The poset of hyperbolic structures on a group $G$ is still very far from being understood and several questions remain unanswered.

In this talk, I will speak about some new results that describe hyperbolic structures on the wreath product $G \wr Z$, for any group $G$.

As a consequence, I answer two open questions regarding quasi-parabolic structures: I will give an example of a group $G$ with an uncountable chain of quasi-parabolic structures and give examples of groups that have finitely many quasi-parabolic structures.