Volume bounds for the canonical lift complement of a random geodesic

TL;DR

This paper establishes a lower bound on the volume of hyperbolic 3-manifolds derived from canonical lifts of generic geodesic curves on surfaces, linking geometric properties to dynamical counting problems.

Contribution

It provides the first known lower volume bounds for these manifolds, connecting geometric and dynamical systems techniques.

Findings

Lower volume bounds in terms of curve length

Reduction of volume estimation to a counting problem

Application of exponential multiple mixing for geodesic flow

Abstract

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.