Currents on cusped hyperbolic surfaces and denseness property

TL;DR

This paper proves the density of rational geodesic currents in the space of geodesic currents on cusped hyperbolic surfaces, extending known results from compact surfaces and exploring convergence and intersection properties.

Contribution

It extends the density property of rational geodesic currents to cusped hyperbolic surfaces and analyzes convergence issues related to geodesics connecting cusps.

Findings

Rational geodesic currents are dense in the geodesic current space for cusped surfaces.

Sequences of weighted closed geodesics can converge to geodesics connecting cusps.

The space of subset currents also exhibits density of rational subset currents.

Abstract

The space $\mathrm{GC} (\Sigma)$ of geodesic currents on a hyperbolic surface $\Sigma$ can be considered as a completion of the set of weighted closed geodesics on $\Sigma$ when $\Sigma$ is compact, since the set of rational geodesic currents on $\Sigma$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(\Sigma )$. We prove that even when $\Sigma$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(\Sigma )$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $\Sigma$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\mathrm{GC}(\Sigma )$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.