This project proposes to use the theory of geodesic currents to study links between the theory of hyperbolic manifolds, rational dynamics and number theory. The project consists of three themes. The first theme is to study metric structures introduced by the PI and Taylor, on Teichmuller space and quasifuchsian space. These structures generalize the classical Weil-Petersson metric on Teichmuller space to quasifuchsian space. In recent related work, McMullen has defined a Weil-Petterson type metric on the moduli of certain dynamical systems, thereby adding another entry to the Sullivan dictionary between hyperbolic geometry and the dynamics of rational maps. The second theme is to study measures introduced by the PI and Dumas, arising from considering the intersection of a ``random geodesic'' with a geodesic laminations on a hyperbolic surface. The measures give the distribution of lengths of geodesic arcs in the complement of the geodesic lamination and are explicitly given by the image of the volume measure on the unit tangent bundle of the surface under a certain measurable real map. These measures closely resemble the Sato-Tate measures studied in number theory and the theory of L-functions which arise by taking the image of Haar volume measure under a real representation. The third theme is to study the torsion of a quasifuchsian manifold. This invariant is the imaginary part of the complex length of the Patterson-Sullivan measure and is a natural invariant to measure the average ``twistedness'' of a quasifuchsian group and an be thought of as a Chern-Simons invariant for quasifuchsian manifolds. In particular, the PI would like to describe how the torsion changes under bending deformations.
This project lies at the intersection of hyperbolic geometry, dynamics and number theory. In each of these areas, natural spaces arise, called Moduli spaces, which describe the possible deformations of a given object. In the case where the object is a hyperbolic surface, the space we consider is the space of all hyperbolic geometries on the surface, called Teichmuller space. We propose to study these Moduli spaces by considering geometric invariants arising from geodesic currents on the objects. These geodesic currents are uniformly random paths on the objects. In the case of the hyperbolic surface we consider the Liouville geodesic current associated with a given surface. By measuring the variation of the length of the Liouville geodesic current as we move around in Teichmuller space, we obtain important geometric information about the surface. By studying these invariants, we hope to elucidate our understanding of the three areas and the connections between them.
