Hyperbolic right-angled Coxeter groups with boundaries as a Sierpiński carpet and a Menger curve

Abstract

Let (W,S) be a hyperbolic right-angled Coxeter system whose nerve is a 1-dimensional strongly co-connected simplicial complex, where “strongly co-connected” is defined in this paper. Then we provide characterizations of the Coxeter group W whose boundary ∂W is a Sierpiński carpet and a Menger curve. Using this result, we construct hyperbolic right-angled Coxeter groups with boundaries as their universal curves. We note that hyperbolic non-right-angled Coxeter groups with boundaries as their universal curves have been constructed in N. Benakli's PhD Thesis [3].