# Counting curves, and the stable length of currents

### Viveka Erlandsson

University of Bristol, UK### Hugo Parlier

University of Luxembourg, Luxembourg### Juan Souto

Université de Rennes 1, France

## Abstract

Let $γ_{0}$ be a curve on a surface $Σ$ of genus $g$ and with $r$ boundary components and let $π_{1}(Σ)↷X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $γ$ of type $γ_{0}$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$, of $π_{1}(Σ)$ the limit

exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on $X$ extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group.

## Cite this article

Viveka Erlandsson, Hugo Parlier, Juan Souto, Counting curves, and the stable length of currents. J. Eur. Math. Soc. 22 (2020), no. 6, pp. 1675–1702

DOI 10.4171/JEMS/953