Higher generation subgroup sets and the virtual cohomological dimension of graph products of finite groups

We introduce panels of stabilizer schemes (K, G*) associated with finite intersection-closed subgroup sets script K of a given group G, generalizing in some sense Davis' notion of a panel structure on a triangulated manifold for Coxeter groups. Given (K, G*), we construct a G-complex X with K as a strong fundamental domain and simplex stabilizers conjugate to subgroups in script K . It turns out that higher generation properties of script K in the sense of Abels-Holz are reflected in connectivity properties of X. Given a finite simplicial graph Γ and a non-trivial group G(v) for every vertex v of Γ, the graph product G(Γ) is the quotient of the free product of all vertex groups modulo the normal closure of all commutators [G(v) G(w)] for which the vertices v, w are adjacent. Our main result allows the computation of the virtual cohomological dimension of a graph product with finite vertex groups in terms of connectivity properties of the underlying graph Γ.