A Teichmuller space for negatively curved surfaces

We first describe the action of the fundamental group of a closed surface sigma$\Sigma$ of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of S1xR$S<^>1\times \mathbf {R}$. Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite-dimensional group and focuses on a family of metrics, relying on a construction of O. Biquard, which is parametrised by the infinite-dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmuller space.