Hyperbolization of the limit sets of some geometric constructions

Abstract

Inspired by the construction of Sierpi\'{n}ski 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 $K$.