Graph Separators and Boundaries of Right-Angled Artin and Coxeter Groups

Abstract

We examine the connection between some vertex separators of graphs and topological properties of CAT(0) spaces acted on geometrically by groups corresponding to graphs. For right-angled Coxeter groups with no $^3$ subgroups (three-flats), we show that the boundary of any CAT(0) space such a group acts on geometrically is locally connected if and only if the presentation graph of the group lacks a certain type of vertex separator. It was known that the presence of such a separator in the presentation graph of any right-angled Coxeter group implies that any boundary of the group is non-locally connected, and so this result fully classifies the right-angled Coxeter groups with no three-flats and locally connected boundary. For right-angled Artin groups, we show that the presence of a type of vertex separator in the presentation graph of the group guarantees that the standard CAT(0) cube complex on which the group acts geometrically has non-path-connected boundary.