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In this paper we provide a procedure to obtain a non-trivial HHS structure ona hyperbolic space. In particular, we prove that given a finite collection$\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is anHHG structure on $G$ that is compatible with $\mathcal{F}$. We will use this toprovide explicit descriptions of the Gromov Boundary of hyperbolic HHS and HHG,and we recover results from Hamenst\"adt, Manning, Trang for the case when $G$is hyperbolic relative to $\mathcal{F}$. Further applications in theconstruction of new HHG will be presented in a subsequent paper.

Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups