Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.
Many natural functionals on closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of hyperbolic length with respect to a fixed metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve functional that guarantees it extends to geodesic currents. It is easy to see that our criterion is satisfied for almost all the known examples of continuous functions on geodesic currents, while also applying to new examples like extremal length.
Given a filling primitive geodesic loop in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the loop'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 loops. 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.
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 for a collection of curves with asymptotic density one. 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.
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