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 fini te intersection-closed subgroup sets H of a given group G, generalizin g in some sense Davis' notion of a panel structure on a triangulated m anifold for Coxeter groups. Given (K, G), we construct a G-complex X with K as a strong fundamental domain and simplex stabilizers conjugat e 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 Gamma and a non-trivial group G (v) for every vertex v of Gamma, the graph product G(Gamma) is the quo tient of the free product of all vertex groups module the normal closu re of all commutators [G(v), G(w)] for which the vertices v, w are adj acent. Our main result allows the computation of the virtual cohomolog ical dimension of a graph product with finite vertex groups in terms o f connectivity properties of the underlying graph Gamma.