Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups

In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an HHG structure on $G$ that is compatible with $\mathcal{F}$. We will use this to provide 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 the construction of new HHG will be presented in a subsequent paper.