In this paper, we prove that the Gromov hyperbolic space $(\Cal X,h)$ which was introduced by Z. Ibragimov and J. Simanyi in [3] is an asymptotically $\operatorname{PT}_{-1}$ space and extend the methods of [3] to the case of uniform Cantor sets, show that the uniform Cantor set is isometric to the Gromov hyperbolic boundary at infinity of some asymptotically $\operatorname{PT}_{-1}$ space.