Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)

Abstract

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpi\'nski's theorem on embedding $1$--dimensional planar compacta into the Sierpi\'nski carpet. See the appendix for a brief erratum.