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Proceedings of the Conference, Geometric and Asymptotic Group Theory with Applications (GAGTA-5)No Access


    Let G be a graph product of a collection of groups and H be the direct product of the same collection of groups, so that there is a natural surjection p : G → H. The kernel of this map p is called a graph product kernel. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

    AMSC: 20F36, 20F65, 20F67


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