340 research outputs found
Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits
In this article, we study the knots realized by periodic orbits of R-covered
Anosov flows in compact 3-manifolds. We show that if two orbits are freely
homotopic then in fact they are isotopic. We show that lifts of periodic orbits
to the universal cover are unknotted. When the manifold is atoroidal, we deduce
some finer properties regarding the existence of embedded cylinders connecting
two given homotopic orbits.Comment: 20 pages, 9 figure
Pseudo-Anosov flows in toroidal manifolds
We first prove rigidity results for pseudo-Anosov flows in prototypes of
toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered
manifold is up to finite covers topologically equivalent to a geodesic flow and
we show that a pseudo-Anosov flow in a solv manifold is topologically
equivalent to a suspension Anosov flow. Then we study the interaction of a
general pseudo-Anosov flow with possible Seifert fibered pieces in the torus
decomposition: if the fiber is associated with a periodic orbit of the flow, we
show that there is a standard and very simple form for the flow in the piece
using Birkhoff annuli. This form is strongly connected with the topology of the
Seifert piece. We also construct a large new class of examples in many graph
manifolds, which is extremely general and flexible. We construct other new
classes of examples, some of which are generalized pseudo-Anosov flows which
have one prong singularities and which show that the above results in Seifert
fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows.
Finally we also analyse immersed and embedded incompressible tori in optimal
position with respect to a pseudo-Anosov flow.Comment: 44 pages, 4 figures. Version 2. New section 9: questions and
comments. Overall revision, some simplified proofs, more explanation
Ideal boundaries of pseudo-Anosov flows and uniform convergence groups, with connections and applications to large scale geometry
Given a general pseudo-Anosov flow in a three manifold, the orbit space of
the lifted flow to the universal cover is homeomorphic to an open disk. We
compactify 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 and 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 implies that pseudo-Anosov flows without perfect fits are quasigeodesic
flows and we show that the stable/unstable foliations of these flows are
quasi-isometric. Finally we apply these results to foliations: if a foliation
is R-covered or with one sided branching in an 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.Comment: 69 pages. Major revision, more explanations and simplified some
simplified proofs. Detailed explanations of scalloped regions, parabolic
points and perfect fit horoballs. 22 figures (3 new figures
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