2016/12/18 by Viveka Erlandsson, Erlandsson, Viveka, Hugo Parlier +3 · 1 citation
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · doi:10.48550/arxiv.1612.05980
openalex publication_date 2016/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let γ0 be a curve on a surface Σ of genus g and with r boundary components and let π1(Σ)\curvearrowright X be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves γ of type γ0 with translation length at most L on X. For example, as an application, we derive that for any finite generating set S of π1(Σ) the limit limL→∞\frac 1L6g-6+2r\γ of type γ0 with S-translation length≤ 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.