Contact Anosov flows on hyperbolic 3–manifolds

Foulon, Patrick¹ ; Hasselblatt, Boris²

¹ 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
² Department of Mathematics, Tufts University, Medford, MA 02155, USA

Geometry & topology, Tome 17 (2013) no. 2, pp. 1225-1252.

Résumé

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.

Détail

DOI : 10.2140/gt.2013.17.1225

Classification : 37D20, 57N10, 57M50
Keywords: Anosov flow, 3–manifold, contact flow, hyperbolic manifold, surgery

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