Constructing Bowditch boundaries of some relatively hyperbolic groups that are homeomorphic to the $n$-dimensional Sierpi\'nski carpet

TL;DR

This paper shows that certain relatively hyperbolic groups with Bowditch boundary homeomorphic to an n-sphere can be restructured to have a Bowditch boundary homeomorphic to an (n-1)-dimensional Sierpiński carpet, revealing new boundary homeomorphism properties.

Contribution

It establishes a method to relate the Bowditch boundaries of relatively hyperbolic groups to the Sierpiński carpet, expanding understanding of boundary homeomorphisms in geometric group theory.

Findings

Relatively hyperbolic groups with n-sphere boundary can have boundaries homeomorphic to the Sierpiński carpet.

Reconstruction of parabolic subgroups leads to new boundary homeomorphisms.

The Bowditch boundary can be transformed to a lower-dimensional Sierpiński carpet boundary.

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.