Jens Harlander, Holger Meinert; Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups. J London Math Soc 1996; 53 (1): 99-117. doi: 10.1112/jlms/53.1.99
We introduce panels of stabilizer schemes (K, G*) associated with finite intersection-closed subgroup sets ℋ 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 ℋ. It turns out that higher generation properties of ℋ 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(υ) for every vertex υ 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(υ), G(w)] for which the vertices υ, 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.