Higher generation subgroup sets and the Σ-invariants of graph groups

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character chi is an element of Hom(G, R) represents a point in the homotopical or homological C-invariants of the group G. Let G be a finite simplicial graph, (G) over cap the flag complex induced by G, and GB the graph group, or 'right angled Artin group', defined by G. We use our result on n-generating subgroup sets to describe the homotopical and homological Sigma-invariants of GG in terms of the topology of subcomplexes of (G) over cap. In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the C-invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by Sigma(1).