Back to WTC Spring 2004

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Francis Bonahon
Quantum Teichmuller Theory

    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.

Martin Bridgeman
Analyticity of the Length Function for Geodesics Currents; An Extension of  the
          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.


Jim Cannon


Maryam Mirzakhani
Ergodic theory of the earthquake flow

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.

Dragomir Saric
Deformations and Self-Maps of the Universal Hyperbolic Solenoid

Abstract in PDF

Pete Storm
Dynamics of the Mapping Class Group Action on the Character Variety

    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.