Given a filling closed geodesic on a hyperbolic surface, one can consider its canonical lift in the projective tangent bundle. Drilling this knot, one obtains a hyperbolic 3-manifold. In this talk we are interested in volume bounds for these manifolds in terms of geometric quantities of the geodesic, such as the hyperbolic length. In particular, we give a volume lower bound in terms of length when the filling geodesics are closed geodesics converging to the Liouville geodesic current. The bound is given in terms of a counting problem in the unit tangent bundle that we solve by applying an exponential multiple mixing result for the geodesic flow.
This is joint work with Tommaso Cremaschi, Yannick Krifka and Franco Vargas Pallete.