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 to a closed disc that inherits a ${\pi }_1$–action. The embedding $P\hookrightarrow {ℍ}^3$ extends continuously to the compactification, and restricts to a surjective ${\pi }_1$–equivariant map $\partial P\to \partial {ℍ}^3$ on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic $3$–manifolds.