  We consider classical Teichmuller theory and the geodesic flow on the
cotangent bundle of the Teichmuller 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 S_g x R
where S_g is a Riemann surface of genus g >1. We construct an explicit York's
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.
