#### Volume 16, issue 1 (2012)

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

### Sérgio Fenley

Geometry & Topology 16 (2012) 1–110
##### Abstract

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.

##### Keywords
pseudo-Anosov flow, Gromov hyperbolic group, ideal boundary, quasigeodesic flow
##### Mathematical Subject Classification 2000
Primary: 37C85, 37D20, 53C23, 57R30
Secondary: 58D19, 37D50, 57M50
##### Publication
Received: 22 May 2009
Revised: 18 April 2011
Accepted: 3 February 2010
Published: 2 January 2012
Proposed: Danny Calegari
Seconded: Benson Farb, Dmitri Burago
##### Authors
 Sérgio Fenley Department of Mathematics Florida State University Room 208, 1017 Academic Way Tallahassee FL 32306-4510 USA