Classical Teichmüller theory and (2+1) gravity

We consider classical Teichmüller theory and the geodesic flow on the cotangent bundle of the Teichmüller space. We show that the corresponding orbits provide a canonical description of certain (2+1) gravity systems in which a set of point-like particles evolve in universes with topology Σg×R where Σg is a Riemann surface of genus g>1. We construct an explicit Yorkʹs slicing presentation of the associated spacetimes, we give an interpretation of the asymptotic states in terms of measured foliations and discuss the structure of the phase spaces.