Given a closed hyperbolic –manifold with a quasigeodesic flow, we construct a –equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal to the lifted flow on has a natural compactification to a closed disc that inherits a –action. The embedding extends continuously to the compactification, and restricts to a surjective –equivariant map on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic –manifolds.
"Quasigeodesic flows and sphere-filling curves." Geom. Topol. 19 (3) 1249 - 1262, 2015. https://doi.org/10.2140/gt.2015.19.1249