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Contact Anosov flows on hyperbolic 3–manifolds

Patrick Foulon and Boris Hasselblatt

Geometry & Topology 17 (2013) 1225–1252

Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an E–transverse Legendrian link that encompasses both the Handel–Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3–manifolds, any of which have remarkable dynamical and geometric properties.

To the latter end we include a proof of a folklore theorem from 3–manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3–manifold.

Dedicated to the memory of William P Thurston

Anosov flow, 3–manifold, contact flow, hyperbolic manifold, surgery
Mathematical Subject Classification 2010
Primary: 37D20
Secondary: 57N10, 57M50
Received: 1 February 2012
Revised: 10 February 2013
Accepted: 13 October 2012
Published: 29 May 2013
Proposed: Danny Calegari
Seconded: Ronald Stern, Leonid Polterovich
Patrick Foulon
Institut de Recherche Mathematique Avancée
UMR 7501 du Centre National de la Recherche Scientifique
7 Rue René Descartes
67084 Strasbourg Cedex
Centre International de Rencontres Mathématiques
163 Avenue de Luminy
Case 916
13288 Marseille Cedex 9
Boris Hasselblatt
Department of Mathematics
Tufts University
Medford, MA 02155