Nonhyperbolic Coxeter groups with Menger boundary

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 $n\ge 5$. The construction depends on a slight extension of Sierpinski’s theorem on embedding 1-dimensional planar compacta into the Sierpinski carpet. We give a simplified proof of this theorem using the Baire category theorem.

A correction to this paper is available.