Right-angled Coxeter groups with n-dimensional Sierpi\'nski compacta as boundaries

TL;DR

This paper identifies a broad class of right-angled Coxeter groups whose boundaries are homeomorphic to n-dimensional Sierpiński compacta, and provides conditions for planar complexes to have Sierpiński curve boundaries.

Contribution

It characterizes when right-angled Coxeter groups have boundaries homeomorphic to n-dimensional Sierpiński compacta, extending understanding of their geometric and topological properties.

Findings

Classified a large family of Coxeter systems with Sierpiński compacta boundaries.

Established necessary and sufficient conditions for planar complexes to have Sierpiński curve boundaries.

Connected the combinatorial structure of nerve complexes to boundary topology.

Abstract

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpi\'nski compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the right-angled Coxeter system whose nerve is L has the visual boundary homeomorphic to the Sierpi\'nski curve.