A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a hyperbolic metric space. Also, the definition can be given as follows, a group $G$ generated by a finite set $X$ is hyperbolic if the Cayley graph $\Gamma(G;X)$ is a hyperbolic metric space. The equivalence of both the definitions can be provided by Svarc-Milnor Lemma.

My question is:

What are the examples of group (finitely generated and infinite) acting on a hyperbolic space cocompactly and fails to be hyperbolic? i.e., the action is not proper.

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ generated by a finite set $X$ is hyperbolic if the Cayley graph $\Gamma(G;X)$ is a hyperbolic metric space. The equivalence of both the definitions can be provided by Svarc-Milnor Lemma.

My question is:

What are the examples of group (finitely generated and infinite) acting on a geodesic hyperbolic space cocompactly and fails to be hyperbolic? i.e., the action is not proper.