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Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. (English) Zbl 1279.37026
Given a general pseudo-Anosov flow in a closed 3-manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. The author constructs a natural compactification of this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, then he shows that the ideal circle of the orbit space has a natural quotient space which is a sphere. This sphere is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold.
The main result of this paper is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. The author concludes that the fundamental group of the manifold is Gromov-hyperbolic and so the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. These results give a new proof that the fundamental group of a closed, atoroidal 3-manifold which fibers over the circle is Gromov-hyperbolic. The main result also implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and that the stable and unstable foliations of these flows are quasi-isometric foliations. If a foliation is \({\mathbb R}\)-covered or with one-sided branching in an aspherical, atoroidal 3-manifold, then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.

MSC:
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57R30 Foliations in differential topology; geometric theory
57M50 General geometric structures on low-dimensional manifolds
58D19 Group actions and symmetry properties
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