Quasigeodesic flows and sphere-filling curves

TL;DR

This paper constructs a sphere-filling curve in the boundary of hyperbolic space for closed hyperbolic 3-manifolds with quasigeodesic flows, generalizing Cannon and Thurston's work.

Contribution

It introduces a method to produce c6_1-equivariant sphere-filling curves in the boundary of hyperbolic space for a broad class of 3-manifolds with quasigeodesic flows.

Findings

Constructed a c6_1-equivariant sphere-filling curve in the boundary of hyperbolic space.

Extended the Cannon-Thurston result to non-fibered hyperbolic 3-manifolds.

Provided a natural compactification of transversals with a continuous boundary map.

Abstract

Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a \pi_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a \pi_1 action. The embedding of P in H^3 extends continuously to the compactification and the restriction to the boundary is a surjective \pi_1-equivariant map from S^1 to S^2_\infty. This generalizes the result of Cannon and Thurston for fibered hyperbolic 3-manifolds.