  Suppose that a countable group $G$ admits a cusp-uniform action on a
hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of
the paper is characterizing the purely exponential growth type of the orbit
growth function by a condition introduced by Dal'bo-Otal-Peign\'e. For
geometrically finite Cartan-Hadamard manifolds with pinched negative curvature
this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In
this case, our result recovers a theorem of Roblin (in a weaker form). Our main
tool is the Patterson-Sullivan measures on the Gromov boundary of $X$, and a
variant of the Sullivan shadow lemma called partial shadow lemma. This allows
us to prove that the purely exponential growth of either cones, or partial
cones or horoballs is also equivalent to the condition of Dal'bo-Otal-Peign\'e.
These results are further used in the paper \cite{YANG7}.
