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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 F of quasi-convex subgroups of a hyperbolic group G, there is an HHG structure on G that is compatible with F. We will use this to provide explicit descriptions of the Gromov Boundary of hyperbolic HHS and HHG, and we recover results from Hamenstädt, Manning, Trang for the case when G is hyperbolic relative to F. Further applications in the construction of new HHG will be presented in a subsequent paper.
Davide Spriano (Wed,) studied this question.