Quasigeodesic flows and sphere-filling curves

Frankel, Steven¹

¹ Department of Mathematics, Yale University, PO Box 208283, New Haven, CT 06520-8283, USA

Geometry & topology, Tome 19 (2015) no. 3, pp. 1249-1262.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.1249

Classification : 57M60, 57M50, 37C27
Keywords: quasigeodesic flows, Cannon–Thurston, pseudo-Anosov flows

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