Quasigeodesic flows and sphere-filling curves

Steven Frankel

doi:10.2140/gt.2015.19.1249

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arXiv · 1210.7050

Quasigeodesic flows and sphere-filling curves

Steven Frankel

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Abstract

Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_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 π_1 action. The embedding of P in H^3 extends continuously to the compactification and the restriction to the boundary is a surjective π_1-equivariant map from S^1 to S^2_\infty. This generalizes the result of Cannon and Thurston for fibered hyperbolic 3-manifolds.

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Steven Frankel. 2012-10-26. Quasigeodesic flows and sphere-filling curves. https://doi.org/10.2140/gt.2015.19.1249

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