There begins to be many hints (volume conjecture, etc...) of a connection between topological quantum field theory and hyperbolic geometry. We investigate a construction which mixes these two fields.

The quantum Teichmuller space is a deformation of the field of rational functions on the Teichmuller space of a surface, namely on the space of hyperbolic metrics on the surface. It turns out that the representation theory of this purely algebraic object is controlled by the same data as a pleated surface in a hyperbolic 3-manifold. As an application we construct invariants of surface diffeomorphisms, by applying this correspondence to geometric data extracted from the hyperbolic metric on the mapping torus of the surface diffeomorphism.

This is joint work with Xiaobo Liu.

Weil-Petersson Metric to Quasi-Fuchsian Space

Ergodic theory of the earthquake flow

We prove that the length function associated with a geodesic current is

analytic on Quasifuchsian space. Using this, we show that a certain length

distortion function associated with the Patterson-Sullivan measure is

analytic on Quasifuchsian space. Taking the second derivative of length

distortion, we obtain a symmetric bilinear two-tensor that extends the

Weil-Petersson metric on Fuschsian space to the whole of Quasifuchsian

space.

In this talk we study the the ergodic properties of the earthquake flow on the bundle of geodesic measured laminations by using a relationship between the earthquake flow and the Teichmuller horocycle flow. We use these results to find the growth of the number of simple closed geodesics on a hyperbolic surface.

Deformations and Self-Maps of the Universal Hyperbolic Solenoid

Abstract in PDF

The mapping class group acts naturally on quasi-Fuchsian space, and this

action extends to an action on the appropriate variety of surface

group representations. We study the dynamics of this action using hyperbolic

geometry, and prove a non-existence result for mapping class group

invariant meromorphic functions on the character variety. This is joint

work with Juan Souto.