Counting curves, and the stable length of currents
Counting curves, and the stable length of currents
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计数曲线、电流稳定长度
DOI：
10.4171/jems/953
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发表时间：
2016
影响因子：
2.6
通讯作者：
J. Souto
中科院分区：
文献类型：
--
作者：
V. Erlandsson;H. Parlier;J. Souto
设$\gamma_0 $是亏格为$g$的曲面$\Sigma$上的曲线，边界分支为$r $，$\pi_1（\Sigma）\curvearrowright X$是度量空间上的离散余紧作用。研究了X上平移长度不超过L的$\gamma_0 $型曲线的个数的渐近性态.例如，作为一个应用，我们推导出对于任何$\pi_1（\Sigma）$的有限生成集$S$，极限$$\lim_{L\to\infty}\frac 1{L^{6 g-6+ 2 r}}\{\gamma\text{ of type }\gamma_0\text { with }S\text{-translation length}\le L\}$存在且为正。主要的新技术工具是，与每条曲线相关联的函数，其稳定长度相对于$X$上的作用延伸到电流空间上的（唯一的）连续和齐次函数。我们证明了这确实是一个挠自由双曲群的任何行动的情况。
Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $\gamma$ of type $\gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $\pi_1(\Sigma)$ the limit $$\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.