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.
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Received: October 18, 1996
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Meier, J., Meinert, H. & VanWyk, L. Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups. Comment. Math. Helv. 73, 22–44 (1998). https://doi.org/10.1007/s000140050044
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DOI: https://doi.org/10.1007/s000140050044