Geometry & Topology

Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry

Sérgio Fenley

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Given a general pseudo-Anosov flow in a closed three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We construct 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, we then show: 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 is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This gives an entirely new proof that the fundamental group of a closed, atoroidal 3–manifold which fibers over the circle is Gromov hyperbolic. In addition with further geometric analysis, the main result also implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and that the stable/unstable foliations of these flows are quasi-isometric foliations. Finally we apply these results to (nonsingular) foliations: if a foliation is R–covered or with one sided branching in an aspherical, atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.

Article information

Geom. Topol., Volume 16, Number 1 (2012), 1-110.

Received: 22 May 2009
Revised: 18 April 2011
Accepted: 3 February 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 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; geometric theory
Secondary: 58D19: Group actions and symmetry properties 37D50: Hyperbolic systems with singularities (billiards, etc.) 57M50: Geometric structures on low-dimensional manifolds

pseudo-Anosov flow Gromov hyperbolic group ideal boundary quasigeodesic flow


Fenley, Sérgio. Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1--110. doi:10.2140/gt.2012.16.1.

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