Reference : Counting curves, and the stable length of currents
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Physical, chemical, mathematical & earth Sciences : Mathematics
Counting curves, and the stable length of currents
Erlandsson, Viveka [> >]
Parlier, Hugo mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit]
Souto, Juan [> >]
[en] Mathematics - Geometric Topology ; Mathematics - Differential Geometry ; Mathematics - Dynamical Systems ; Mathematics - Group Theory
[en] 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.
28 pages, 6 figures

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