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
–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.
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