Counting curves, and the stable length of currents

Abstract

Let ${\gamma }_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let ${\pi }_1(\Sigma )\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $\gamma$ of type ${\gamma }_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$, of ${\pi }_1(\Sigma )$ 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.