Abstract. We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character χ ∈ Hom(G, ) represents a point in the homotopical or homological Σ-invariants of the group G. Let G be a finite simplicial graph, G the flag complex induced by G, and GG 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 Σ-invariants of GG in terms of the topology of subcomplexes of 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 Σ-invariants for a family of groups whose higher invariants are not determined -either implicitly or explicitly -by Σ 1 . Mathematics Subject Classification (1991). 57M07, 20F36.

Given a finite simplicial graph g, the graph group Gg is the group with generators in one‐to‐one correspondence with the verticles of g and wuth relations stating that two generators commute if their associated vertices are adjacent in g. The Bieri–Neumann–Strebel invariant can be explicitly described in terms of the original graph g and hence there is an explicit description of the distribution of finitely generated normal subgroups of Gg with abelian quotient. We construct Eilenberg‐MacLane spaces for graph groups and find partial extensions of this work to the higher‐dimensional invariants.

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A b s t r a c t . We describe the homological and homotopical Σ-invariants of graph groups in terms of topological properties of sub-flag complexes of finite flag complexes. Bestvina and Brady have recently established the existence of FP groups which are not finitely presented; their examples arise as kernels of maps from graph groups to Z . Since the Σ-invariants of a group G determine the finiteness properties of all normal subgroups above the commutator of G, our Main Theorem extends the work of Bestvina and Brady. That is, our Theorem determines the finiteness properties of kernels of maps from graph groups to abelian groups. Applications of this result are indicated.Given a finite simplicial graph G the corresponding graph group, or 'rightangled Artin group', has generators corresponding to the vertices of G, where two generators commute if and only if they are adjacent in G. The graph G is the defining graph and the corresponding graph group is denoted GG. For example, if the defining graph G is the 1-skeleton of an octahedron, then the graph group GG is the direct product of three copies of F 2 . (The class of graph groups includes all finite direct products of free groups.) Graph groups have finite K(π, 1)'s which can be constructed by glueing tori associated to cliques in G [MV1]. We denote the K(GG, 1) by KG, and mention that its universal cover KG is CAT(0).Subgroups of graph groups provide examples of infinite groups exhibiting one kind of 'finiteness' but not another. Recall that a group G is F m if and only if there is a K(G, 1) with finite m-skeleton. The properties F 1 and F 2 are topological reformulations of the two most common finiteness conditions, finite generation and finite presentation. On the other hand, a group G is FP m if Z, thought of as a trivial ZG-module, admits a projective resolution with finitely generated m-skeleton. (See Chapter VIII of [Br] for background on finiteness properties of infinite groups.)