Counting curves, and the stable length of currents

TL;DR

This paper investigates the asymptotic count of curves on surfaces with bounded translation length in a metric space, establishing a limit law and extending stable length functions to currents for hyperbolic groups.

Contribution

It introduces a novel extension of stable length functions to currents for hyperbolic groups and analyzes the asymptotic growth of curves with respect to translation length.

Findings

The limit of the normalized count of curves exists and is positive.

The stable length function extends uniquely to a continuous, homogeneous function on currents.

The results apply to actions of torsion-free hyperbolic groups.

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.