#### Volume 19, issue 3 (2015)

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Quasigeodesic flows and sphere-filling curves

### Steven Frankel

Geometry & Topology 19 (2015) 1249–1262
##### 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 $P↪{ℍ}^{3}$ 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.

##### Keywords
quasigeodesic flows, Cannon–Thurston, pseudo-Anosov flows
##### Mathematical Subject Classification 2010
Primary: 57M60
Secondary: 57M50, 37C27