For all $R\in\mathbb{R}$, set
$$\Gamma(R) = \{g\in\Gamma \text{ such that } d(v_0,gv_0)\leq R\}.$$
Recall that $\Gamma(R)$ is a finite set. Let $D = \sup_{v\in T} d(v_0,v)$.
Show that, for all $s\geq 0$,
$$\frac{1}{C}|\Gamma(\operatorname{arcch}(s)-D)|\cdot|P_k\cap T| \leq |P_k(s)| \leq |\Gamma(\operatorname{arcch}(s)+D)|\cdot|P_k\cap T|.$$