Quasigeodesic flows and sphere-filling curves

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.