  A graph product kernel means the kernel of the natural surjection from a
graph product to the corresponding direct product. 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.
