Sergio Fenley, Florida State University
Thursday October 27, 4:00PM,
Herman Brown 227
What can a flow say about (the large scale or asymptotic
structure of) a manifold?
Pseudo-Anosov flows are extremely common amongst
3-manifolds and they are intimately related to the topology.
In this talk we show they are also strongly connected with the
asymptotic structure of the universal cover M~ and the large
scale geometry of M~. Given a pseudo-Anosov flow, the lift to
the universal cover has orbit space which is homeomorphic to a
plane. It turns out one can always compactify the orbit space
with an ideal circle so that the union is a closed disk.
With an additional (and very common) condition on the flow,
we can show that the ideal circle of the flow has a quotient R
which is an ideal boundary of M~ and produces a compactification
of M~. This quotient is a 2 sphere and the fundamental group acts
in this sphere. The action in R has excellent properties: it is a
uniform convergence action. By a result of Bowditch this implies
that the fundamental group G of M is Gromov hyperbolic, R is
homeomorphic to the ideal boundary of G and the action in R is
conjugate to the action of G in its boundary. In this way the
large scale geometric properties of the group and the universal
cover are completely described using only the dynamics of the
pseudo-Anosov flow in this situation. There are consequences for
metric properties of flows and foliations.