Constructing Bowditch boundaries of some relatively hyperbolic groups that are homeomorphic to the $n$-dimensional Sierpiński carpet

Abstract: In this paper we prove that if some relatively hyperbolic groups have Bowditch boundary homeomorphic to the $n$-sphere, then they are also relatively hyperbolic with respect to another set of parabolic subgroups and its Bowditch boundary is homeomorphic to the $n-1$-dimensional Sierpi\'nski carpet.