Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

Abstract.

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character $\chi \in \rm {Hom}(G, \Bbb {R})$ represents a point in the homotopical or homological $\Sigma$-invariants of the group G. Let $\cal {G}$ be a finite simplicial graph, $\widehat {\cal {G}}$ the flag complex induced by $\cal {G}$, and $G \cal {G}$ the graph group, or 'right angled Artin group', defined by $\cal {G}$. We use our result on n-generating subgroup sets to describe the homotopical and homological $\Sigma$-invariants of $G \cal {G}$ in terms of the topology of subcomplexes of $\widehat {\cal {G}}$. 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 $\Sigma$-invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by $\Sigma$ ¹.