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On the orbit function of discrete groups in negative curvature. (Sur la fonction orbitale des groupes discrets en courbure négative.) (French) Zbl 1008.20040
Let $$\Gamma$$ be a non-elementary discrete group of isometries of a metric space $$X$$ and $$O$$ be a fixed point in $$X$$. Denote by $$N_\Gamma(R)$$ the number of points of the orbit point $$O$$ with respect to $$\Gamma$$, which belong to the ball with center $$O$$ and radius $$R$$. Set $$\delta(\Gamma)=\overline\lim_{R\to\infty}(\log N_\Gamma(R)/R)$$. Theorem. $$\log N_\Gamma(R)/R\to\delta(\Gamma)$$ as $$R\to\infty$$.

##### MSC:
 20F69 Asymptotic properties of groups 37C35 Orbit growth in dynamical systems 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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