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.

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We prove the homological part of the Σm‐conjecture for metabelian groups G of finite Prüfer rank: if G is of type FPm then the complement of the higher homological invariant Σm(G; Z) introduced by R. Bieri and B. Renz [10] is given by the formula conv⩽m Σ1(G; Z)c = Σm(G; Z)c, where conv⩽m Σ1(G; Z)c is the union of the convex hulls of all subsets of at most m elements of Σ1 (G; Z)c ⊆ Hom(G; R).

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