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Ideal boundaries of pseudo-Anosov flows and uniform
convergence groups with connections and applications to
large scale geometry
Sérgio Fenley
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Geometry & Topology 16 (2012)
1–110
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Abstract
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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.
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Keywords
pseudo-Anosov flow, Gromov hyperbolic
group, ideal boundary, quasigeodesic flow
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Mathematical Subject Classification
Primary: 37C85, 37D20, 53C23, 57R30
Secondary: 37D50, 57M50, 58D19
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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
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