Complete characterizations of hyperbolic Coxeter groups with Sierpi\'nski curve boundary and with Menger curve boundary

TL;DR

This paper characterizes hyperbolic Coxeter groups with Gromov boundaries homeomorphic to the Sierpiński and Menger curves, using nerve-based criteria and integrating existing results.

Contribution

It provides complete, nerve-based characterizations of hyperbolic Coxeter groups with specific fractal boundaries, expanding understanding of their geometric and topological properties.

Findings

Characterization of Coxeter groups with Sierpiński curve boundary

Characterization of Coxeter groups with Menger curve boundary

Integration of existing results for comprehensive criteria

Abstract

We give complete characterizations (in terms of nerves) of those word hyperbolic Coxeter groups whose Gromov boundary is homeomorphic to the Sierpi\'nski curve and to the Menger curve, respectively. The justification is mostly an appropriate combination of various results from the literature.