# Currents on cusped hyperbolic surfaces and denseness property

### Dounnu Sasaki

Gakushuin University, Tokyo, Japan

## Abstract

The space $GC(Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $GC(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $GC(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ 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 $GC(Σ)$. 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.

## Cite this article

Dounnu Sasaki, Currents on cusped hyperbolic surfaces and denseness property. Groups Geom. Dyn. 16 (2022), no. 3, pp. 1077–1117

DOI 10.4171/GGD/688