Quasigeodesic flows and sphere-filling curves

Given a closed hyperbolic 3–manifold M with a quasigeodesic flow, we construct a π₁ –equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on ℍ³ has a natural compactification to a closed disc that inherits a π₁ –action. The embedding P→ℍ³ extends continuously to the compactification, and restricts to a surjective π₁ –equivariant map ∂P→∂ℍ³ on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic 3–manifolds.