings

Flows are quasigeodesic iff orbit homotopy bounds are finite

Fundamental group acts as a uniform convergence group on flow boundary

Flow ideal compactification is homeomorphic to Gromov compactification

Abstract

In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover. We prove that such flows are quasigeodesic if and only if there is an upper bound, depending only on the flow, to the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow. The main ingredient is a proof that under the boundedness condition, the fundamental group of the manifold acts as a uniform convergence group on a flow ideal boundary of the universal cover. We also construct a flow ideal compactification of the universal cover and prove it is equivariantly homeomorphic to the Gromov compatification. This implies the quasigeodesic behavior of the flow. The flow ideal boundary and flow ideal compactification are constructed using only the structure of the flow.