Hyperbolization of the limit sets of some geometric constructions

Inspired by the construction of Sierpinski carpets, we introduce a new class of fractal sets. For a such fractal set K, we construct a Gromov hyperbolic space X (which is also a strongly hyperbolic space) and show that K is isometric to the Gromov hyperbolic boundary of X. Moreover, under some conditions, we show that Con(K) and X are roughly isometric, where Con(K) is the hyperbolic cone of K.