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Free Seifert pieces of pseudo-Anosov flows

Thierry Barbot and Sérgio R Fenley

Geometry & Topology 25 (2021) 1331–1440

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.

pseudo-Anosov flows, torus decomposition, Seifert pieces
Mathematical Subject Classification 2010
Primary: 34D23, 37D05, 37D20, 37D50, 57R30
Secondary: 34C25, 34C45, 37C10, 57M50, 57M60
Received: 4 February 2017
Revised: 15 March 2019
Accepted: 4 June 2020
Published: 20 May 2021
Proposed: Anna Wienhard
Seconded: David Gabai, Mladen Bestvina
Thierry Barbot
LMA Avignon University
Campus Jeah-Henri Fabre
Sérgio R Fenley
Department of Mathematics
Florida State University
Tallahassee, FL
United States