Non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic

TL;DR

This paper proves that non R-covered Anosov flows in hyperbolic 3-manifolds are quasigeodesic, and establishes the existence of quasigeodesic flows and related properties in these manifolds.

Contribution

It demonstrates that non R-covered Anosov flows are quasigeodesic and shows that hyperbolic 3-manifolds support quasigeodesic flows up to a double cover, with new properties of foliations and invariant curves.

Findings

Non R-covered Anosov flows are quasigeodesic.

Hyperbolic 3-manifolds support quasigeodesic flows up to a double cover.

Stable and unstable foliations have the continuous extension property.

Abstract

The main result is that if an Anosov flow in a closed hyperbolic three manifold is not R-covered, then the flow is a quasigeodesic flow. We also prove that if a hyperbolic three manifold supports an Anosov flow, then up to a double cover it supports a quasigeodesic flow. We prove the continuous extension property for the stable and unstable foliations of any Anosov flow in a closed hyperbolic three manifold, and the existence of group invariant Peano curves associated with any such flow.