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 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.