Quasigeodesic flows and sphere-filling curves

Frankel, Steven

doi:10.2140/gt.2015.19.1249

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Quasigeodesic flows and sphere-filling curves

Steven Frankel

Geometry & Topology 19 (2015) 1249–1262

DOI: 10.2140/gt.2015.19.1249

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 to a closed disc that inherits a $π_{1}$ –action. The embedding $P ↪ ℍ^{3}$ extends continuously to the compactification, and restricts to a surjective $π_{1}$ –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

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

Volume 19, issue 3 (2015)