Counting curves, and the stable length of currents
Viveka Erlandsson, Hugo Parlier, Juan Souto
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 \lim_{L\to\infty}\frac 1{L^{6g-6+2r}}|\{\gamma \: \text {of type} \: \gamma_0 \: \text {with} \: S\text {-translation length} \: \le L\}| 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.