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

    Camp, Wes Alan
    : https://etd.library.vanderbilt.edu/etd-03252013-161827
    http://hdl.handle.net/1803/11306
    : 2013-04-15

    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.
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