| We consider groups defined by graphs. These include right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products of groups. We define an operation on finite graphs, called co-contraction. By showing that cocontraction of a graph induces an injective map between graph products of groups, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the graph products of any non-trivial groups on those graphs contain hyperbolic surface groups. By applying this to the special case of right-angled Artin groups, we answer a question raised by Gordon, Long and Reid negatively.;We also give a family of right-angled Artin groups that do not contain hyperbolic surface groups. Let A(Gamma) denote the right-angled Artin group defined by a graph Gamma. Using transversality, any pi 1-injective map from a compact surface S to the standard Eilenberg-MacLane space XGamma of A (Gamma) can be realized as a cubical map for some cubical structure on S. We examine the transversely oriented simple closed curves and the properly embedded arcs dual to this cubical structure. As a result, we prove that A(Gamma) does not contain a hyperbolic surface group for each Gamma in an inductively defined family F of graphs. F is shown to contain each chordal graph, as well as each bipartite graph without any induced cycle of length at least 5. |
