Universal circles for quasigeodesic flows

We show that if $M$ is a hyperbolic $3$ –manifold which admits a quasigeodesic flow, then $π_{1} (M)$ acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic $3$ –manifold without a quasigeodesic flow, answering a long-standing question of Thurston.

Keywords

quasigeodesic flows, universal circles, laminations, Thurston norm, 3-manifolds

Mathematical Subject Classification 2000

Primary: 57R30

Secondary: 37C10, 37D40, 53C23, 57M50

References

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Publication

Received: 15 June 2004

Revised: 10 September 2006

Accepted: 25 October 2006

Published: 29 November 2006

Proposed: David Gabai

Seconded: Benson Farb, Walter Neumann

Authors

Danny Calegari
Department of Mathematics California Institute of Technology Pasadena CA, 91125 USA

Volume 10, issue 4 (2006)

Danny Calegari

Abstract