  Given a filling primitive geodesic curve in a closed hyperbolic surface one
obtains a hyperbolic three-manifold as the complement of the curve's canonical
lift to the projective tangent bundle. In this paper we give the first known
lower bound for the volume of these manifolds in terms of the length of generic
curves. We show that estimating the volume from below can be reduced to a
counting problem in the unit tangent bundle and solve it by applying an
exponential multiple mixing result for the geodesic flow.
