Let $ X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $ \Gamma < Isom(X)$ a discrete group of isometries of $ X$ . On can define the orbital function as follows

$ $ N_{\Gamma}(x,\rho) := \sharp \{ \gamma \in \Gamma \ , \ d(x, \gamma \cdot x) \le \rho \} $ $

where $ d$ is the distance on $ X$ induced by the metric. We will denote by $ V(x, \rho)$ the volume of a ball centered at $ x$ of radius $ \rho$ .

All the examples of orbital functions I have in mind (action of $ \mathbb{Z}$ or $ \mathbb{Z}^2$ on the plane, fundamental groups of finite volume hyperbolic manifolds and their abelian covers, convex-cocompact group actions…) satisfy that the following function

$ $ \frac{N_{\Gamma}(x, \rho)}{V(x, \rho)} $ $

is equivalent when $ \rho \to \infty$ to a decreasing function. So that I wondered if this property may be more general. Here is then my question, does the previous property always occur ?

One can relax a bit the previous property as follow.

There exists $ \rho_0 \in \mathbb{R}_+$ such that there exists a constant $ C>0$ such that for all $ \rho_2 \ge \rho_1 \ge \rho_0$ we have :

$ $ \frac{N_{\Gamma}(x, \rho_2)}{V(x, \rho_2)} \le C \hspace{0.3cm} \frac{N_{\Gamma}(x, \rho_1)}{V(x, \rho_1)} \ .$ $

From the homogeneity that enjoys the orbit $ \Gamma \cdot x$ , it seems quite natural to me that it actually holds, at least when the group is finitely generated, but I don’t know how to get into it.

Does someone has either a reference/an idea toward a proof or a counter example to this ?

Thank you for your reading.