Sergio Fenley, Florida State University

What can a flow say about (the large scale or asymptotic structure of) a manifold?

Thursday October 27, 4:00PM, Herman Brown 227
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.