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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 π1–equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on 3 has a natural compactification to a closed disc that inherits a π1–action. The embedding P3 extends continuously to the compactification, and restricts to a surjective π1–equivariant map P 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
References
Publication
Received: 22 July 2013
Revised: 25 February 2014
Accepted: 26 July 2014
Published: 21 May 2015
Proposed: David Gabai
Seconded: Benson Farb, Danny Calegari
Authors
Steven Frankel
Department of Mathematics
Yale University
PO Box 208283
New Haven, CT 06520-8283
USA