Free Seifert pieces of pseudo-Anosov flows

Barbot, Thierry¹ ; Fenley, Sérgio R²

¹ LMA Avignon University, Campus Jeah-Henri Fabre, Avignon, France
² Department of Mathematics, Florida State University, Tallahassee, FL, United States

Geometry & topology, Tome 25 (2021) no. 3, pp. 1331-1440.

Résumé

We prove a structure theorem for pseudo-Anosov flows restricted to Seifert-fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration such that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A nonperiodic Seifert-fibered piece is called free. In a previous paper (Geom. Topol. 17 (2013) 1877–1954) we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. Here we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blowup of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. In the proof we introduce almost $k$–convergence groups and prove a convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we carefully define what is a hyperbolic blowup.

Détail

DOI : 10.2140/gt.2021.25.1331

Classification : 34D23, 37D05, 37D20, 37D50, 57R30, 34C25, 34C45, 37C10, 57M50, 57M60
Keywords: pseudo-Anosov flows, torus decomposition, Seifert pieces

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