Volume 17, issue 2 (2013)

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Contact Anosov flows on hyperbolic 3–manifolds

Patrick Foulon and Boris Hasselblatt

Geometry & Topology 17 (2013) 1225–1252
Abstract

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